Tracing All the Connections
Wittgenstein on Internal and External Relations
JAKUB MÁCHA
Der Denker gleicht sehr dem Zeichner. Der alle Zusammenhänge nachzeichnen
will.
Wittgenstein, Ms 153a, p. 90v[2], 1931
Contents
Contents .............................................................................................................vii
Preface and acknowledgements.........................................................................ix
List of abbreviations .........................................................................................xiii
I. Introduction......................................................................................................1
1. Wittgenstein’s method of analysis: “I’ll teach you differences.”.................1
2. Why relations matter .................................................................................... 6
3. What is wrong with the internal/external distinction...................................9
II. Prelude ..........................................................................................................15
4. Hegelianism and British idealism .............................................................. 15
5. Russell and Moore...................................................................................... 31
III. Wittgenstein’s early writings.......................................................................43
6. Definitions of the internal/external distinction in the early writings.........43
7. The Doctrine of External Relations............................................................ 53
8. The nature of simple objects ......................................................................62
9. The picture theory ...................................................................................... 73
IV. Wittgenstein’s later writings ........................................................................93
10. Definitions of the internal/external distinction in the later writings........93
11. Intentionality........................................................................................... 119
12. Reason, motive, and cause .....................................................................136
13. Rules and their applications ...................................................................156
14. Mathematics ...........................................................................................166
15. Colors .....................................................................................................181
16. The standard meter .................................................................................200
17. Aspect-seeing and philosophy of psychology........................................217
18. Aesthetics and art....................................................................................230
V. Conclusion...................................................................................................240
19. Internal relations as imperatives.............................................................241
20. The maxim of no reflexive uses of internal relations.............................246
Bibliography.....................................................................................................252
Index.................................................................................................................261
Preface and acknowledgements
I follow Wittgenstein’s term ‘album’ in the Preface of the Philosophical Investigations,
so this book can be labeled an album too—meaning an album of
themes, items, problems, or questions that have caught my interest while reading
Wittgenstein’s writings. It is my own album of Wittgenstein’s remarks that I
have collected in a certain order that, I hope, reveals an important strand in
Wittgenstein’s thinking that has been neglected in Wittgenstein scholarship thus
far.
My plan or strategy is to look at Wittgenstein’s writings from a certain perspective—from
a perspective that focuses on the distinction between internal and
external relations. A relation is internal if it is unthinkable that its terms should
not possess it, and it is external otherwise.1
This book is intended to capture the
landscape (re)presenting Wittgenstein’s engagement with this distinction. Seen
from this perspective, this distinction appears to be one of the most fundamental
distinctions that Wittgenstein drew in his writings. I have assembled Wittgenstein’s
remarks in such a way that the fundamental character of this distinction
will become apparent.
Although the character of this study is exegetical, I try to keep a slight distance
from his writings. I have assembled Wittgenstein’s remarks in a different way
than he himself (or his editors) did and have placed emphasis on some problems
that may be tangential to his concerns. I admit that I have tried to extract a
workable philosophical view or, rather, a coherent set of views from Wittgenstein’s
Nachlass.
It is often proclaimed by so-called resolute readers that Wittgenstein did not
provide any philosophical theory (of language, mind, perception and so on). His
achievements have to be seen as residing more in a therapeutic approach to philosophy.
Although I am sympathetic to such readings, I must insist that there
still remains something in Wittgenstein’s philosophy (the early as well as the
1
TLP 4.123.
later) that can be called a theory. This attempt at setting out a theory is neither
about the world nor about knowledge nor about language. It is a theory of how
to analyze a philosophical text in order to get rid of any philosophical problems
that emerge due to the unsurveyable character of natural languages. It is Wittgenstein’s
method of analysis. Wittgenstein is, in my view, the real godfather of
analytic philosophy. Despite the fact that the goal of logical or philosophical
analysis has shifted from the kind of Tractarian ‘concept-script’ of the Tractatus
to surveyable representations in his later philosophy, the main traits of his analytical
method remained unchanged throughout his philosophical career. The
main thesis that I am advancing in my book is that Wittgenstein’s method of
analysis rests on the distinction between internal and external relations. I
do not hesitate to call Wittgenstein’s method of analysis a kind of philosophical
theory, although this clashes with Wittgenstein’s desire not to offer theories.
Drawing this distinction is not theoretically neutral. It presupposes various
views, primarily concerning the nature of modality (necessity, thinkability, or
conceivability). This theory is not, however, a theory of a primary order. It introduces
a procedure for how to deal with other philosophical theories—a sort
of transcendental theory.
The present book has the following structure: it proceeds chronologically in its
main outline. Part II summarizes the philosophical background against which
the distinction between internal and external relations emerged. Hegel and
Bradley are addressed in Chapter 4. Russell and Moore—Wittgenstein’s direct
teachers and colleagues—are the subject of Chapter 5. Part III is devoted to
Wittgenstein’s early writings, i.e., to the texts that precede the Tractatus LogicoPhilosophicus
and to the Tractatus itself. Chapter 6 distills the definition of the
notions of internal and external relations from these texts. The subsequent chapters
in this part are all highly interlinked. You can read them in any order and
you can skip some of them. If you are not concerned with Wittgenstein’s early
writings or with Wittgenstein’s philosophical development, feel free to skip
Parts II and III entirely. Part IV deals with Wittgenstein’s later writings from
1929 up to his death in 1951. Its structure is similar to the previous part. Chapter
10 provides some definitions of internal and external relations in these texts.
The following chapters explore various themes from Wittgenstein’s later philosophy
in which the distinction between internal and external relations is im-
portant.2
The subsequent chapters are independent of each other, although there
are some connections and continuities. So, for instance, Chapter 12 continues
the discussion of Intentionality from Chapter 11. I do not explicitly distinguish
between the so-called middle Wittgenstein of the early 1930s, the Wittgenstein
of the Philosophical Investigations, and the third Wittgenstein. But at some
points, there are recognizable shifts in Wittgenstein’s thinking that need to be
taken into consideration. Accordingly, sections 11.1–11.3 discuss Wittgenstein’s
account of intentionality from the early 1930s, whereas section 11.4 offers a
somewhat more complex analysis from The Blue Book. Chapter 13 ‘Rules and
their applications’ is subdivided into two sections. The first one is devoted to
Wittgenstein’s notion of a rule that can be traced back to his calculus model of
language from the early 1930s. The second section pays attention to the final
stage of Wittgenstein’s thinking, which is sometimes labeled the third Wittgenstein.
The concluding Part V gives the rationale for Wittgenstein’s method of
analysis based on the distinction between internal and external relations. Please
feel free to skip the footnotes, which provide additional thoughts or references
to thoughts that lie outside the main concerns of this book. Sometimes I bring
up parallels and analogies to the ideas of other thinkers and philosophers that
Wittgenstein might have not been directly acquainted with or that emerged later
after his death.
A major part of this book was written during my stay at the Wittgenstein Archives
at the University of Bergen in 2013, where I had many opportunities to
discuss some of the content of this book with members of the Archives and with
numerous guests. I am most grateful to Professor Alois Pichler, Director of the
Wittgenstein Archives, and I would like to thank him for his friendly hospitality
throughout my stay, which has made this book possible. I would also like to
thank the following people for their helpful comments and useful corrections of
the earlier versions of some of the material presented here: James Klagge, Her-
2
In the text, I will touch upon a considerable majority of the themes from Wittgenstein’s
later philosophy. There are, however, themes that I will pass over. The internal/external distinction
does not apply in Wittgenstein’s treatment of ethics or religion—or I have not found
any substantial employment of it there. The distinction does apply, however, to Wittgenstein’s
thoughts about certainty (the so-called hinge propositions that express internal relations
whereas proper propositions express external relations). I will address these ideas in
§10.3 in more detail without devoting a complete chapter to them.
bert Hrachovec , Peter Baumann, Modesto M. Gómez Alonso, Wilhelm Krüger,
Alois Pichler, Deirdre Smith, Ondřej Beran, Dinda Gorlée, Peter Hacker, Gisela
Bengtsson, Christian Erbacher, Maja Jaakson, Bernard Linsky, Kenneth Blackwell,
Sorin Bangu, Karl-Friedrich Kiesow. My apologies to anyone I may have
forgotten.
During the years 2011–2013, I received support from the Czech Science Foundation
(Grant № P401/11/P174), for which I am most grateful. I also wish to
thank the Faculty of Arts, Masaryk University, for a period of research leave,
for generous support, and for a stimulating working environment.
List of abbreviations
AWL Wittgenstein’s Lectures: Cambridge 1932–1935. Ed. A. Ambrose.
Oxford: Blackwell, 1979.
BBB The Blue and Brown Books. Oxford: Blackwell, 1958.
BT The Big Typescript. TS 213. Ed. and trans. C. G. Luckhardt und
M. A. E. Aue. Oxford: Blackwell, 2005.
CV Culture and Value. Ed. G. H. von Wright in collaboration with H.
Nyman, trans. P. Winch. Oxford: Blackwell, 1980.
LA Lectures and Conversations on Aesthetics, Psychology and Religious
Belief. Ed. C. Barrett. Oxford: Blackwell, 1966.
LFM Wittgenstein’s Lectures on the Foundations of Mathematics,
Cambridge 1939. Ed. C. Diamond. Sussex: Harvester Press,
1976.
LWL Wittgenstein’s Lectures: Cambridge, 1930–1932. Ed. Desmond
Lee. Oxford: Blackwell, 1980.
LWPP I Last Writings on the Philosophy of Psychology. Vol. 1. Ed. G. H.
von Wright and H. Nyman, trans. C. G. Luckhardt and M. A. E.
Aue. Oxford: Blackwell, 1982.
LWPP II Last Writings on the Philosophy of Psychology. Vol. 2. Ed. G. H.
von Wright and H. Nyman, trans. C. G. Luckhardt and M. A. E.
Aue. Oxford: Blackwell, 1992.
M Wittgenstein’s Lectures in 1930–33. In: PO, pp. 46–114.
MDC M. O’C. Drury. Conversations with Wittgenstein. In: Recollections
of Wittgenstein. Ed. R. Rhees. Oxford: Blackwell, 1981, pp.
112–189.
Ms Wittgenstein’s Nachlass. The Bergen Electronic Edition. Oxford:
Oxford University Press, 1998–2000, manuscript.
NB Notebooks 1914–16. Ed. G. H. von Wright and G. E. M.
Anscombe, trans. G. E. M. Anscombe. Oxford: Blackwell, 1961.
OC On Certainty. Ed. G. E. M. Anscombe and G. H. von Wright,
trans. D. Paul and G. E. M. Anscombe. Oxford: Blackwell, 1969.
PG Philosophical Grammar. Ed. R. Rhees, trans. A. J. P. Kenny. Oxford:
Blackwell, 1974.
PGL Wittgenstein’s Lectures on Philosophical Psychology 1946–7. Ed.
P. T. Geach. New York: Harvester, 1988.
PI Philosophical Investigations. Ed. G. E. M. Anscombe and R.
Rhees, trans. G. E. M. Anscombe. Oxford: Blackwell, 1958.
PO Philosophical Occasions 1912–51. Ed. J. Klagge and A. Nordmann.
Indianapolis: Hackett, 1993.
PPO Public and Private Occasions. Ed. J. Klagge and A. Nordmann.
Lanham, Boulder, New York, Oxford: Rowman and Littlefield
2003.
PR Philosophical Remarks. Ed. R. Rhees, trans. R. Hargreaves and
R. White. Oxford: Blackwell, 1975.
RFM Remarks on the Foundations of Mathematics. Ed. G. H. von
Wright, R. Rhees, G. E. M. Anscombe, revised edition. Oxford:
Blackwell, 1978.
ROC Remarks on Colour. Ed. G. E. M. Anscombe, trans. L. L. McAlister
and M. Schättle. Oxford: Blackwell, 1977.
RPP I Remarks on the Philosophy of Psychology. Vol. I. Ed. G. E. M.
Anscombe and G. H. von Wright, trans. G. E. M. Anscombe. Oxford:
Blackwell, 1980.
RPP II Remarks on the Philosophy of Psychology. Vol. II. Ed. G. H. von
Wright and H. Nyman, trans. C. V. Luckhardt and M. A. E. Aue.
Oxford: Blackwell, 1980.
TLP Tractatus Logico-Philosophicus. Trans. D. F. Pears and B. F.
McGuinness. London: Routledge & Kegan Paul, 1961. [Sometimes
using trans. by Ramsey and Ogden or my trans.]
Ts Wittgenstein’s Nachlass. The Bergen Electronic Edition. Oxford:
Oxford University Press, 1998–2000, typescript.
UW Ursache und Wirkung: Intuitives Erfassen / Cause and Effect: Intuitive
Awareness. In: PO, pp. 370–426.
VW The Voices of Wittgenstein. By Ludwig Wittgenstein and Friedrich
Waismann, ed. G. Baker. London: Routledge, 2003.
WWK Ludwig Wittgenstein and the Vienna Circle. Conversations recorded
by Friedrich Waismann. Ed. B. McGuinness. Oxford:
Blackwell, 1979.
Z Zettel. Ed. G. E. M. Anscombe and G. H. von Wright, trans. G. E.
M. Anscombe. Oxford: Blackwell, 1967.
All references to Wittgenstein’s manuscripts and typescripts are quoted from
Wittgenstein’s Nachlass. The Bergen Electronic Edition. Oxford: Oxford University
Press, 1998–2000 and referenced according to the von Wright catalogue
G. H. von Wright, Wittgenstein. Oxford: Blackwell, 1982 by Ms or Ts number
followed by page number.
The original punctuation and spelling is retained in quotations. Double quotations
marks are used for quotations and titles of papers. Single quotation marks
are used within quotations, to indicate that what is being referred to is the expression
appearing within the quotes, and sometimes for emphasis. If not stated
otherwise, italics are in the original.
I. Introduction
1. Wittgenstein’s method of analysis: “I’ll teach you
differences.”
What is the method of analysis? Traditionally, at least since Kant, philosophical
analysis has aimed at resolving a whole into its parts. The godfathers of analytic
philosophy—Frege, Russell, and Moore—understood the concept of analysis in
this sense. In the manuscript the Theory of Knowledge, Russell takes analysis to
be “the discovery of the constituents and the manner of combination of a given
complex”3
.
The important thing here is that in the works of these philosophers, another
conception of analysis is implicit. With the aid of a formal language, analysis
now reveals the underlying logical form of language, which might be obscured
in ordinary language. In other words, starting with the surface grammar4
of ordinary
language, analysis aims at uncovering its ‘logical form’. Russell’s theory
of descriptions is a good example here. Unlike with proper names, definite descriptions
that appear in the position of a grammatical subject have to be analyzed
away. The surface ‘grammar’, however, does not discriminate between
proper names and definite descriptions. Thus, a logically relevant distinction
might get lost in the surface grammar of ordinary language. The aim of logical
analysis is to uncover and fix such logically relevant distinctions.
3
Russell, 1984, p. 119.
4
Wittgenstein distinguishes between surface grammar and depth grammar: “In the use of
words one might distinguish ‘surface grammar’ from ‘depth grammar’. What immediately
impresses itself upon us about the use of a word is the way it is used in the construction of
the sentence, the part of its use—one might say—that can be taken in by the ear.—And now
compare the depth grammar, say of the word ‘to mean’, with what its surface grammar
would lead us to suspect. No wonder we find it difficult to know our way about.” (PI §664)
Although this metaphor comes from Wittgenstein’s later philosophy, it fits the Tractatus as
well.
2
Wittgenstein follows Russell exactly on this point: “Russell’s merit is to have
shown that the apparent logical form of the proposition need not be its real
form.”5
For Wittgenstein, as he interprets it, the theory of descriptions deals
with one particular logical distinction (namely, the distinction between proper
names and definite descriptions). His goal—at least in the Tractatus—was to
clarify the logic of our language, which means developing a logically adequate
language (whose grammar would be captured by a concept-script or a Be-
griffsschrift).6
One thing is to provide a logical notation for our natural language
and another to construct a new formal language. The latter was Frege’s aim, the
former Wittgenstein’s. In what follows I will frequently speak of a logically adequate
language. This expression should cover both a formal language and a
natural language whose logical form is transparent. In such a language, the surface
grammar would match the logical grammar and thus the distinction between
them would effectively collapse. The idea of a single logically adequate
language, which would embody the general form of every proposition, was
abandoned in Wittgenstein’s later work. Logical analysis is no longer like a
chemical analysis which enables us to see the hidden structure of a proposition;
what remained is the idea that some expressions have to be substituted for others
in order to clear away any misunderstandings caused by false analogies between
the different regions of language:
Our investigation is therefore a grammatical one. Such an investigation sheds light on
our problem by clearing misunderstandings away. Misunderstandings concerning the
use of words, caused, among other things, by certain analogies between the forms of
expression in different regions of language.—Some of them can be removed by sub-
5
TLP 4.0031, Ogden/Ramsey’s trans.
6
This follows Kuusela’s (2011b, p. 134) interpretation of the Tractatus: “Rather than putting
forward a theory or a doctrine about logic, or gesturing at ineffable truths, Wittgenstein’s
goal in the Tractatus is to introduce a particular logical notation, a concept-script—or
at least an outline of (some central principles governing) such a notation. […] This notation,
the principles of which the Tractatus’s purpose is to make understandable, is then the expression
of the logical insights of the early Wittgenstein. This means that these logical insights
don’t find their expression in (paradoxically nonsensical) theoretical true/false assertions.
Rather, they are embodied or built into the notation […].”
3
stituting one form of expression for another; this may be called an ‘analysis’ of our
forms of expression, for the process is sometimes like one of taking a thing apart.7
False analogies and ipso facto philosophical misunderstandings may be caused
by ambiguous words and sentences. It was Wittgenstein’s intention to tackle
such logical ambiguities.8
To do so, firstly, one has to detect an ambiguity and,
secondly, there has to be a rule or a test to resolve the ambiguity. Sometimes it
is enough simply to point out that an expression that is causing problems is ambiguous.
But even then this presupposes a generic logical distinction that makes
it possible to detach the separate meanings. This distinction would be a key that
allows us to say that a given expression means this as opposed to that. In this
book, I am going to examine one important distinction in Wittgenstein’s
works—namely, the distinction between internal and external relations.9
But I would like to assert an even stronger claim—that the distinction between
internal and external relations is one of the most fundamental distinctions that
drives Wittgenstein’s method of analysis. The most conclusive evidence stems
from the Tractatus where Wittgenstein writes that he introduces “these expressions
in order to indicate the source of the confusion between internal relations
and relations proper (that is, external relations), which is very widespread
among philosophers.)”10
Later, he continues with the thinking that philosophy
or metaphysics has obscured the distinction between conceptual and factual
statements: “Philosophical investigations: conceptual investigations. The essential
thing about metaphysics: it obliterates the distinction between factual and
conceptual investigations.”11
The distinction is employed here in order to be
able to indicate a certain widespread confusion among philosophers. It allows
7
PI §90.
8
Examples of such an ambiguity that are given by Wittgenstein include the word “is” which
“figures as the copula, as a sign for identity, and as an expression for existence” (TLP
3.323). In §49 of the Philosophical Investigations he gives the example of a sign ‘R’ or
‘Red’ that may sometimes be a word and sometimes a proposition. It may be the name of
the color of a colored square or it may be a proposition expressing that there is one red
square. See 15.2 for my discussion of this example.
9
These sections draw on my paper Mácha, 2012b.
10
TLP 4.122.
11
Z §458; RPP I §949. See also Kuusela 2011a, p. 604.
4
us to ask about the philosophical claim: does the author mean an internal relation
or an external relation here?12
The general lesson behind Wittgenstein’s method of analysis remains unchanged.
Two forms of expression are identified that look the same in ordinary
language. The aim of analysis is to show, however, that they are different. Here
is the most explicit expression of this conception of analysis (reported by Dru-
ry):
Hegel seems to me to be always wanting to say that things which look different are
really the same. Whereas my interest is in showing that things which look the same
are really different. I was thinking of using as a motto for my book a quotation from
King Lear: ‘I’ll teach you differences.’13
Hegel’s leading methodological maxim is to unify all differences. Wittgenstein
wants to see himself as turning Hegel on his head, which might be an allusion
to Marx’s treatment of Hegel as Marx’s intention was the “turning of Hegel on
his head”14
. But Marx still stood, in a sense, within Hegel’s system. Marx’s
philosophical method is basically the same as Hegel’s. The real—as it were methodical—turning
of Hegel on his head comes only with Wittgenstein.
The radicalism of Wittgenstein’s approach—in comparison with Frege or Russell—consists
in his insisting on this general methodological principle and resisting
the urge to formulate any underlying philosophical theory. Wittgenstein’s
claim that “All philosophy is a ‘critique of language’”15
indicates that
philosophy—a philosophy of the future which is invoked in the Tractatus
12
McManus (2006, p. 98) argues that “[i]nternal relations emerge in contexts of misunderstanding”.
This sounds correct to me if it means that the distinction is used in order to cope
with misunderstandings. McManus argues, however, that the distinction itself is a kind of
misunderstanding and that internal relations are meant to be thrown away. Obviously, if
there were no misunderstandings, we would not need this distinction. But we cannot use
internal and external relations in order to cope with misunderstandings and at the same time
think that this distinction itself is a misunderstanding. Cf. Chapter 3 below.
13
MDC, p. 157.
14
In fact, Marx writes in the “Afterword” to the second edition of his Capital the following:
“With him [Hegel] it is standing on its head. It must be turned right side up again, if you
would discover the rational kernel within the mystical shell.” (Marx, 1967, p. 12)
15
TLP 4.0031.
5
6.53—is neither about the world and its essential features, nor is it about the
essential features of language. What all philosophy is actually about is how to
deal with other accounts of the world and language. Philosophy is a secondorder
or transcendental theory as indicated in my Preface above.16
16
For an illustration of this point see Chapter 8, where I give two main accounts of simple
objects: the de re (or metaphysical) view and the de dicto (or semantic) view. On the former
interpretation, Wittgenstein provided an alternative account of modalities that overcame
Russell’s and Moore’s extreme pluralism (and their Doctrine of External Relations). On the
latter interpretation, Wittgenstein provided only a methodological principle, leaving it undecided
what the actual forms of reality are.
6
2. Why relations matter
The traditional distinction between essential and accidental properties is well
known. It rests on the commonsense intuition that if some properties of an entity
were taken away from it, it would no longer be the same entity. Such properties
are essential to the entity. Things get complicated, however, when we consider
that some properties are relational properties (whose characterization involves
reference to another thing or things). Relations can be seen as generalized
properties (i.e., properties are unary relations). The crucial question is:
does an analogous distinction apply to relations?
It is comprehensible when two things (particulars, e.g., material objects) are related
by a relation that is accidental (e.g., the relation of being close to each
other). But are we able to imagine two things that cannot but be related by a
certain relation? Why would we then treat these two things as separate objects if
we cannot conceptually separate them? Such a relation must be internal in the
sense of holding among parts of a whole. Even then, there is some doubt as to
whether such a picture is not self-contradictory.17
Do internal relations hold between
particulars at all or are all relations merely external? The latter claim is
sometimes call the Doctrine of External Relations.
There are several historical factors leading up to this subject matter.18
At the beginning
of the 20th century, philosophers like Russell and Moore tried to solve
the problem of the unity of a proposition by employing a unifying relation between
the proposition (or its constituents) and the judging mind. This means
that the mind ensures that a proposition (for instance, ‘The river is moving’) is
not merely a conglomerate of elements (river, to be, moving), but a coherent
whole. Such a relation is on the one hand a constituent of the proposition and on
the other hand something that ensures the unity of the whole proposition. It
turned out that external relations could not do this unifying job, because nothing
unifies the unifying relation with its terms.
A similar problem emerges if one takes facts to be entities that make propositions
true. Initially, Russell and Moore did not distinguish between them. Prop-
17
See §5.1 for arguments that it is self-contradictory.
18
See Hochberg & Mulligan (2004, pp. 7–9) for an overview.
7
ositions were identical with the facts that make them true (this is the identity
theory of truth). The constituents of a proposition are not linguistic entities. So
among the constituents of the proposition ‘The apple is red’ are a certain apple
and the color red. But later on Russell and Moore were forced to distinguish between
judgments and the facts that make them true. Then, however, there
emerges the question about the nature of the relation between a proposition and
a fact that makes it true. And again, it turned out that external relations are bad
candidates for this task, because then all truths would be accidental.
The third cluster of problems concerns Bradley’s regress argument, which leads
to the view that there are no relations at the ontological base and finally to ontological
monism. If there are no relations in any ontologically significant sense,
then reality is a single whole, i.e., ontological monism is the ultimate account of
reality.
The notion of an internal relation might be helpful in saving pluralism. If two
terms are internally related, there is no need for any further unifying relations.
The fact that these two things are internally related is grounded in these very
things, in their natures. The internal relation does not add anything to the complex
consisting of these two terms and is in this sense only an apparent rela-
tion.19
There is, furthermore, no need for any relation connecting a term with
the internal relation, because internal relations are grounded in the natures of
their terms. Hence, the infinite regress of instantiations of additional unifying
relations is avoided.
The price to pay for this turn is divorcing the linguistic form (i.e., the surface
form) from the logical form (i.e., the true form) of predication. As already discussed
in the previous chapter, such a discrepancy must be surmounted by logical
analysis. Another problem is what the ultimate form of predication, i.e., the
general form of the proposition, actually determines. Does this form reflect any
(necessary) properties of the world or is it imposed during the course of logical
analysis?20
Why do internal and external relations in Wittgenstein matter so much? The distinction
between internal and external relations imposes a metaphysical burden
19
See the next chapter.
20
The former alternative is envisaged in §8.1, the latter one, in turn, in §8.2.
8
or—as one would say today—imposes ontological commitments with respect to
the nature of reality and the nature of modality. This is obvious in Bradley, Russell,
and Moore. Wittgenstein employs the internal/external distinction primarily
as a heuristic tool, eventually freeing it from any metaphysical burdens—if
not in the Tractatus,21
then certainly in his later work. The general lesson I
would like to draw is how a metaphysical distinction—far from being regarded
as nonsensical—can be transformed into and employed as an analytical heuristic
tool.22
Wittgenstein was still wholeheartedly maintaining Frege’s Third Principle
(“Never to lose sight of the distinction between concept and object.”)23
when he wrote: “Philosophical investigations: conceptual investigations. The
essential thing about metaphysics: it obliterates the distinction between factual
and conceptual investigations.”24
This is, for me, the thing that analytic philosophy is most significantly about.
21
The notion of internal and external relations has metaphysical commitments if we interpret
the Tractatus metaphysically (as was done in §8.1).
22
This transition resembles Kant’s transformation of metaphysical principles into regulative
ones.
23
Frege, 1980, p. xxii.
24
Z §458.
9
3. What is wrong with the internal/external distinction
There is something inherently misleading about the distinction between internal
and external relations. The first thing is that this terminology is historically laden.
It had been used by Wittgenstein’s predecessors and contemporaries to formulate
various philosophical theories and doctrines. The expressions ‘internal
relation’ and ‘external relation’ had been used in several different senses in philosophy.
Rorty, following A. C. Ewing, summarizes the historical provenance at
the beginning of the 20th century:
[T]he meanings given to “internal” ranged from a very weak sense, in which to say
that a relation R which X bore to Y was internal to X meant merely that “R makes a
real difference to X,” to a very strong sense, in which it meant that “from a
knowledge of Y and R we could infer with logical necessity that X possesses a certain
determinate or relatively determinate characteristic other than the characteristic of
standing in the relation in question.”25
Even philosophers within a single tradition have occupied the opposite poles of
this range. Moore subscribed rather to the weak sense of ‘internal relation’
while Wittgenstein’s notion of an internal relation is closely connected to logical
necessity. Such ambiguities have distorted the discussion of this topic. It is,
then, somewhat surprising that Wittgenstein took the distinction between internal
and external relations as one of the central concepts of the Tractatus. At the
beginning of the 1930s, Moore reported that
[Wittgenstein] used it “only because others had used it”; and he proceeded to give a
slightly different formulation of the way in which the expression had been used, viz.
“A relation which holds if the terms are what they are, and which cannot therefore be
imagined not to hold”.26
In Wittgenstein’s later writings, the internal/external distinction seems not to be
as central as in the Tractatus. However, this distinction is still there and what
has changed is rather its verbal expression. Instead of “internal relation” Witt-
25
Rorty, 1967, p. 337; see also Ewing, 1934, pp. 117–142.
26
M, p. 86.
10
genstein now uses “grammatical relation” or “conceptual relation” and instead
of “external relation” he uses “factual relation”.27, 28
Such ambiguities affect more or less every philosophical notion. But there is a
second and more serious reason why the difference between internal and external
relations may be misleading. Distinguishing between internal and external
relations suggests that there must be some external relations and some internal
relations among all relations. But this is misleading according to Wittgenstein.
The class of all relations is not divided into external and internal relations.
Wittgenstein says occasionally that only external relations are proper while internal
relations are called improper.29
Internal relations are, strictly speaking,
not relations at all. The modifier ‘internal’ operates like ‘fake’ or ‘apparent’. It
is like the class of horses, which is not divided into real ones and wooden ones.
We can, of course, distinguish between internal and external relations, but we
have to keep in mind that only external relations are proper relations. Wittgenstein
expressed these concerns about this distinction at the beginning of the
1930s. He is reported as saying that internal relations are “entirely different
from other relations” and that “the expression ‘internal relation’ is misleading”
because internal relations and external relations are categorially different; they
“belong to different categories.”30
What exactly is the source of this categorial difference? There is almost pervasive
evidence in Wittgenstein’s Nachlass indicating that the difference stems
from the nature of the things that are related. Internal and external relations
27
See Ch. 10 for textual support for this shift.
28
I think that Read (1997) did not notice this terminological shift when summarizing “the
‘career’ of ‘internal relations’ in Wittgenstein’s thought […] as follows: (1) ‘Tractatus’ period:
Internal relations are important, and ineffable. (2) ‘Middle period’: Internal relations are
extended in RANGE somewhat, but after what might be judged to be initial vacillation,
Wittgenstein again becomes increasingly inclined against their expression. (3) c. 1933–
1939: Internal relations largely drop out. (4) Mature philosophy. Internal relations vanish
ALMOST TOTALLY […].” Sodoma (2014) argues on the other hand that the concept of an
internal relation shows a line of continuity in Wittgenstein’s thinking.
29
Cf. TLP 4.122: “I introduce these expressions in order to indicate the source of the confusion
between internal relations and relations proper (external relations).” See also VW, p.
237; WVC, p. 55.
30
M, pp. 85 & 87.
11
have categorially different relata. Internal relations hold between concepts (or
properties, qualities, universals) while external relations hold between objects
(or particulars). This delimitation is implicit in Wittgenstein’s early writings,
but is obscured by his shifting uses of the words “object”, “property”, and “relation”.
In the later texts, these shifting uses are anchored in the notion of a language-game.
What is a factual (external) relation in one language-game can be
a grammatical (internal) relation in another language-game.31
We have to clarify now in what sense internal relations are said to be improp-
er.32
Something is an improper X if it has most but not all of the characteristic
properties of X, but still resembles X. So which essential characteristics of relations
do internal relations lack?33
A proper relation should relate two (or more)
distinct terms. These terms should be distinct objects of thought. We should be
able to refer to them independently of each other. However, the crux of the assertion
of an internal relation is to say that its terms are not independent of each
other. These terms are indistinguishable in the respect that is captured by the
internal relation. Then, however, we lose the reason34
for taking these terms to
be distinct entities. In an important sense, these terms are partially or wholly
identical. Every internal relation wears its identity on its sleeve, which may give
rise to all the perplexities and paradoxes of identity.35
These paradoxes emerge
primarily if we consider the identity or dependence of objects as opposed to the
identity and dependence of concepts. Saying that two objects are identical or
not independent is prima facie baffling because we can then ask whether there
are two objects or only one. On the other hand, saying that two concepts are
31
See §§10.2 and 10.3.
32
TLP 4.126.
33
McManus speaks aptly of “the oxymoronic quality of ‘internal relation’” (2006, p. 65; cf.
pp. 77 & 99).
34
Cf. Fichte’s “ground of distinction” which is discussed in ch. 20. There might, however,
be another reason for taking the terms to be independent entities that is not captured by the
internal relation.
35
This consideration can be amended for internal properties. Suppose we predicate of an
object X that it has a property p, i.e., p(X). In order to refer to X, it must be a distinct object
of thought. But if the property p is an internal one, we cannot think of the object X not having
the property p. The problem would be avoided if p(X) expressed an internal relation between
the concepts X and p.
12
identical is easily comprehensible; it might mean that they are synonymous, coreferential
or the like.
Wittgenstein once called identity the very Devil.36
It is no exaggeration to say
that identity posed one of the main problems that Wittgenstein struggled with
throughout his philosophical career. In an early notebook entry, he tells us: “To
say of two classes that they are identical means something. To say it of two
things means nothing.”37
Later in the Tractatus we are told: “That identity is not
a relation between objects is obvious.”38
Wittgenstein was preoccupied with the
Law of Identity in his later work as well. In one of the most important passages
of the Philosophical Investigations, where the problem of rule-following culminates,
he writes:
“A thing is identical with itself.”—There is no finer example of a useless proposition
[…]. “Every coloured patch fits exactly into its surrounding” is a rather specialized
form of the law of identity.39
The idea behind this remains unchanged. Identity is not a relation between (material)
objects. In this sense, internal relations cannot be relations between objects
either. This is also the reason why internal relations are improper for Witt-
genstein.
We do not find any explicit argument against internal relations in Wittgenstein.
Obviously, he was not as hostile towards this notion as Russell and Moore
were.40
Resolute readers of Wittgenstein, however, tend to see internal relations
as nonsensical, as a kind of misunderstanding, as an incoherent notion. But
where exactly does the problem lie? Is it something problematic about the internal
relations themselves or is it problematic when they get confused with external
relations? Diamond argues for the latter option when she says that the
problematic thing about a logical (i.e., internal) relation is “that we may misun-
36
NB, p. 123.
37
NB, p. 4.
38
TLP 5.5301, Ogden/Ramsey’s trans.
39
PI §216.
40
In Part II, I will present and evaluate the various arguments against internal relations given
by Bradley, Russell, and Moore. These authors are much more explicit in arguing why
internal relations are improper.
13
derstand its role”41
McManus seems to argue that any asserted internal relations
are inherently muddled even if they are not confused with external relations:
… the person who hears an ‘internal relation’ assertion as telling him something has
to be confused, and that is what someone who asserts such assertions is getting at.
These assertions are moves within a person’s confusion […].42
Internal relations for McManus only have a signaling function, indicating that
something is not quite right about our language and understanding. Although
this account may find some support in the Tractatus, it is wholly inconsistent
with Wittgenstein’s later work in which internal relations are grammatical relations.
All mathematics, for instance, is based on internal relations.43
Does the
McManus account imply that every expression of a grammatical rule or every
arithmetical statement is a move within one’s confusion? In Part IV of this
book, and especially in the concluding Part V, rather different bounds of sense
for the expressions of internal relations will be proposed. A statement of an internal
relation is nonsensical when it is used reflexively without any underlying
external relation that would account for a difference between the relata. Such
statements are, in fact, expressions of the pure identity ‘a is a’. I will call this
methodological principle the maxim of no reflexive uses of internal relations.
The argument here proposes that assertions of internal relations—when intended
and recognized as such—may have a positive function in clearing up the
confusion. When one arrives at an asserted internal relation, the assertion may
be taken as a kind of a reminder or as an imperative. An internal relation assertion
may thus be intended to remind one that there is a logical or grammatical
rule that should have been followed. But it may also be intended as an invitation
to amend or to improve one’s manner of expression. An asserted internal
relation may be understood as an invitation to incorporate the relation into one’s
logical notation (in the Tractatus) or into the grammar of a language-game (in
the later philosophy). Chapter 19 sums up several ways of introducing new
grammatical rules, such as defining and learning a new expression, introducing
41
Diamond, 2002, p. 276.
42
McManus, 2006, p. 62. Cf. a similar statement quoted in fn. 12 above.
43
Cf. §14.
14
a new standard of measurement, proving a mathematical statement, presenting a
novel work of art, and much more.
If an imperative has been successfully carried out and the confusion surmounted,
there is then no point repeating the imperative. There is no point ordering
someone ‘Come here!’ if they are already here. If the logical grammar of our
language is clear, there is no point repeating any logical or grammatical rules.
As McGinn formulates it: “once Wittgenstein’s remarks have achieved what
they are intended to achieve, they can be completely left behind.”44
Obviously,
in a logically adequate language like that which is strived for in the Tractatus,
an internal relation could not even be expressed. And analogously, if a language-game
were wholly surveyable, there would be no need to express its
grammatical rules. The distinction between internal and external relations collapses
away eventually when there is no confusion anymore and therefore no
need for any logical analysis.
44
McGinn, 2009, p. 13.
II. Prelude
The aim of Part II is to present possible sources of Wittgenstein’s thinking in
terms of internal and external relations. While he claims in the Tractatus to settle
“the disputed question ‘whether all relations are internal or external’,”45
Wittgenstein is obviously referring to the dispute over relations which was led
by Russell and Moore on the one hand and by the so-called British idealists, especially
by Francis Bradley, on the other.
Russell ascribed to Bradley the view that all relations are internal while he himself
advocated the opposite view that all relations are external. I am going to
argue in Chapter 5 that Russell’s interpretation of Bradley’s position is wrong.
Wittgenstein followed Russell in his assessment of the core of the debate, but in
doing so he did not embrace Russell’s Doctrine of External Relations. One must
conclude that Wittgenstein was closer to Bradley than to Russell or Moore.
Moreover, Wittgenstein was more sensitive to Bradley’s arguments concerning
the nature of relations in his early philosophy and some Bradleyan (or Hegelian
points) points can be found in his later philosophy as well.46
4. Hegelianism and British idealism
Idealism can be characterized very roughly as a mind-dependence of reality or a
concept-dependence of objects. Let me give a voice to some critics of idealism
who are important in the present context. Russell understands idealism as “the
doctrine that whatever exists, or at any rate whatever can be known to exist,
must be in some sense mental.”47
Moore says that “Modern Idealism, if it asserts
any general conclusion about the universe at all, asserts that it is spiritu-
al.”48
To put it in terms of relations: reality is internally related to mind or expe-
45
TLP, 4.1251, Ogden/Ramsey’s trans.
46
See §§7 & 17.1.
47
Russell, 1912, p. 58.
48
Moore, 1922, p. 1.
16
rience.49
All objects are conceptually related and these relations are essential to
its relata, i.e., to these objects. This statement undermines the very distinction
between concepts and objects.
British idealism—a philosophical movement at the turn of the 19th and 20th
centuries—is not easy to characterize in terms of a single doctrine or a single
philosophical strategy or method. With some reservations one can say that the
British idealists were, although rather unorthodox, followers of Hegel.50
The
exponents of the early phase of analytic philosophy—Russell, Moore, and to
some extent Wittgenstein—did not sharply distinguish between Hegel’s views
and the views of his British followers.
Nevertheless, a good starting point for our historical presentation of the distinction
between internal and external relations would be Hegel. In §1 I contrasted
Wittgenstein’s method of analysis with Hegel’s synthetic method. Hegel’s
method of dissolving (‘aufheben’) all conceptual distinctions leads ultimately to
monistic ontology. Hegel was indeed an adherent of the Doctrine of Internal Relations.
He writes in his Logic: “Everything that exists stands in correlation, and
this correlation is the veritable nature of every existence”51
. To adequately understand
the veritable nature (i.e., the essence) of every single thing, one has to
understand its relations to every other thing and, in the end, to the whole, to the
Absolute. To put the doctrine in negative terms: we cannot isolate or abstract
one single thing out of the whole and understand it adequately in isolation.52
The crucial point of this doctrine is the requirement for adequate understanding.
Our practical requirements for adequate understanding are limited. On the one
hand, we can get along with obviously lower requirements for adequate understanding
in daily practice. One can manage to understand what a table is with
knowledge of a restricted number of its relations to other things. On the other
hand, one cannot understand what a particular table is without any understanding
of several other things. One cannot understand what a table is without un-
49
Cf. Candlish, 2007, p. 42.
50
Cf. Rockmore: “British idealism, which was unorthodox, does not follow any standard
idealist model.” (2005, p. 31) and “They were idealists first and Hegelians, if they were Hegelians
[…], only afterwards.” (2005, p. 35).
51
Hegel, 1968, p. 235.
52
Cf. Kain, 2005, pp. 4–6.
17
derstanding what tables are for, i.e., without understanding its role in human
practice.
Practical considerations can provide hints both in favor of and against the Doctrine
of Internal Relations. Wittgenstein is led by practical considerations in his
reflections about the doctrine. For his and Hegel’s argument in favor of the doctrine,
see §17.1.
4.1. Bradley’s metaphysics
We do not know to what extent Wittgenstein was acquainted with Bradley’s or
any other idealist’s works.53
It may be the case that Wittgenstein read only those
parts of Bradley’s works that were quoted by Russell or Moore. Because Wittgenstein’s
affinities to Bradley’s views concerning the nature of relations are
apparent, we have to address them in this study. Before doing so, I am going to
briefly sketch the outlines of Bradley’s metaphysics.
Bradley’s ontology consists of one single substance, which is the Absolute or
Reality-as-a-whole. From an epistemological point of view, he then distinguishes
between Appearance and Reality.54
Reality is everything, including our
Appearances. Appearances are abstractions from Reality; they are partial views
or aspects of Reality. Here comes the analogy to a painting by Candlish:
Appearances thus contribute to Reality in a fashion analogous to the ways in which
segments of a painting contribute to the whole work of art: detached from their background,
they would lose their significance and might in isolation even be ugly; in
context, they can themselves be beautiful and make an essential contribution to the
beauty and integrity of the whole.55
53
In a letter to Moore dated 7 May 1914, Wittgenstein confesses that he copied his BA dissertation
from Bernard Bosanquet, another exponent of British idealism. This dissertation
has not survived. Whether or not an exaggeration, this indicates that Wittgenstein might
have been directly acquainted with some of the idealists’ texts.
54
The expressions ‘Appearance’ and ‘Reality’ are capitalized in Bradley’s idiosyncratic
sense.
55
Candlish, 2007, p. 23.
18
Everything human beings experience, every judgment, every object, every relation,
becomes a mere Appearance. Appearances are not individual or independent
substances. The only substance is the Absolute or Reality. Appearances are
not substances because their being depends on something else—namely, of
course, Reality. Although Appearances are real (in the sense that they belong to
Reality), they are epistemologically deficient. Appearances are, however, not
wholly real; they are real to the extent that they qualify Reality. Bradley says of
Appearances that they are unreal and not real: so, for instance, causality and
change are unreal, time and space are unreal, bodies are unreal, nature is unre-
al56
, but relations are also unreal.57
One can conclude that Appearances are both
real and unreal.
The (amount of) Reality of Appearances is a matter of degree. This is to understand
that human knowledge is always a matter of degree. Bradley conceives of
our knowledge in terms of judgment or predication. No judgment is quite real;
every judgment depends on something else. In his terminology: “[T]he condition
of the assertion must not fall outside the judgement.”58
We can try to include
such conditions into the judgment, but we never succeed entirely. For example,
take the judgment:
This apple is red.(1)
This judgment depends on many factors: on the meanings of its constituents, on
the context of the deictic act of the speaker, and on other relevant factors. One
can try to include all of this in the judgment:
This apple is red where ‘apple’ means ‘pomaceous fruit of the apple tree’,(2)
‘red’ means ‘the color of blood’, and I am referring to the object lying on
the table in front of me.
Although this judgment is more accurate than the previous one, the problem
remains. (2) depends on even more conditions than (1). Hence, we can never
56
Bradley, 1897, pp. 61, 205, 286, 297, and passim.
57
This conclusion is a mistake. Bradley uses the expression ‘unreal’ in the sense ‘not wholly
real’ while one would expect that ‘unreal’ or ‘not real’ means ‘not real at all’. I think this
misunderstanding underlies the whole debate over relations.
58
Bradley, 1914, p. 252.
19
make (1) quite true.59
The upshot is that “in the end, no possible truth is quite
true.”60
Bradley dismisses traditional two-valued logic: every truth is one of degree.
Of course, our everyday judgments can be said to be either true or false.
The distinction between true and false is a matter of degree in the end. The adverbial
term “in the end” is used here in Bradley’s idiosyncratic sense with respect
to the Reality of a judgment. This view is sometimes called the Doctrine
of Degrees of Truth.
What is judgment for Bradley? Judgment cannot be identified with any linguistic
entity like sentences. Judgment consists in taking an idea—a mere idea as
Bradley puts it61
—and saying that the idea is related to Reality. The ultimate
subject of every judgment is Reality: “All judgement […] predicates its idea of
the ultimate Reality.”62
The idea is indicated by the grammatical subject and the
predicate. Bradley calls the grammatical subject a “special” or “limited” subject.
In any judgment ‘S is P’ we are asserting, in fact, S(R)P, which is to read:
“[In] Reality, being what it is, P can qualify S and together they both qualify the
larger reality which is their condition.”63
The content of a judgment always exceeds
the content of its special subject. That is why no judgment can be wholly
true. “The judgement, as it stands, can therefore […] be both affirmed and denied.
It remains conditional and relative only.”64
Every judgment, like (1) for
instance, can be true in some context and false in another.
It is further important to make clear how Bradley understands the notion of
truth. What is his theory of truth? We have seen that judgment (2) can be truer
than judgment (1) (provided that there is indeed a red apple on my table). This
is to understand that (2) is nearer to Reality than (1). In Bradley’s idealist metaphysics
there is, however, no room for any kind of correspondence. Appearance
is not a pendant of Reality. A judgment is true to the extent that its special subject
together with its special predicate is identical with its ultimate subject,
59
This account of judgment resembles Quine’s meaning holism. Every judgment depends
on sense-giving and empirical content; and one cannot distinguish between these two com-
ponents.
60
Bradley, 1897, p. 544.
61
Bradley, 1897, p. 366. See Ferreira, 1999 , p. 20.
62
Bradley, 1914, p. 253.
63
Ferreira, 1999, p. 27.
64
Bradley, 1914, p. 254.
20
which is Reality. This means that a judgment is true insomuch as it is identical
with Reality. Bradley can be said to be an adherent of the identity theory of
truth. Since no judgment is wholly true, no judgment is wholly identical with
Reality. When a judgment is becoming truer, i.e., more identical with Reality,
Reality then gradually swallows its character. The judgment disappears in Reality.
It is like committing a “happy suicide” as Bradley puts it.65
Bradley’s account
of truth, thus, turns out to be ultimately eliminativist.66

After the nature of Appearances comes the definition of Reality-as-a-whole.
Every attempt to characterize Reality is, however, condemned to fail because
such an attempt has to be carried out within the framework of judgment.67
Judgments, however, are Appearances which are abstractions from Reality. Reality
transcends our intellect. But this is only a negative characterization. Bradley,
however, strives for a positive account of Reality or the Absolute.
How can Bradley characterize Reality without using the framework of judgment?
There is a kind of human experience which lies outside any judgment.
Bradley calls this kind of experience “feeling” or the “feeling base”: “This Reality
is present in, and is my feeling; and hence, to that extent, what I feel is the
all-inclusive universe.”68
The point is that feeling is free from any distinctions
and relations.69
Bradley goes on to argue that conceptual distinctions and, what
is most important, relations are unreal in order to come to his most famous thesis,
namely ontological monism: “Reality is one.”70
If relations belonged to Appearance
only, the ultimate Reality would be a single and uniform whole. Bradley’s
argument for the unreality of relations (in Bradley’s idiosyncratic sense) is
the subject of the next section.
65
Bradley, 1897, p. 173.
66
See Candlish & Damnjanovic, 2011, sec. 2.2.
67
Cf. Candlish, 2007, p. 43: “Bradley was forced to resort to metaphor whenever it came to
setting out what reality is like.”
68
Bradley, 1897, p. 253.
69
See more about Bradley’s account of feeling in §17.1.
70
Bradley, 1897, pp. 511, 519, 522, 533, and passim.
21
4.2. Bradley on relations
Bradley’s published views on relations went through several changes or, more
precisely, amendments over time. His account of relations in Appearance and
Reality is, to say the least, obscure. The clearest account is to be found only in
his posthumously published essay “Relations”, though this had no impact on the
debate over internal and external relations.71
It is, therefore, no surprise that
Russell and Moore misunderstood Bradley’s views to some extent.
Bradley provided several arguments in favor of the unreality of relations. Only
some of them make use of the distinction between internal and external rela-
tions.72
Bradley’s main argument for the unreality of relations proceeds in two
steps: he argues that (i) external relations are unreal and (ii) internal relations
are unreal. The conclusion is, then, that all relations are unreal. First, I am going
to give an overview of Bradley’s understanding of the internal/external distinction.
Then I will provide a definition of external relations followed by an argument
against their reality. Finally, I will present a definition of internal relations
and an argument against their reality.
Bradley called relations external or internal73
in at least two senses. In an informal
sense, a relation is external to a judgment if it is not part of that selfsame
judgment. An internal relation is, then, part of this judgment. ‘Internal to’ means
in this sense ‘to be part of’ whereas ‘external to’ means ‘not to be part of’.74
These definitions include hardly any metaphysical significance. However, Bradley
came up with a much deeper notion of internal and external relations than
this.
71
In my discussion, I shall make use of Bradley’s later works, for they can be taken as clarifications
of his earlier views on relations.
72
Bradley’s most famous argument against the reality of relations from Appearance and
Reality (1897, Ch. III) makes no use of this distinction (although the expression “internal
relation” occurs there; see my explanation below in the main text).
73
Bradley sometimes uses the expression “intrinsic” instead of “internal”.
74
See Candlish, 2007, p. 147 for more details. Manser (1982, p. 192) argues that if ‘internal’
means ‘part of the judgment’, then the Doctrine of Internal Relations is tantamount to Frege’s
context principle. Each part of a judgment is internal to it, i.e., each part has meaning
only in the context of the whole judgment.
22
The externality and internality of relations is for Bradley a matter of degree. A
relation can be partly internal and partly external. A merely internal relation and
a merely external relation are the limiting cases of a scale. Bradley’s argument
leads, however, to the conclusion that these limiting points are impossible: “No
relation is merely intrinsic or external, and every relation is both [of these]”75
.
Here is Bradley’s most explicit definition of an external relation:
What should we mean […] by a relation asserted as simply and barely external? We
have here, I presume, to abstract so as to take terms and relations, all and each, as
something which in and by itself is real independently. 76
Bradley defines here a relation that is “simply and barely external”, i.e., wholly
external (and not internal at all). A relation is external if this relation and its
terms are independently real. Let us take a relational fact aRb. Relation R is external
if R and its terms a and b are independently real. We have to take the expression
‘independently real’ in Bradley’s sense. How do we find out whether a
thing is capable of existing independently of another thing? In a polemic note to
Russell, Bradley proposes the following criterion:
I am still in doubt as to the sense in which according to Mr. Russell relations are external.
The terms are to contribute nothing, and so much I understand. But I still do
not know whether Mr. Russell takes the relations apart from any terms to be thinka-
ble.77
Hence, a thing that is capable of existing independently must be thinkable inde-
pendently.78
Relation R would be, thus, external if we were able to think of this
relation R without relating its terms a and b and furthermore without these
terms being related by R. In other words, we must be able to think of a and b as
independent objects and to think of the relation R without relating it to any particular
terms. In today’s fashionable terminology, an external relation can be
75
Bradley, 1935, p. 667; see also p. 641: “Every relation […] has a connection with its
terms, not simply internal or external, must in principle be both at once.”
76
Bradley, 1935, p. 642.
77
Bradley, 1914, p. 291. Cf. also p. 237f.: “And the external relations themselves, if they are
to be absolute, must, I suppose, be thinkable apart from any terms.”
78
I highlight Bradley’s stress on defining the internal/external distinction in terms of their
thinkability because Wittgenstein used a similar definition in the Tractatus. See §6.2.
23
thought of as a case of relatedness without relation. Neither the relation nor the
terms can do the task of relating. Neither of them can be thought of as the reason
for the existence of the relational fact aRb:
And we must, if so, assume that their coming or being together in fact, and as somehow
actually in one, is due in no way to the particular characters of either the relations
or the terms […] Undeniably the fact is somehow there, but in itself it remains
irrational as admitting no question as to its ‘how’ or ‘why’. Or, if you insist on a reason,
that would have to be sought neither in the terms nor the relation, but in a third
element once more independently real and neither affecting, nor again affected by, either
the relation or the terms. 79
This explanation already contains a germ of an argument against the possibility
of external relations. This argument is known as Bradley’s Regress Argument.
The gist of the argument is as follows: suppose a relational fact aRb where R is
an external relation. Making use of spatial metaphors, we can say that R must
be something that exists between its terms. We may now wonder how R is connected
to these terms or how R affects them. In order to explain this we have to
postulate the new dyadic relations R1 and R2 as doing this job.80
R1 relates a and
R and R2 relates R and b. The original relational fact aRb is now decomposed
into two relational facts aR1R and RR2b. Relations R1 and R2 are, however, external
relations (for if they were not, R itself would not be a merely external relation).
The original problem of explaining how a merely external relation can
relate its terms still remains unsolved. It is clear that postulating other external
relations runs into an endless regress.
There is another strategy for resisting this regress, however. Suppose that the
terms a and b have different aspects and that some of these aspects are not essential
to these terms. Let a1 be an accidental aspect of a. It is thinkable that a
lacks the aspect a1 and so let us call this case a’ (i.e., where a is deprived of a1).
The aspect a1 can be compatible with relation R. The reason why R relates a can
thus be explained by reference to a1. Recall that this scenario is possible even
though R is an external relation and a is an independent object. The same ex-
79
Bradley, 1935, p. 642.
80
We may alternatively postulate a triadic relation R’ that aims to relate R, a, and b in the
relational fact R’(R, a, b). The argument would be the same as in the case of two dyadic relations
R1 and R2.
24
planation can be presented with regard to the term b. Hence, the relational fact
aRb can be explained by a’s aspect a1 and b’s aspect b1.
The price that we pay for this explanation is the disruption of the unity of the
terms. Term a is now composed of a’ and a1. The aspect a1 is, in fact, a relational
property of a. Aspect a1 becomes an external relation between a’ and R.
We have to ask now how a’ is related to a1. The regress is here again because
a’s “unity disappears, and its contents are dissipated in an endless process of
distinction.”81
This strategy thus transposes the original problem of explaining
the relational fact aRb into the problem of explaining the unity of the terms.82
What are the conclusions of Bradley’s regress argument? The argument shows
prima facie that wholly external relations are self-contradictory and thus impossible.
What about relations which are partly external and partly internal? Although
such relations are not wholly contradictory, they contain a contradiction.
This means, however, that they are unreal, i.e., they belong to Appearance. And
finally, does this argument lead to the Doctrine of Internal Relations? It depends
how we understand this doctrine. If the doctrine says that all relations are merely
internal, then the argument yields no such conclusion. If, however, what is
meant by the doctrine is that all relations (if there were any) must be partly internal,
we can then accept this conclusion.

In order to prove the unreality of all relations, Bradley still has to show that
there are no merely internal relations. The definition of internal relation is as
follows:
Relations would be merely internal if, the terms being taken as real independently,
each in itself, the relations between them (as a class, or in this or that particular case)
in fact arose or were due merely to the character of the terms as so taken.83
Two features of internal relations follow from this definition: First, internal relations
(in contrast to external relations) cannot stand alone without relating to
81
Bradley, 1897, p. 31.
82
Cf. Ferreira, 1999, pp. 110–116, Vallicella, 2002, and Candlish, 2007, pp. 169–171 for indepth
discussions of this argument.
83
Bradley, 1935, p. 665.
25
their terms. Bradley has stressed this feature again and again: an internal relation
“essentially penetrates the being of its terms” or must “affect, and pass into,
the being of its terms”84
. There is no internal relatedness without relation. Second,
an internal relation holds in virtue of the character of its terms. The character
(or nature) of a term must be understood as all its non-relational proper-
ties.85
We might further say that these terms together make up a systematic or
genuine unity. A change of one term would modify the relation and the other
term. On the one hand, the terms must be conceivable as different entities and,
on the other hand, they are parts of a perfect unity. Internally related things can
be described as a perfect “identity-in-difference”.86
Let me proceed to Bradley’s argument against merely internal relations. As we
have already seen, Bradley employs spatial metaphors in order to characterize
the relationship between relations and their terms. Although merely external relations
are ‘between’ their terms, they fail to be ‘together’ with their terms on
pain of falling into an infinite regress of postulating other relations. Internal relations
show the opposite defect. They are ‘together’ with their terms. They fail,
however, to be ‘between’ them. In Bradley’s words:
An actual relation […] must possess at once both the characters of a ‘together’ and a
‘between’, and, failing either of these, is a relation no longer. Hence our terms cannot
make a relation by passing themselves over into it bodily. For in that event their individuality,
and with it the required ‘between’, would be lost. All that we could have
left would be another form of experience, no longer relational, which qualifying directly
our terms would have ceased to be terms. 87
A wholly internal relation is wholly ‘together’ with its terms. Such an internal
relation is supervenient on all the properties of its terms. The relation, thus, can
have no other properties but those that are supervenient on the properties of its
84
Bradley, 1897, pp. 392 & 364.
85
This understanding of the notion of character follows from Bradley’s example of a relation
between a red-haired man and red-hairedness. If this relation were wholly internal, we
“could from the nature of red-hairedness reconstruct all the red-haired men.” And more generally,
if all things were internally related, “you could start internally from any one character
in the Universe, and you could from that pass to the rest.” (Bradley, 1897, p. 580)
86
Cf. Ferreira, 1999, p. 116.
87
Bradley, 1935, p. 644.
26
terms. This means, however, that a wholly internal relation is not ‘between’ its
terms. Vallicella makes this point aptly: “They [internal relations] are ‘between’
their terms as relations, but not ‘between’ their terms as internal.”88
The problem
is that if the terms stood in a wholly internal relation, we could not distinguish
between them, i.e., they would lose their individuality. Then, however, if
there were no distinct terms, there would be nothing to relate. The (originally
binary) relation would become a property of the whole, consisting of both
terms.89
Let me restate the point in a more formal way. As argued in the case of external
relations, a relation must somehow be connected with its terms or at least some
aspects of its terms. Let aRb be a relational fact and R an internal relation.
Above, we considered the possibility of a case where R was an external relation
that is compatible with a’s aspect a1. In our present case, where R is an internal
relation, the aspect a1 must be comprised of the whole term a. The same holds
for b’s aspect b1. The relational fact aRb actually becomes a non-relational
predication R(ab) or
ab is R.(3)
As we have seen in the previous section (p. 18), no predicative judgment can be
wholly true. There is always an external condition on its truth. The judgment (3)
always qualifies a part of Reality. The property R is external to the whole of ab,
or in other words, ab is not wholly identical with R. The judgment (3) has to be
restated as
ab is an R.(4)
And this judgment in fact has the form
Reality is such that ab is an R.(5)
What seems to be a perfect unity, a perfect system, or a perfect whole, is always
an abstraction from a larger part of Reality. We can consider two examples of a
systematic unity: a living organism and a perfect work of art. We may presume
88
Vallicella, 2002, p. 8.
89
In other words, if all relations were merely internal, all relations would be reducible to
properties. This consideration might lead Russell to a critique based on the fact that not all
relational statements are reducible to the subject-predicate form. See §5.1.
27
that a living organism exhibits a significant degree of systematic unity. Each
part of the organism has a function within the whole and each part depends on
each other part. An analogous consideration may be applied to a great work of
art. Each part (tone, brushstroke, syllable, and so on) fits together with other
parts and with the whole. There is nothing superfluous. But no living organism
is thinkable entirely outside its environment or biotope; and no work of art can
be taken as such outside of its society and culture.90
Hence, if one claims that
this is a living organism or that is a great work of art, these judgments always
qualify a larger part of reality which is not expressed in the surface form (4) of
these judgments. There is always an external condition, e.g., the environment or
culture. Every living organism is only a living organism within its environment;
every work of art is only a work of art within its culture.91
The conclusion is that wholly internal relations are impossible, for judgments
expressing them cannot avoid external conditions. We may be tempted to think
of Reality as a perfectly internally related system, as a perfect identity-indifference.
But then the terms of such relations would have to be independent
substances which would be opposed to these relations. These oppositions are,
however, abstract differences that cannot be wholly true. Bradley can conclude,
therefore, that “‘internal’ relations, though truer by far than external, are, in my
opinion, not true in the end.”92
Merely internal as well as merely external relations
are self-contradictory. Every relation is both internal and external93
and
belongs to Appearance. In other words, all relations are unreal.

In the remainder of this section, I want to focus on the issue of what the terms
of relations in question are, or more precisely, what kinds of terms are supposed
to be related here. Do we speak about relations between particulars (objects,
things, and so on) or between universals (concepts)? Bradley is sometimes unclear
concerning the nature of related terms when he is speaking, for instance,
about the relation between practice and life, desire and will, imagination and
play, or humanity and the universe. These examples suggest that relations be-
90
Wittgenstein advocated similar views concerning works of art and culture. See §18.
91
See Ferreira, 1999, p. 118 for more examples.
92
Bradley, 1914, p. 312.
93
Bradley, 1935, p. 667.
28
tween universals are his main focus. However, Bradley speaks of relations between
things or objects, especially when the internal/external distinction is em-
ployed.94
We have to realize that if the unreality of relations should result in ontological
monism, i.e., in the view that Reality is one, then the term must be realities, at
least prima facie. The view that Reality is one can be taken negatively as meaning
that Reality is not made up of particular building blocks (atoms, objects,
things, and so on). We are interested in the purported relations between these
building blocks.
Furthermore, if one accepted Bradley’s arguments that merely internal and
merely external relations are self-contradictory, then one could not give any examples
of such relations. All examples would then be negative examples of relations
that are not merely internal or external.95
It is surprising, then, that some
commentators of Bradley’s works have offered plenty of examples of internal
relations. So Wollheim, for instance, says that “the relation ‘being married to
someone’ is internal to him [a husband]. Similarly the relation of being disloyal
to one’s country is internal to a traitor”96
. In a similar manner, Pears claims that
A relation is internal if the proposition attributing it to an individual is true a priori.
[…] [T]he proposition that a particular husband is married, or a particular wife is
married, is true a priori, because it is guaranteed by definition.97
To reduce internal relations to a priori truths or to truths that are guaranteed by
definition is an ignoratio to Bradley’s arguments.98
Truths by definition are
conceptual truths. Internal relations would hold, then, between concepts, e.g.,
between the concepts of husband and wife. As we have seen above, even such
94
Cf., for example, Bradley, 1914, p. 176: “A relation exists only between terms, and those
terms, to be known as such, must be objects.”
95
See the example of a relation between a red-haired man and red-hairedness quoted in
footnote 85.
96
Quoted in Manser, 1982, p. 182.
97
Quoted in Manser, 1982, p. 183.
98
As we will see below, Moore and Wittgenstein offered arguments against the possibility
of expressing internal relations.
29
seemingly internal relations cannot escape external conditions and, thus, are
both internal and external.
Do, then, the relations under consideration hold between particulars? Some authors
have interpreted Bradley’s writings in this way with some textual sup-
port.99
If this were so, how could we account for Bradley’s example of a relation
between red-hairedness and a red-haired man? Red-hairedness is a universal
property and if we insisted on its particularity, we would be forced to accept
something like the trope theory, which postulates concrete properties.100
Even
though this line of argument is not wholly unpromising, there is, I think, a less
complicated and simpler explanation available.
As indicated at the beginning of this chapter, the distinction between concepts
and objects is not an either-or choice in the idealist framework. As Appearances,
concepts as well as objects are abstractions from Reality. That is to say: Reality
unites concepts with objects. In Bradley’s words: “The immanent Reality,
both harmonious and all-comprehending, demands the union of both its characters
in the object.”101
An object’s character, as we already know, is comprised of
all properties of the object. A character abstracted from the object can be taken,
then, as a concept. At any rate, objects are continuous with concepts. To put it
another way, the distinction between objects and concepts is a matter of Appearance.
As argued above, relations are Appearances, but their terms are Appearances
too. They are all “abstractions and mere ideal constructions”102
. Every
term is, in Bradley’s system, always partly an object and partly a concept,
and these parts are connected.
By having an entity which is always a continuously between an object and a
concept, we can abstract away an object, leaving a concept (or rather a part of
the object’s character), and we get a term of an internal relation. Such a relation
is, however, internal to a certain degree; it is, at best, more internal than external.
If we abstracted away a great deal of the object’s character, we would get
99
Manser (1982) argues that Bradley’s treatment of relations in his early Principles of Logic
is focused on relations between particulars.
100
In such a scenario, we can consider the relation between Hugo and Hugo’s redhairedness,
where Hugo’s red-hairedness is a concrete property.
101
Bradley, 1914, p. 226.
102
Bradley, 1914, p. 151.
30
an abstract object that can be a term of an external relation. Once again, such a
relation would only be more external than internal, for we cannot abstract from
the character completely. All these examples of ‘conceptual’ relations given by
Wollheim and Pears are, thus, examples of relations that are internal rather than
external.
I have discussed the issue of the nature of terms in detail, since this becomes
central for Wittgenstein. The distinction between concepts and objects is available
to him (although with some reservations in the Tractatus). In his later
works, the nature of related terms determines the internality or externality of a
relation. Conceptual relations are internal, while external relations hold between
objects.103
103
See §10.2.
31
5. Russell and Moore
Both Russell and Moore offered several arguments against the views and doctrines
involving internal relations that they (more or less justifiably) attributed
primarily to the idealists and to Bradley. By rejecting these doctrines, they introduced
their own claims, mostly involving external relations. This rough description
may give the impression that one side favored internal relations over
external relations and the other side the other way around. The nature of their
disagreement is, however, more complex.
5.1. Arguments against internal relations
We have to clarify what exactly Russell and Moore are arguing against. What
are the claims they attribute to the idealists’ camp and which they then subject
to analysis in turn? Russell and Moore attribute to their opponents the so-called
Axiom (or Doctrine or Dogma) of Internal Relations. This is how Russell formulates
the axiom: “Every relation is grounded in the natures of the related
terms.”104
Let us suppose that Russell and Bradley more or less agree in their
definitions of internality or, at least, that their definitions are compatible.105
The
Axiom of Internal Relations thus reads:
Every relation is internal.(6)
As we have seen above in §4.2, the internality of a relation is a matter of degree
for Bradley. This axiom can mean either
Every relation is merely internal (i.e., not external at all).(7)
or it can mean
Every relation is internal to some degree (i.e., not entirely external).(8)
104
Russell, 1907, p. 160.
105
The expression “is grounded in the natures of the related terms” might be taken to mean
something similar to Bradley’s expression “is due merely to the character of the terms”.
Schaffer (2010, p. 349) argues that there was one dominant sense of the notion of internality
invoked by Bradley, Bosanquet, and Moore. This is Schaffer’s formal reconstruction of this
notion: R is internalessential =df (x)(y) if Rxy then necessarily ((x exists ↔ Rxy) (y exists ↔
Rxy)). (I have simplified Schaffer’s definition by considering binary relations only.)
32
Only the latter, weaker claim can be attributed to Bradley; moreover, it is only
an intermediate claim in his proof of the unreality of all relations. But neither
Russell nor Moore really noticed that internality is a matter of degree and,
hence, they could not distinguish between (7) and (8). For them, a relation is
either internal or external.106
This is a serious flaw in their discussion. Russell
and Moore actually provide arguments against (7), as we are going to see. Then,
however, they draw their conclusions as if they have provided arguments
against (8). They imply that some or even all relations are merely external.
Let us now move to Russell’s and Moore’s arguments in more detail. The first
argument was elaborated by Russell primarily in The Principles of Mathematics.
As we saw in §4.2, an internal relation can be reduced to a property of the
whole of all its terms.107
Hence, an internal relation R that holds between a and
b can also be reduced reducible to the property R’ of the whole of ab. The relational
fact aRb is, then, equivalent to the subject-predicate fact R’(ab).108
The
whole (ab) is, however, equivalent to (ba).109
As Russell puts it: “(ab) is sym-
106
The either-or distinction between internal and external relations is salient in Moore’s article
“External and Internal Relations” (1919). Internal relations are opposed here to relations
that are ‘purely’ or ‘merely’ external. Russell in his texts (1903, 1907) also opposes the
fact of internal relations being purely or merely external relations on the one hand, and on
the other hand he also argues for “the axiom of internal relations [being] incompatible with
all complexity” (1907, p. 168). Does, then, ‘internal’ here mean ‘merely internal’ or ‘not
entirely external’?
107
This reducibility was so important for Russell that he made it the defining characteristic
of internal relations. A relation is internal if and only if it is reducible to a property; and it is
external otherwise. See Russell, 1918, pp. 140f. Many years later, in 1959, Russell wrote to
J. O. Wisdom: “As for Hegel, I can still understand why people were Hegelian. The matter
was concerned with the axiom of internal relation according to which a relation between
two terms consists of adjectives of both or an adjective of the whole of which both compose.
If this is accepted, the rest of Hegel’s philosophy follows, as in Bradley.” (Russell,
1959).
108
This conviction is an instance of a broader argumentative strategy of Russell’s. He
thought that the whole philosophical and logical tradition was flawed in its commitment to
the subject-predicate view of propositions and neglected the relational form (and was unable
to admit the reality of relations in the end). See, for example, Russell, 1918, p. 127 or Candlish,
2007, pp. 132–136.
109
This equivalence does not hold for Wittgenstein. At TLP 3.1432 he indicates how a relation
(regardless of its symmetry) can be eliminated from a relational fact aRb. In the fact ab,
33
metrical with regard to a and b, and thus the property of the whole will be exactly
the same”110
in the case of aRb as in the case of bRa. This means, however,
that internal relations must be symmetrical. From the Axiom of Internal Relations
it follows, therefore, that all relations are symmetrical. Russell argues
now that because there are—as a matter of fact—asymmetrical relations (like
the relation of something being greater than something), the Doctrine of Internal
Relations hence cannot be true.
Russell’s objection turns around the problem of explaining the evident asymmetry
of some relations (given that the Axiom of Internal Relations holds true).
If we take the weaker reading of the axiom (8), there is an obvious way of explaining
the asymmetry. On this reading, every relation may be partly internal
and partly external and a relation gets its asymmetry due to its partial externality.
Russell’s objection holds only for relations that are wholly internal; and
therefore he has to presuppose the stronger reading of the Axiom of Internal Relations
(7) in his argument. Furthermore, Russell’s argument is in the end based
on our commonsense conviction that there are asymmetrical relations like
‘greater than’. Bradley would also admit that there are such relations, but they
belong to Appearance only. He would hardly admit, however, that the Axiom of
Internal Relations (7) holds true for Appearances, i.e., that every relation is
merely internal in Appearance.
As we saw in §4.1, the grammatical or apparent form of a judgment is, for
Bradley, not equivalent to its real form. The ultimate subject of every judgment
(subject-predicate or relational) is Reality-as-a-whole. The grammatical subject
is only a limited or special subject and the grammatical predicate is only a limited
predicate. Let us take the subject-predicate judgment ‘Hugo is tall’. Its real
form is expressed as ‘Reality is such that Hugo is tall’. The same transposition
can also be applied to the relational judgment:
Hugo is taller than Guido.(9)
which is in fact
Reality is such that Hugo is taller than Guido.(10)
it is said that a stands to b in a certain relation. The fact ba expresses a different relation
than the fact ab.
110
Russell, 1903, p. 225.
34
This judgment is, however, different from
Reality is such that Guido is taller than Hugo.(11)
In other words, although the ultimate subject of every judgment is Reality, the
difference between the limited subject and the limited predicate is preserved.
The limited predicate of (10) is ‘taller than Guido’, not ‘Guido’; and the limited
predicate of (11) is ‘taller than Hugo’.111
The reason why this is so is the (partial)
externality of every judgment. No judgment is capable of a complete unification
of its limited subject with its limited predicate and of this semi-unified
whole with Reality. Hence, Bradley’s account of judgment is able to handle
asymmetrical relations in the same way as it is able to handle symmetrical rela-
tions.

Let us turn now to the next argument. The previous argument was based on the
commonsense belief that there are asymmetrical relations. But Bradley’s ultimate
claim that there are, in the end, no relations would therefore be disproved
by the existence of any relation (whether symmetrical or asymmetrical). If, as
Russell says, “the axiom of internal relations is equivalent to the assumption of
ontological monism and to the denial that there are any relations”112
, then the
existence of any relation would refute Bradley’s position. And Bradley is willing
to admit this:
Asymmetrical relations are said to disprove Monism, because Monism rests on simple
inherence [i.e., predication] as the only way in which there is [any] ultimate reali-
ty.
The argument, if right, is improperly limited – because any relations, if so, disprove
Monism.
But Monism does not rest on simple inherence as the one form of reality. It even (in
my case) says that that form is unsatisfactory (see Appearance).
111
See Ferreira, 1999, p. 197.
112
Russell, 1907, p. 163. Cf. footnote 107 above.
35
In short, far from admitting that Monism requires that all truths can be interpreted as
the predication of qualities of the whole, Monism with me contends that all predication,
no matter what, is in the end untrue and in the end unreal …113
Bradley argues in Appearance and Reality114
that relations presuppose qualities
and vice versa. The subject-predicate form presupposes the relational form,
namely a relation between the subject and the predicate. No commonsense belief
can prima facie prove or disprove anything about Reality, because commonsense
beliefs are about Appearances only. This argument can also be seen
as a reductio ad absurdum of the previous argument that invoked asymmetrical
relations.
What the disagreement is really about is the ontological status of relations, and
not the way they appear to us. Relations played an important role in Russell’s
accounts of judgment and predication115
at that time. Russell had proposed several
theories aimed at explaining the unity of the proposition by invoking a binary
relation between the other constituents of the proposition or a multiple relation
between the other constituents and the judging mind. Here it is important
that such a relation is a real constituent of the proposition among the other
terms. If relations were unreal, Russell’s binary or multiple relation theories of
judgment would then be unsatisfactory. These theories would explain predication
by referring to a relation which is not wholly real. Such a relation must be
explained by something else. Something would therefore be missing in Russell’s
account if Bradley were right. The root of the disagreement between Russell
and Bradley lies also in their general theories of judgment.

The third and last argument I shall now focus on is the most promising one in
my eyes. The previous arguments focused on whole classes of relations (on
asymmetrical relations or on all relations). This argument focuses on a single
relation, namely the relation between a part and the whole. If the Axiom of Internal
Relations holds true, this relation must be internal as well.
113
Bradley, 1935, pp. 670, 672. Quoted in Candlish, 2007, p. 165.
114
Bradley, 1897, Ch. III.
115
We can take these two terms to be synonymous here.
36
The part-to-whole relation is, of course, an asymmetrical relation, i.e., it is different
from the whole-to-part relation which is a typical example of an internal
relation. A particular stamp collection, for example, would no longer be the
same collection if some stamps were taken away from it. The inverse relation
(part to whole) does not seem to be an internal one. A particular stamp remains
the same stamp whether it is a part of a collection or not.116
Both Russell and Moore admit that the whole-to-part relation is internal: “every
relational property of the form ‘having this for a spatial part’ is ‘internal’ in our
sense”117
. They argue, however, that the opposite part-to-whole relation is not
internal, but purely external. This position is, in fact, a fundamental feature of
ontological atomism. Each object is what it is regardless of its placement
among other objects. The nature of each object is independent of the relations it
bears to other objects.
The argument aims to demonstrate that if a part were internally related to the
whole, then the distinction between part and whole would collapse. Let me first
state Russell’s brief version of this argument:
In a “significant whole,” each part, since it involves the whole and every other part, is
just as complex as the whole; the parts of a part, in turn, are just as complex as the
part, and therefore just as complex as the whole. Since, moreover, the whole is constitutive
of the nature of each part, just as much as each part is of the whole, we may
say that the whole is part of each part. In these circumstances it becomes perfectly
arbitrary to say that a is part of W rather than that W is part of a.118
If two objects were internally related, we could reconstruct the nature of one
object from the nature of the other object.119
If a part were internally related to a
whole, this part would be as complex as the whole. Then, however, this part
would contain the whole. The entire situation of a whole having parts would be
unintelligible, which leads us to the conclusion that there is no complexity
whatsoever.
116
But a particular stamp might be more valuable if it is a part of a certain miniature sheet. I
am thankful to Maja Jaakson for providing me with this example.
117
Moore, 1919, p. 288.
118
Russell, 1907, p. 154.
119
See Bradley’s example of a red-haired man in footnote 85, p. 15.
37
Let me restate this argument in Moore’s terms. He argues in §22 of the Principia
Ethica that internal relations between a part and the whole are unintelligible.
Suppose there were an organic whole whose parts would not be what they are
but for the existence of the whole. These parts would also be internally related
to the whole. Moore argues that this situation is self-contradictory. The parts of
such a whole cannot be distinct objects of thought. For if we tried to name a
part of the whole W by a predicate a, the whole would figure in the definition of
this predicate. When we assert that something is a, we could also assert that the
same thing is W. This means, however, that W is a part of a. This is contradictory
unless W is identical with a. This implies that we cannot successfully assign
a name to an internal part of a whole.
I think that this is by far the best argument that Moore and Russell offered
against internal relations. Moreover, I think that Bradley and the other idealists
would accept this line of argument and echoes of it can be found in Wittgenstein’s
Tractatus as well as in his principle that internal relations can only be
shown, but not said (the Doctrine of Showing). Moore concludes from this argument
that the part-to-whole relation must be external. Furthermore, I maintain
that the argument is valid not only for the part-to-whole relation, but for a
large class of internal relations between objects. This argument does not affect
those internal relations that are analytic in the sense that they hold in virtue of
being in language. That is also the case for the whole-to-part relation, which is
internal in spite of this argument. This argument nevertheless allows for the important
conclusion that all relations between objects are external. This is the
Doctrine of External Relations.
Moore concluded his extreme pluralism from this argument against the intelligibility
of internal relations. This argument is, however, a double-edged sword.
External relations are, for Bradley, not an alternative to internal relations. External
relations are even more unintelligible or unreal than internal ones, a view
which Bradley supports by his regress argument.120
Bradley’s radical conclusion
is that there are, in the end, no relations at all; and thus no distinct objects that
could be related. The one and only object is Reality-as-a-whole. Hence, this
third argument cannot help us to resolve the choice between pluralism and mon-
ism.
120
See §4.2.
38
5.2. In favor of external relations
The rejection of the Axiom of Internal Relations leads us to the opposite view
that there are some external relations. And there is indeed enough textual support
for this claim in Russell and Moore. Russell says, for instance, that
there are such facts as that one object has a certain relation to another, and such facts
cannot in general be reduced to, or inferred from, a fact about the one object only together
with a fact about the other object only.121
Moore says quite explicitly that “some relations are purely external”122
and similar
claims can be found in Russell too. I would like to focus now on a different
claim that has often been ascribed123
to Russell and Moore, namely the claim
that
All relations are (merely) external.(12)
Russell, in his early paper “The Classification of Relations” (1899), indeed
wrote explicitly that “all relations are external”124
. But on a closer look at his
argument, we see that he derives this claim from the irreducibility of relations
to pairs of predicates of their terms. No relation aRb is, for Russell, reducible to
the pair of predicates α(a) and β(b). This is so because every relation presupposes
the relations of diversity between their terms. Such a pair of predicative
statements cannot guarantee that the terms a and b would be distinct.125
Russell
stresses that he is retaining Bradley’s phrasing in describing his view that all
relations are external. If so, however, then from this argument it follows only
that there are no relations which are purely internal or that all relations are partly
external. We have here an instance of the first argument against internal rela-
121
Russell, 1907, p. 161, my italics.
122
Moore, 1919 , p. 289.
123
See, for example, Hylton, 1990, p. 121; Proops, 2002; Potter, 2009, p. 13; McGinn, 2006,
p. 178.
124
Russell, 1899, p. 143.
125
As we will see, the relation of diversity is plays an important role in Wittgenstein’s analysis
of internal relations. He condemns reflexive internal relations (that is, relations of the
form aRa) to be unintelligible.
39
tions that has already been presented in the previous section. Russell only denies
(7) by his argument, but he continues his argument as if he has denied (8).
The general claim (12) must be taken with additional qualifications. According
to Bradley, Russell “defends a strict pluralism, for which nothing is admissible
beyond simple terms and external relations.”126
Russell protests, however, that
this is not what he himself means by the Doctrine of External Relations. He derives
this internal/external distinction from the notion of reducibility, as we saw
above. However, he says, even within the same text (The Philosophy of Logical
Atomism), that
Particulars [belong to] an inventory of the world, that each of them stands entirely
alone and is completely self-subsistent. It has the sort of self-subsistence that used to
belong to substance, except that it usually only persists through a very short time, so
far as our experience goes. That is to say, each particular that there is in the world
does not in any way logically depend upon any other particular.127
This quotation may give rise to the impression that there are only mutually independent
particulars. These particulars are simples. Any complexity is to be
located in complexes, i.e., in facts. If so, no internal relations between simples
would be admissible and hence all relations between simples would be external.
Russell is indeed committed to this doctrine. This doctrine is, however, restricted
to relations between simples (let us call this the restricted Doctrine of External
Relations as opposed to the unrestricted doctrine mentioned above). Relations
between complexes and facts128
are not affected by this doctrine. For instance,
the relation between a whole and its parts may still be internal in spite of
the doctrine.
As we can see, the main motivation for the ascription of the doctrine that all relations
are external lies in Russell’s (and Moore’s) general metaphysical insights
rather than in their direct arguments. There is, furthermore, another general
view that might motivate the unrestricted Doctrine of External Relations.
Let us proceed to the case of relations between propositions. Like Bradley,
126
Quoted in Russell, 1918, p. 142.
127
Russell, 1918, p. 30.
128
For Russell, facts are always complexes; they are unities of two or more simples or other
complexes.
40
Moore and Russell held, at a particular stage of their thinking, that (true) propositions
are identical with the facts they express. The identity of a fact is constituted
solely by the fact itself. This view is sometimes labeled the identity theory
of truth. If, however, there were no internal relations between objects, then there
would be no internal relations between propositions. A fact could happen to be
related to another fact; but this relation could not be constitutive of this very
fact. The consequence is that the relation of logical entailment is also an external
relation. But then Russell and Moore would be committed to the view that
logical relations are accidental; propositions are what they are independently of
the logical relations in which they stand. Although this view seems to be unpromising,
let us try to make some sense of it.
This account, I would like to argue, does not imply that logical entailment is an
external relation.129
The restricted Doctrine of External Relations together with
the identity theory of truth implies that all relations between propositions that
directly stand for facts are external. But propositions that directly stand for (or
are identical with) facts are just elementary propositions. Thus, Moore and Russell
are committed to the view that all relations between elementary propositions
are external.
This view does not, however, render logic external. Logic is, for Russell explicitly,
about complexes—or molecular propositions which are actually identical
with complexes. In his manuscript Theory of Knowledge, Russell writes: “Belief
in a molecular proposition gives what is most distinctive in the process of
inferring”130
. At least some relations between complexes or structures are clearly
internal. Consider the whole-to-part relations which are obtained between a
complex and its part (which may also be a complex as well). So, therefore, the
relations of inference between (p ∨ q) and q can be taken as an instance of the
whole-to-part relation. In §7.3 I am going to show that Wittgenstein fully developed
this idea in the Tractatus.
To sum up the various doctrines of external relations that Russell and Moore are
committed to:
129
Proops (2002) wrongly takes Russell to be committed here to the (in this context quite
absurd) view that the relation of logical entailment is an external one.
130
Russell, 1984, p. 105.
41
There are external relations.(13)
All relations between simples are external.(14)
All relations between elementary propositions are external.(15)
In Chapter 7, I shall argue that Wittgenstein endorsed in the Tractatus the existential
claim (13) and the universal claim (15). He, however, admitted that there
might be internal relations between simple objects. (14) cannot, therefore, be
attributed to Wittgenstein.
III. Wittgenstein’s early writings
6. Definitions of the internal/external distinction in the early
writings
The aim of this chapter is to investigate how Wittgenstein defines the distinction
between internal and external relations in his early texts, i.e., his preTractarian
texts, and, of course, in the Tractatus itself. We have to make clear
how Wittgenstein defines this distinction before it can be employed to explain
or illuminate various problems. Hence, only remarks that may contribute to our
understanding of the notion of internal and external relations are going to be
addressed in this chapter. Remarks that employ the internal/external distinction
in order to throw light on other problems will be addressed in subsequent chap-
ters.
6.1. Pre-Tractarian texts
In Wittgenstein’s earliest texts, the Notes on Logic, which he dictated to or
elaborated with Russell, there is no occurrence of the internal/external distinction.
The first occurrence is to be found in Wittgenstein’s notes that he dictated
to G. E. Moore in Norway in April 1914. Here is Wittgenstein’s very first remark
concerning the distinction:
Internal relations are relations between types, which can’t be expressed in propositions,
but are all shewn in the symbols themselves, and can be exhibited systematically
in tautologies. Why we come to call them “relations” is because logical propositions
have an analogous relation to them, to that which properly relational propositions
have to relations.131
Many ideas are expressed in this remark. Let us try to untangle them. The terms
of internal relations are types of entities. This means that internal relations are
relations holding between universals. Wittgenstein will stand by this feature of
131
NB, pp. 116f.
44
internal relations throughout his philosophical career and later it becomes even
more important than it is in his early texts.132
A second idea is that internal relations cannot be expressed in propositions; they
can only be shown in the notation of a logically adequate language (a kind of
Begriffsschrift). The external/internal distinction is, thus, closely connected to
the distinction between saying and showing from the outset. These two distinctions
are, however, not identical. All internal relations can only be shown, but
surely not everything that can be shown can be conceived of as a case of an internal
relation. Or at the very least, it is not straightforwardly clear how to apply
the internal/external distinction in ethics or to Wittgenstein’s reflections about
the sense of the world.
There is also a close connection here between internal relations and logical tautologies.
Again, all internal relations can be transformed into tautologies. Internal
relations share with logical tautologies their strict necessity. Moreover, the
holding of an internal relation can be proven systematically just as logical tautologies
can be deduced from axioms. So, if proposition p is related to proposition
q by an internal relation R, there must be a systematic way of transforming
p into q. In other words, there must be a formal operation O that transforms p
into q so that O(p) ↔ q is a tautology.
Internal relations are opposed to proper relations which can be expressed by
means of a proposition. Internal relations are (somehow) improper; whereas external
relations are relations of a proper nature. The external relation that, for
example, Wittgenstein admired Frege can be expressed in a (proper, i.e., empirical)
proposition ‘Wittgenstein admired Frege’. This is analogous to an internal
relation being shown in a logical proposition, i.e., in a tautology. In De Morgan’s
law ¬( P ∧ Q ) ↔ ¬P ∨ ¬Q, for instance, the internal relation between ¬(
132
There will be one important exception: on what I call the de re interpretation of simple
objects, Wittgenstein allows for the internal properties of Tractarian objects which determine
an object’s combinatorial possibilities with other objects. These internal properties are,
in effect, relational properties; hence there are internal relations between simple objects. But
more about this later in §8.1. The subsequent section 8.2 addresses the competing interpretation
of simple objects, which is called de dicto and does not require any internal relations
between simple objects.
45
P ∧ Q ) and ¬P ∨ ¬Q (where P and Q are propositions) is shown. The systematic
operation of transforming one of these expressions into the other occurs here.
Wittgenstein insists that internal relations hold among universals. This is, however,
often obscured by his mode of expression. Consider the following remark:
“The agreement of two complexes is obviously internal and for that reason cannot
be expressed but can only be shewn.”133
Taken literally, this remark makes
the claim for an internal relation between two complexes, including complexes
of particular things. This would, however, be a misunderstanding. Two complexes
agree on two conditions: they have the same structure and their simple
parts mutually agree. To have the same structure is tantamount to having the
same form. Hence, there must be agreement between the forms of the complexes,
which is something universal. The second condition amounts to identity between
the concepts that describe the simple parts (identity is thus one case of an
internal relation). This is so because two simple entities can be the same only in
the sense that they fall under the same general concept. Hence, this remark can
be read as claiming that an internal relation of identity holds between concepts,
but it does not commit Wittgenstein to relations between objects.
This idea of internal agreement between complexes can now be extended to the
idea of partial agreement. Wittgenstein says that “The proposition that is about a
complex stands in an internal relation to the proposition about its component
part.”134
Following on from the elaboration in the previous chapter, this remark
says that a description of the form of a complex stands in an internal relation to
a description of the form of its part. In short, the form of a complex contains a
form of its part. This is to say merely that a complex contains its part. We could
also conclude that for Wittgenstein (and likewise for Moore135
) the relation between
a whole and its part is always an internal one. The internal agreement of
two complexes is thus rendered as a special case of the whole-to-part relation.
Before moving any further, let us stop for a moment and illustrate how to transform
this whole-to-part relation into a logical tautology. Let us suppose a complex
consists of two parts M and N. This can be described as ( m ∧ n ), where
the proposition m stands for ‘There is M here’ and n for ‘There is N here’. Here
133
NB, p. 9.
134
NB, p. 43; see also TLP 3.24.
135
See §5.
46
we have a tautology that ( m ∧ n ) → n. To put this briefly, the fact that there are
M and N here implies that there is N here.136
There is also another important essential feature of internal relations, namely
that there are “no hypothetical internal relations”137
. Wittgenstein’s explanation
of this is as follows: “If a structure is given and a structural relation to it, then
there must be another structure with that relation to the first one. (This is involved
in the nature of structural relations.)”138
In the light of Wittgenstein’s
equating of internal relations with structural relations in the Tractatus139
, we do
not need to be disturbed by Wittgenstein’s shifting terminology here. Internal
relations are necessary in the sense that if there is one structure and an internal
relation that relates this structure to another structure, there must necessarily be
the other structure (or structures in the case of relations with more members).
This means that the internal relation gives an unequivocal way of transforming
the former structure into the latter one. In terms of formal operations, this feature
means that if a structure s and an operation O are given, then there is an
unequivocal way for this operation to be applied to the structure, i.e., it is possible
to calculate O(s) unequivocally.
Wittgenstein’s account of the distinction between internal and external relations
that has been outlined here is very sketchy. Fundamentally, it says only that internal
relations express a sort of tautology that obtains between universals140
and that one term can be transformed into the other by a formal operation. This
account is incomplete for at least two reasons: it is restricted solely to binary
relations and there is no specification for how to transform an internal relation
into a logical tautology. This is supplemented in the Tractatus (or in those manuscripts
from which the Tractatus was composed141
).
136
The reverse transformation would not work. The conjunction of ‘There is M here’ and
‘There is N here’ does not entail that M and N are parts of the whole, for one would need
set theory and predicate logic.
137
NB, p. 19.
138
NB, p. 19.
139
TLP 4.122.
140
Sentences are among them, for any sentence contains at least one universal.
141
Remarks concerning internal and external relations from the so-called Prototractatus do
not substantially differ from the ones in the Tractatus.
47
6.2. The Tractatus
In the Tractatus Wittgenstein gives an explicit definition of the notion of an internal
property:
A property is internal if it is unthinkable that its object should not possess it.142
Wittgenstein supplies two addenda (in parentheses) to this definition. The first
is an example of an internal relation:
This shade of blue and that one stand, eo ipso, in the internal relation of lighter to
darker. It is unthinkable that these two objects should not stand in this relation.143
The second addendum is:
Here the shifting use of the word ‘object’ corresponds to the shifting use of the words
‘property’ and ‘relation’.144
As in the previous section, we should not take this definition too literally in the
sense that internal relations hold exclusively among Tractarian simple objects.
The expression ‘object’ in the definition should be understood without any
commitments as a term or a relatum. Next, the definition of internal property is
suitable for defining internal relations too. Hence, a relation is internal if it is
unthinkable that its objects (or terms, or relata) should not possess it.
Before examining this definition, let me say a few words about the example
Wittgenstein gives. The internal relation between two color shades is paradigmatic
for him. He repeatedly uses this example in his later texts, most prominently
in his conversations with members of the Vienna Circle145
or in his last
manuscripts on colors.146
The relata of this internal relation are neither concrete
things that have these colors, nor expressions denoting these colors. We can
take as an example two stones. One has a midnight-blue-colored surface, the
142
TLP 4.123.
143
TLP 4.123.
144
TLP 4.123. See §8.1 for further discussion.
145
VW 2003, p. 238.
146
ROC I, §1; Ms 176, p. 1r.
48
other one is sky-blue. The relation of being darker between these two stones is
an external one. It is thinkable that the color of the first stone may fade so that
this stone will cease to be darker than the other stone. The same can be said of
the relation between the expressions ‘midnight-blue’ and ‘sky-blue’. These have
concrete occurrences (as color patches) on the pages of this book, and it is
thinkable that the color of some of these patches could fade. And finally, the
lightness relation between the color shades ‘midnight-blue’ and ‘sky-blue’ is an
internal one. The reason for this internality could be that, for instance, in the
HSV color space sky-blue has a brightness of 92%, whereas midnight-blue has
a brightness of 44%. It is unthinkable that 92 is less than 44. Note that no psychological
considerations can be brought in as a counterexample here. If someone
nevertheless insisted that midnight-blue is not darker than sky-blue, they
must either understand some other colors under these terms, or understand the
relation of being darker in some other sense, i.e., their intuitive notion of darkness
and brightness would be different from that of the HSV color space. Internal
relations can thus be exhibited in true mathematical statements and, in the
end, in tautologies.147
The crucial notion in the definition used here is that of thinkability, or rather of
unthinkability (‘Undenkbarkeit’). We can rule out the psychological notion of
thinkability which Wittgenstein certainly did not have in mind. There are (at
least) two other ways to understand this notion. A straightforward reading
would be that ‘unthinkable’ means ‘impossible’ or ‘inconceivable’. A proposition
is impossible if one cannot conceive or entertain this proposition. But there
are no impossible propositions, so we cannot say of a proposition that it is impossible
or unthinkable. What is meant here is rather this: every attempt to put
certain signs together must fail. These signs cannot be assembled into a sign
expressing a proposition, i.e., into a propositional sign. Hence, although we
cannot say of a proposition that it is unthinkable, we can say of certain signs
that when put together, they cannot express any proposition.
A relation is internal if it is impossible that its relata could not be related by this
selfsame relation. This definition can be rephrased by saying that a relation is
internal if it is necessary for its relata to stand in this relation. This is in agreement
with the previous definition which states that internal relations can be sys-
147
Cf. TLP 6.22.
49
tematically exhibited in tautologies. A not holding of an internal relation is just
as unthinkable as it is unthinkable that a tautology is not true.
There is, however, a more technical understanding of the notion of thinkability.
Wittgenstein says that “‘A state of affairs is thinkable’: what this means is that
we can picture it to ourselves.”148
Since no pictures are true a priori,149
no tautologies
are thinkable in this technical sense. If so, the notion of an internal relation
would be self-contradictory, for internal relations are supposed not to be
unthinkable, but as they cannot picture a state of affairs they are unthinkable as
well.150
A possible way out is to point out that Wittgenstein, strictly speaking, makes
only the following claim: “What is thinkable is possible too.”151
We cannot,
however, infer from this claim that what is necessary is unthinkable. Wittgenstein
speaks of the thinkability of states of affairs and leaves open for now
whether there are other realms of thinkability or unthinkability. The immediately
following remarks, however, may shed light on this issue. Here we are going
to delimit what cannot be thought: “We cannot think anything unlogical”.152
Something illogical is simply something which contradicts logical laws; and
this is the same as claiming that a contradiction is true. And again, this delimitation
does not touch upon the question of whether logical tautologies are thinkable
or unthinkable. It would be strange, however, to claim that we cannot think
or entertain both contradictions and tautologies. This interpretation is further
supported by Wittgenstein’s claim that “Propositions can represent the whole of
reality.”153
Such propositions are, precisely, logical tautologies.
These considerations lead to the conclusion that Wittgenstein’s own technical
definitions of thinkability and unthinkability allow for tautologies being thinkable
and contradictions being unthinkable. By the same token, internal relations
can be located within the realm of the thinkable just as logical tautologies can—
148
TLP 3.001.
149
TLP 2.225.
150
See Proops, 2002 for arguments in favor of this conclusion.
151
TLP 3.02.
152
TLP 3.03. I use Ogden/Ramsey’s trans. here which is closer to the German original: “Wir
können nichts Unlogisches denken”.
153
TLP 4.12.
50
except that they touch the borderline between the thinkable and the unthinkable.
Hence, the Tractarian definition of internal relation does not need to be rendered
self-contradictory.
This definition of internal properties and relations occurs in the context in
which Wittgenstein describes what propositions and reality have in common,
and this is their logical form. But a proposition cannot represent its logical
form. In order to be able to talk (in a certain sense) about logical forms (and
about their parts, i.e., their formal properties), Wittgenstein introduces the saying/showing
distinction and the closely connected external/internal distinction.
Tractarian objects and states of affairs have formal properties just as (complex)
facts have structural properties. These properties can be relational properties or
they can be mutually related. Wittgenstein now stipulates that these properties
and relations are internal:
Instead of ‘structural property’ I also say ‘internal property’; instead of ‘structural relation’,
‘internal relation’.154
These terminological stipulations allow us to broaden the scope of inquiry. This
talk of structural or formal properties and relations can be taken to be the same
as talk of internal relations. So when Wittgenstein talks of, for instance, “the
structure of a picture,”155
we can take him to be talking of the internal properties
of (or internal relations within) a picture.156
Another example is the remark from
the Notebooks discussed a few pages above where Wittgenstein talks of “the
nature of structural relations”157
.
154
TLP 4.122.
155
TLP 2.15.
156
The notion of an internal relation is ambiguous here, reproducing the ambiguity of the
notion of a picture. Internal relations within a real picture express its structure, which is an
arrangement of its parts. (The red apple is to the left of the jug.) ‘Internal’ means ‘inner’
here. Internal relations within a logical picture are relations among its logical parts, which
are concepts. Internal relations are conceptual relations. An analogous ambiguity is to be
found in Fr. Bradley where ‘internal’ means inter alia ‘to be part of’ a judgment. Cf. the beginning
of §4.2. I am indebted here to Ondřej Beran.
157
NB, p. 19, 25.10.1914.
51
Wittgenstein’s definition of the internal/external distinction is supposed to
match or follow the definitions given by the idealists on the one hand and by
Russell and Moore on the other. This is clearly Wittgenstein’s aim when he
writes: “I introduce these expressions in order to indicate the source of the confusion
between internal relations and relations proper (external relations), which
is very widespread among philosophers.”158
We can surmise therefore that the
confusion Wittgenstein had in mind was the obscuring of the internal/external
distinction in the surface grammar of our natural language. Nevertheless, several
remarks later he writes: “Here we have the answer to the vexed question
‘whether all relations are internal or external’.”159
Wittgenstein is, of course,
referring to the controversy about relations as examined in Chapters 4 & 5.
What is the answer Wittgenstein gives us exactly? The answer, which is spread
over the remarks ranging from 4.122 to 4.1251, is that the obtaining of an internal
relation cannot be expressed (or said or asserted) by means of propositions.
The obtaining of internal relations can only be shown160
in propositions that are
concerned with the relevant relata. So Wittgenstein would side with Russell and
Moore here that all relations are indeed external, but with the reservation that
all relations that can be expressed by a proposition are external. He admits,
however, contra Russell and Moore, that internal relations are nevertheless not
nonsensical. They are part of the symbolism161
of a logically adequate language
and they are thus shown in such a symbolism.
What I have presented thus far is, of course, not a solution to the problem as to
whether all relations are internal or external. I take it here that Wittgenstein is
proposing a terminological stipulation that allows him to enter into conversation
with his predecessors by subsuming the problem in question under his distinction
between saying and showing. Any answer to the question whether there are
internal relations at all, or what their status is, depends (in the Wittgenstein of
158
TLP 4.122.
159
TLP 4.1251.
160
Wittgenstein uses the expression “zeigt sich” (TLP 4.122) here which is translated by
Pears/McGuinness as “makes itself manifest”. Ogden and Ramsey translate it as “shows
itself”. This translation does not obscure the connection between the saying/showing distinction
and the external/internal distinction.
161
TLP 4.4611.
52
the Tractatus) on an understanding of the idea that something can be shown, but
not said.
6.3. Summary
The following (though not necessarily independent) characteristics of internal
relations can be found in Wittgenstein’s early texts:
(i) Internal relations are such that it is unthinkable (or impossible) that their
relata do not possess them.
(ii) Internal relations hold between concepts or universals.
(iii) Internal relations can be exhibited in tautologies.
(iv) The identification of a term of an internal relation is, eo ipso, the identification
of all other terms. This characteristic, of course, does not apply to
internal properties.
(v) The external/internal distinction is an instance of the more general saying/showing
distinction.
(vi) Internal relations can also be labeled as structural or formal relations.
53
7. The Doctrine of External Relations
Like Russell and Moore, Wittgenstein was committed to the Doctrine of External
Relations, but in a different sense than his mentors. Although he shared with
Moore and Russell the view that all relations between elementary propositions
are external, he also allowed for internal relations between propositions and
simple objects. Moreover, he sided with Bradley on the issue of the unreality of
all relations. Finally, I shall argue in this chapter that despite the Doctrine of
External Relations, Wittgenstein conceives of logical entailment as being based
on internal relations and, hence, as necessary.
7.1. Elimination of signs for relations
Wittgenstein would have accepted Moore’s argument against the intelligibility
of internal relations, which was discussed earlier in §5.1. Wittgenstein introduces
the internal/external distinction “in order to indicate the source of the confusion
between internal relations and relations proper (external relations)”162
. This
implies that internal relations are somehow improper for Wittgenstein.
I shall now argue that signs for relations cannot occur in fully analyzed propositions.
One of the aims of the analysis proposed in the Tractatus is to eliminate
any signs for relations. The immediate conclusion of this claim is that signs for
relations are not names, and relations are not simple objects. One of the tenets
of the picture theory is that relations between the elements of propositional
signs signify relations between the elements of a signified fact. In the Notes on
Logic Wittgenstein writes: “Thus facts are symbolised by facts, or more correctly:
that a certain thing is the case in the symbol says that a certain thing is the
case in the world.”163
By this remark we can understand that the symbolizing
fact and the symbolized fact have to be in accordance. They have to contain the
same combinations of things, where the things from the symbolizing fact (i.e.,
the names or signs for complexes) correspond to the things from the symbolized
fact. This insight—surely quite revolutionary—is sometimes called the symbolic
or linguistic turn. At this point it is important to say that relations within the
162
TLP 4.122.
163
NB, p. 96.
54
symbolizing fact express relations within the symbolized fact. The picturing relationship
is a relation between facts.
This remark from the Notes is further developed in the Tractatus at 3.14s where
Wittgenstein argues that “[a] propositional sign is a fact.”164
Relations between
things should be signified by spatial relations between the elements of a propositional
sign: “the spatial arrangement of these things will express the sense of
the proposition.”165
Hence, propositional signs are capable of representing relations
by their spatial structure, i.e., by the spatial arrangement of their elements—in
a logically adequate notation.
The key remark, which expresses Wittgenstein’s eliminavistic strategy, then
immediately follows:
Instead of, ‘The complex sign “aRb” says that a stands to b in the relation R’, we
ought to put, ‘That “a” stands to “b” in a certain relation says that aRb.’166
Let us assume for a moment that the signs ‘a’ and ‘b’ stand for simple objects,
i.e., that they are names. In this case, then, a sign for a relation is contrasted
with a sign for an object. I take it here that Wittgenstein is proposing a refinement
to our mode of expression. It is inadequate to analyze the complex sign
‘aRb’ into its simple parts ‘a’, ‘b’, and ‘R’. The sign for the relation ‘R’ should
now be eliminated from our notation or should disappear in the process of analysis.
That which in a logically inadequate notation is signified by aRb, should
be signified in a logically adequate notation by a certain concatenation of the
names ‘a’ and ‘b’. This specific relation is signified by a spatial arrangement of
‘a’ and ‘b’ within the propositional sign.
The relation between ‘a’ and ‘b’ is a structural relation within the propositional
sign. ‘Structural relation’ is synonymous for Wittgenstein with ‘internal relation’.
The sign for relation R, which might have been an internal or an external
relation, has to be eliminated in favor of an internal relation within the signifying
fact. Wittgenstein expresses exactly this idea in one of his conversations
164
TLP 4.14.
165
TLP 3.1431.
166
TLP 3.1432.
55
with Waismann. Wittgenstein says there that external relations give us only incomplete
descriptions of a situation and then goes on to say:
If we describe the state of affairs completely, the external relation disappears. But we
must not believe that there is then any relation left. Apart from the internal relation
between forms that always obtains, no relation need occur in the description, and this
shows that in fact relational form is nothing essential; it does not depict anything. 167
This is no argument against the intelligibility of relations, though. Wittgenstein
maintains, however, that signs for relations can be eliminated in logical analysis
and are therefore not necessary in a logically adequate notation. Then, however,
according to Wittgenstein’s version of Occam’s maxim, if a sign is not being
used, it is meaningless.168
No sign for relations is used in a logically perfect
language. Therefore, signs for relations are meaningless in such a language.
In order to complete the argument that relations cannot be the constituents of
facts along with objects, we need to show that internal relations cannot be
named, i.e., that there are no names for internal relations. This insight is tantamount
to the earlier claim by Wittgenstein that there is “no name which is the
name of a form”169
. We have to realize that a name of a logical form would be a
logical constant. This claim is, hence, a part of the meaning of Wittgenstein’s
fundamental idea that there are no logical constants.170
I follow Gregory Landini
here in equating this fundamental idea with the Doctrine of Showing.171
What arguments could Wittgenstein offer in favor of these fundamental claims
or doctrines?
The Doctrine of Showing is, however, not a philosophical thesis. It is, rather, a
sort of guiding philosophical principle which is related to Frege’s distinction
167
WWK, pp. 54f.
168
TLP 3.328.
169
NB, p. 105.
170
See NB, p. 37: “My fundamental thought is that the logical constants are not proxies.”
And also TLP 4.0312: “My fundamental idea is that the ‘logical constants’ are not repre-
sentatives”.
171
Landini, 2007, p. 79. This idea is related to the general interpretation of the Tractatus
which I follow in the present book—namely that Wittgenstein’s goal in the Tractatus is to
outline a logically adequate notation (a Begriffsschrift) where all the internal relations are
shown. Cf. §1 and note 6 therein.
56
between concept and object, or between unsaturated and saturated expressions.
The claim that there cannot be any name of a logical form can be taken as the
claim that no saturated expression can refer to something unsaturated since a
logical form is always unsaturated.172
We can even move on to identifying the
notions of a logical constant and of an internal property and relation.173
Neither external nor internal relations can be named in a logically adequate language.
The conclusion of this complex argument is that relations cannot be the
constituents of facts. They are not a part of the substance of the world and thus
are not real. Wittgenstein can be regarded as siding with Bradley. Relations are
unreal for Bradley in the sense that they are not real substances. They are not
independent entities; their truth (i.e., whether they are real) depends on something
else. They are, thus, not wholly true. For Wittgenstein, relations are unreal
in the sense that they are not part of the substance of the world.174
There is a long and as yet unresolved dispute concerning whether Tractarian
objects include relations. As noted above, I take the argument just presented as
evidence that relations cannot be Tractarian objects. All Tractarian objects are,
hence, exclusively particulars.175
7.2. All relations between states of affairs are external
One of the most important doctrines endorsed in the Tractatus is that elementary
propositions are logically independent of each other. At 5.134 Wittgenstein
writes: “One elementary proposition cannot be deduced from another.” The justification
for this is that there are no logical relations between two situations.176
172
See Potter, 2009, pp. 116f.
173
Cf. Landini, 2007, p. 85: “[T]he logical constants include all and only notions with logico-semantic
content.” If such notions express a property or a relation, these would be, of
course, internal.
174
Following Candlish (2007, p. 130) I use the term ‘unreal’ in order to indicate Wittgenstein’s
affinity to Bradley. Wittgenstein, however, does not employ this term. At TLP 2.06 he
defines reality as the “existence and non-existence of states of affairs”. If relations are not
parts of states of affairs, they can be conceived of as being unreal. The affinity of both authors
is further undermined by their different accounts of substance.
175
This line of thought is taken up in §8.2.
176
TLP 5.135.
57
Wittgenstein gives a more accurate expression of this independence at 2.062,
where he claims that the existence (or non-existence) of one state of affairs (i.e.,
an atomic fact) is independent from the existence (or non-existence) of another.
So given that one state of affairs exists, it is thinkable (conceivable) that another
state of affairs exists and it is also thinkable that it does not exist.
An internal relation relates terms that we cannot even conceive of not being related
by this very relation. If two things are internally related, it is unthinkable
that one of these things exists and the other does not. The existence of internally
related things is not independent. But states of affairs are independent in precisely
this way. Therefore, they cannot be internally related. There can be no
internal relations between states of affairs; to put it another way, all relations
between states of affairs are external.
Wittgenstein does not give any further justification for his Doctrine of External
Relations. As is well known, Wittgenstein abandoned this view in his 1929 paper
“Some Remarks on Logical Form”. However, several attempts have been
made by commentators to justify Wittgenstein’s commitment to this doctrine.
As we saw in §5.2, Russell and Moore were committed to this doctrine too.
Landini provides a detailed argument showing that Wittgenstein arrived at this
doctrine by the study of Russell’s “recursive correspondence theory [of truth]
according to which only ‘atomic’ facts (as it were) are truth-makers.”177
M.
McGinn argues that
the demand for the independence of elementary propositions arises out of Wittgenstein’s
conviction that propositional logic is the essence of all representation as
such.178
This is, of course, a conviction that Wittgenstein shared with Frege and Russell.
If logic and, in particular, logical inference are essentially about complex propositions—as
I shall argue below—then elementary propositions, which lack any
structure from the point of view of the propositional calculus, must be logically
independent. And finally, Michael Kremer argues that the logical independence
177
Landini, 2007, p. 52. For the whole argument, see pp. 40–53.
178
McGinn, 2006, p. 142.
58
of elementary propositions is related to Wittgenstein’s view that names stand
for simple objects.179
I do not disagree with these arguments. Wittgenstein was surely inspired by
Russell in this respect; and the logical independence of elementary propositions
is interconnected with other Tractarian insights as well. I shall follow Kremer
and M. McGinn and provide a text-immanent justification of this doctrine from
within the Tractatus. Suppose we have two states of affairs that are not logically
independent. Suppose furthermore that these two states of affairs are combinations
of the simple objects abc and cd. Then, however, one could ask what leads
us to think that ‘abc’ and ‘cd’ are elementary propositions. The dependence of
these two states of affairs has to stem from the combinatorial properties of their
components, i.e., from the forms of the objects they are composed of. For instance,
abc implies cd because the form of c is such that c must be combined
with d. If this were so, any analysis of a complex proposition into a truthfunction
of elementary propositions ‘abc’ and ‘cd’ would be flawed. The correct
analysis of the situation in question would be that it consists of a state of affairs
abcd, not of two states of affairs abc and cd. The logical relation (the implication,
for instance) between ‘abc’ and ‘cd’ has to be incorporated into the elementary
proposition ‘abcd’.180
The existence of two elementary propositions that are not logically independent
indicates that there is something wrong with our analysis. Either one or both are
complex propositions and they then need to be further analyzed into their elementary
components. Or their dependence lies in the forms of the objects they
describe and in such a case they are combined into a single elementary proposi-
tion.
179
“[A] name is a name of a simple, in virtue of which logical relations hold between propositions
involving it and propositions involving other names; ‘a’ names a complex just in
case propositions of the form ‘φ(a)’ imply propositions of the form ‘ψ(b)’ for some b’s (constituents
of a); ‘a’ names a simple just in case propositions of the form ‘φ(a)’ do not imply
propositions of the form ‘ψ(b)’ for any b’s. Thus the mutual independence of elementary
propositions is a consequence of Wittgenstein’s conception of a simple name.” (Kremer,
1997, p. 98)
180
“An elementary proposition really contains all logical operations in itself.” (TLP 5.47)
59
The conclusion to this is that presumptive elementary propositions that are not
logically independent are not elementary propositions at all. The justification
for the Doctrine of External Relations between states of affairs or elementary
propositions is, in my view, terminological. Elementary propositions and states
of affairs, which are asserted by elementary propositions, are externally related,
because these notions are so defined.
7.3. Logical relations are internal
We now proceed to the relation of logical entailment and logical necessity in
general. In claiming that all relations are external, Moore makes one important
exception: the one relation that he takes to be internal is the previously mentioned
whole-to-part relation.181
This internal relation is crucial to the method of
philosophical analysis which aims to explain a complex whole in terms of its
parts. Since (due to the Doctrine of External Relations) a complex is no more
than a concatenation of its parts, nothing gets lost in the process of analysis.
Thus, just as Wittgenstein allows for the elimination of the relation R in 3.1432,
the sign for the complex which relates its parts together may be eliminated here.
I now want to argue for something which Moore merely suggests, which Russell
more or less endorsed in the Theory of Knowledge manuscript, but which is
only fully developed in the Tractatus. The main idea is that (propositional) logic
is concerned with molecular propositions. More specifically, logic is concerned
with the forms of molecular propositions and the relations between these forms
which are, of course, internal. Although Moore has not really succeeded in explaining
the nature of logical necessity,182
I believe this explanation would have
been available to him too.183
First, Wittgenstein conceived of the method of analysis as described above. At
TLP 2.0201 he writes that a statement about a complex can be decomposed into
a statement about its parts together with a proposition describing the complex.
181
See Moore, 1919, p. 51; cf. Hylton, 1990, p. 143; Proops, 2002, p. 304.
182
See Hylton, 1990, pp. 143–152.
183
Russell expressed this idea in the “Introduction” to the Tractatus: “Thus the whole business
of logical inference is concerned with propositions which are not atomic. Such propositions
may be called molecular.” (TLP, Introduction)
60
We must not confuse the original unanalyzed proposition about the complex
with the proposition describing the form of the complex which comes out in the
process of analysis. Wittgenstein says at TLP 3.24 that “A proposition about a
complex stands in an internal relation to a proposition about a constituent of the
complex.” The proposition about a complex is, of course, the original unanalyzed
proposition. This proposition stands in an internal relation to an elementary
proposition that it contains. This internal relation, which holds among the
parts of a complex, is the proposition that describes the form of the way the
parts are related together. The relation between a complex and its parts is, therefore,
the internal relation that (propositional) logic is concerned with. Logic is
concerned with the forms of propositions. In an adequate logical notation, the
sign for the complex, which aims at describing the form of the complex, should
be resolved or eliminated. The form of a molecular proposition should then be
clear from the proposition itself.
At TLP 5.47 Wittgenstein introduces his slightly complicated reasoning about
molecular propositions: the only way to construct molecular propositions is by
applying a function to an argument. This, however, involves all of the logical
constants.184
This is what all propositions have in common with each other and
this is the general form of the proposition. The general form of the proposition
is, therefore, the form of the construction of a (molecular) proposition. The general
form of the proposition is the only general primitive sign that logic deals
with.
To put all of these considerations together now: logic deals with the general
form of a proposition which expresses, in effect, a form of how logical operations
are applied to elementary propositions. This means that logical space (i.e.,
a space of all possible propositions) is delimited by the general form of the
proposition. Although there are no internal relations between elementary propositions,
there could be molecular propositions that share the same form or the
form of one proposition could be a part of the form of another.185
Logical rela-
184
I understand here that Wittgenstein means that logical constants stand for logical operations
such as conjunction, negation, or the Sheffer stroke.
185
TLP 5.122. Consider, for example, the following inference: (p|q)|(p|q), p|p ⊨ q (see TLP
5.1311). The truth-grounds of the consequent are, in a definite way, contained among the
truth-grounds of the antecedent.
61
tions are, in fact, instances of the whole-to-part relation.186
Thus, there could be
internal relations that hold between molecular propositions. Necessary logical
relations like entailment can be explained internally in just this way. The consequence
of this is that logical relations can be read off the forms of propositions
alone. Rules of inference are thus rendered superfluous.187

To conclude, Wittgenstein held in his early writings, like Moore and Russell,
that all relations between elementary propositions are external. Following
Moore and Russell, he took internal relations to be inexpressible by means of a
proposition; but he sided with Bradley in conceiving of all relations, internal as
well as external, as being unreal. Wittgenstein’s conclusion was that relations
should be built into our logical notation where they express or show themselves.
Wittgenstein claimed he had resolved the question of whether all relations
are internal or external.188
The answer is that all the relations that can be
expressed in a proposition are indeed external, and internal relations can be
shown in a logically adequate notation. It is, perhaps, no accident that Wittgenstein
first came up with his fundamental saying/showing distinction in the Notes
dictated to G. E. Moore where the internal/external distinction’s earliest occurrence
is also to be found.
186
Cf. McManus (2006, p. 146): “One might say that logical relations are ‘internal’, in that
the deduced proposition is, in a sense, part of the proposition from which it is deduced.”
187
Cf. TLP 5.13–5.133.
188
TLP 4.1251.
62
8. The nature of simple objects
Russell and Moore denied any internal relations between both simple objects
and atomic facts. For Wittgenstein, however, objects are not independent in the
sense that only189
certain combinations of objects are possible in a state of af-
fairs.190
These possibilities and impossibilities lie in the objects themselves; and
Wittgenstein calls this the form of independence and the form of dependence.
This form is a part of the nature of an object.191
To know an object, one must
know all of its internal properties.192
The internal properties which make up the
form of an object are always relational properties, because they involve the possibility
of being combined with other objects. The relation of being actually
combined between two objects is an external one because it is thinkable that
these two objects might not actually be combined. However, the opposite relation
of not actually being combined might be an internal one if there were two
objects that cannot be combined in a state of affairs. The conclusion of this argument
is that Wittgenstein (unlike Russell and Moore) allows for internal relations
between simple objects.193
Wittgenstein uses the metaphor of a chain to clarify this sort of dependence: “In
a state of affairs objects fit into one another like the links of a chain.”194
How do
we actually understand this metaphor? Objects are connected without any connecting
relations—without any glue—as we know from the previous chapter.
The links of a chain are connected without any further elements. Objects hang
together in virtue of their formal (i.e., internal) properties. But still, it is not
clear how these properties can be conceived. A natural suggestion might be that
189
That certain combinations of objects are impossible follows from Wittgenstein’s examples
given at 2.0131. Certain objects (specks in the visual field) must have some color; other
objects (notes) must have some pitch and so on. It is thus impossible that these objects stand
alone in states of affairs.
190
Cf. TLP 2.012ff.
191
See TLP 2.0123.
192
See TLP 2.01231.
193
Schaffer (2010) provides a complex argument that “failure of free recombination is the
modal signature of an integrated monistic cosmos” (pp. 342 & 355). If his argument is
sound and if we take the Tractarian objects metaphysically as the ultimate constituents of the
world, then metaphysical monism follows from the Tractarian ontology.
194
TLP 2.03.
63
Tractarian objects are like Democritean atoms which have tiny hooks and barbs.
Accordingly, the internal properties of the Tractarian objects would be based on,
or derived from, their shapes. Some objects would fit together because their
shapes are compatible and some would not. However, this view faces a serious
objection: why should we take shapes to be internal properties and not exter-
nal?195
Saying that simple objects have hooks and barbs implies that they have
spatial parts, which casts doubt upon the very idea of a simple object. Then,
however, we are forced to take this explanation metaphorically. Simple objects
have hooks in some metaphorical sense. But what is this sense? The metaphor
of the chain has, then, merely been replaced by the metaphor of having hooks
and barbs.
In what follows I would like to provide two different answers to these questions.
According to the so-called metaphysical view, internal relations between
simple objects lie in the objects themselves. They are de re necessary and our
language (or a logically adequate language) should mirror these necessities. On
the other hand, according to the so-called anti-metaphysical or resolute view,
internal relations between simple objects emerge during logical analysis. They
hold in virtue of language and are thus de dicto.
8.1. The de re view
This account of the internal properties of simple objects has been favored by
adherents of the so-called traditional or metaphysical interpretation of the Tractatus.
So Peter Hacker says: “The logical (metaphysical) forms of states of affairs
are language independent—de re possibilities do not depend upon our descriptions
of them.”196
The most vigorous advocate of this view is— in my view
—Raymond Bradley197
in his book The Nature of All Being (1992). He interpreted
Wittgenstein’s early philosophy as advancing a robust Leibnizian ontology
of possible worlds with S5 modal logic. Here is the most explicit statement
195
Cf. Candlish & Basile, 2013: “[I]t is all but clear that Wittgenstein’s logico-ontological
atoms can be said to possess a form; surely, they differ from Democritean atoms in that they
lack material properties (cf. 2.0331 and 2.0232).”
196
Hacker, 2001, p. 171.
197
Not to be confused with Francis Bradley, whose views were discussed in previous chap-
ters.
64
of Bradley’s view concerning simple objects: “the logical necessity which binds
an internal property to its possessor is de re necessity.”198
This idea is further
developed:
[T]he formal property of combinatorial potential […] is “essential” to (2.011), “part
of the nature of” (2.0123(2)), or “internal” to (2.01231) its possessors. Hence it is a
de re necessary modal property of all metaphysically simple objects.199
What does it mean exactly that internal properties of simple objects are de re
modalities? We may follow Hacker in saying that these modalities are language
independent. More precisely, the formal properties of combinatorial potential or
powers of metaphysically simple objects are not derived from the combinatorial
powers of names which correspond to these objects. It is in fact the other way
around. The internal properties of names mirror the internal properties of simple
objects.
The main motivation for this approach is to be found in the following remark
from the Notes Dictated to G. E. Moore:
Thus a language which can express everything mirrors certain properties of the world
by these properties which it must have; and logical so-called propositions shew in a
systematic way those properties.200
This claim is not just restricted to the internal properties of simple objects. The
wording suggests, rather, that the claim is about the logical forms of states of
affairs and the facts which determine the logical forms of logical propositions.
This reading is further vindicated in the Tractatus, 4.121: “Propositions cannot
represent logical form: it is mirrored in them. […] Propositions show the logical
form of reality.” I think, however, that these claims imply that the internal properties
of names201
mirror the internal properties of simple objects. The essential,
i.e., the internal properties of simple objects are their combinational powers
within possible states of affairs.202
It is obvious that the primary entities are
198
Bradley, 1992, p. 81.
199
Bradley, 1992, p. 84; see also p. 24.
200
NB, p. 108.
201
I take visual and audial properties of names to be external.
202
See TLP 2.011.
65
states of affairs and that their constituents—simple objects—are derivative. The
nature of the pictorial relationship now indicates that if logical propositions
mirror the logical forms of facts, then the internal properties of names mirror
the internal properties of simple objects.
Bradley finds three different meanings or rather “shifting use[s]” for the term
‘object’ in the Tractatus. Wittgenstein introduces his shifting uses as follows:
“Here the shifting use of the word ‘object’ corresponds to the shifting use of the
words ‘property’ and ‘relation’.”203
Bradley interprets this remark as concerning
the respects in which an object is simple. In his reading, objects can be said to
be simple in a metaphysical, semantic, and epistemic sense.204
Objects are metaphysically
simple if they are the ultimate constituents of the world. Wittgenstein
says, for instance, that “we feel that the WORLD must consist of ele-
ments.”205
An object is semantically simple if “[i]ts composition becomes completely
indifferent”206
during logical analysis. The structure of the object does
not affect any logical entailment. Here is Wittgenstein’s example:
from “All men are mortal” and “Socrates is a man” there follows […] “Socrates is
mortal” which is obviously correct although I equally obviously do not know what
structure is possessed by the thing Socrates or the property of mortality. Here they
just function as simple objects.207
Obviously, the structure of an object may sometimes affect logical entailment
and sometimes it does not need to. The same object may sometimes function as
a simple and sometimes as a complex. The decisive factor here is how the object
is described. Then, however, this notion of simplicity is de dicto.
According to the epistemic interpretation, simple objects are phenomenological
items like points of a visual field. They are akin to Russell’s objects of acquaintance.
There is some textual evidence for this. Wittgenstein says, for example,
that it seems to be possible that “patches in our visual field are simple
203
TLP 4.123. The context of this remark is discussed in §6.2.
204
It is far from clear whether the shifting use concerns the simplicity of objects. See the
next section for another suggestion.
205
NB, p. 62.
206
NB, p. 69.
207
NB, p. 69.
66
objects”208
or “[w]hen we see that our visual field is complex we also see that it
consists of simpler parts.”209
The crucial question now is whether the epistemic
notion of a simple object is more fundamental than the metaphysical or semantic
one. If so, the decision concerning whether something is a simple object or a
complex would be independent of their descriptions. This means, however, that
the epistemic notion of simplicity is de re. I do not want to address all the arguments
for the epistemic interpretation. I would like, rather, to point out that
all the textual evidence allows that although some visual patches may be simples,
other things may be simples too. This implies that the fundamental criterion
of simplicity may be metaphysical or semantic. A visual patch, or more generally,
an object of acquaintance is a simple if it is metaphysically simple or if it
is referred to by a name.
Therefore, in conclusion, the textual evidence for the epistemic interpretation is
very weak. The metaphysical interpretation, which renders the notion of simplicity
to be de re, is more substantial. It is based, however, on remarks from the
Notebooks. The most important expression of the metaphysical view comes
from the early Notes which express a rather immature version of Wittgenstein’s
later views. Moreover, there are yet more arguments that disadvantage the metaphysical
interpretation relative to the semantic one.
In the Tractatus, Wittgenstein says that the “only necessity that exists is logical
necessity.”210
If there were any de re modalities, as the metaphysical interpretation
suggests, logical necessity would include de re modalities too. Bradley is,
then, forced to the claim that “the logical necessity which binds an internal
property to its possessor is de re necessity.”211
But, as argued in §7.3, the main
idea of the Tractarian logic is that propositional logic is concerned with molecular
propositions. This main idea is in apparent tension or, rather, conflict with
the idea that the de re possibilities of the combination of simple objects are logical
possibilities. The fundamental idea of the Tractatus is that logical constants
do not represent.212
This idea can be taken as saying that there are no logical
208
NB, p. 64.
209
NB, p. 65.
210
TLP 6.37.
211
Bradley, 1992, p. 81.
212
Cf. fn. 170, p. 42.
67
properties.213
Bradley’s account, thus, is incompatible with the fundamental
Tractarian idea.
The metaphysical interpretation of simplicity issues in the de re account of the
internal properties of simple objects. Besides the de re internal properties there
are de dicto internal properties of complexes. In his later philosophy, Wittgenstein
conceives of internal relations as being strictly de dicto. Moreover, he
makes this insight into one of his main tools for the analysis of language. This
metaphysical interpretation forces us to see a deep discontinuity in Wittgenstein’s
thinking. This puts it at a disadvantage in the light of the fact that there is
another interpretation available that conceives internal properties of simple objects
as de dicto. In the next section I shall provide evidence in favor of the semantic
interpretation and the de dicto notion of simplicity.214
8.2. The de dicto view
The most convincing evidence for the de dicto account of simples is to be found
in the Notebooks entries. Here Wittgenstein writes:
we do not infer the existence of simple objects from the existence of particular simple
objects, but rather know them—by description, as it were—as the end-product of
analysis, by means of a process that leads to them.215
Simple objects are the references of names after some logical analysis has been
carried out. What counts as a simple object is thus determined by the process of
logical analysis. Such an analysis does not need to be completed in order to result
in a logically adequate language. The metaphysical view tends to see simple
objects as references of names in a logically adequate language (and a language
is logically adequate if it mirrors, inter alia, the internal properties of
metaphysically simple objects). The semantic or de dicto view sees the notion
213
Cf. Landini (2007, p. 281): “Wittgenstein’s objects (logical atoms) have no logically essential
properties at all. Indeed, the Tractarian Grundgedanke states that there are no logical
properties and relations.”
214
Even Bradley (1992, p. 71) admits hesitantly that the semantic sense is more fundamental
than the metaphysical one.
215
NB, p. 50.
68
of simplicity relative to the actual process of logical analysis. To be simple is
not a property of an entity; it is, rather, its function. Some objects can function
as simples in the process of logical analysis: “It functions as a simple object.
(What does that mean?)”216
The criterion of simplicity was mentioned earlier:
“Its composition becomes completely indifferent. It disappears from view.”217
Hence, if the composition of a thing is not important at a certain stage of logical
analysis, this thing only functions as a simple object at that stage. Thus, metaphysically
complex objects, such as for instance a book or a man, can fulfill the
role of a simple object.
Wittgenstein gave another criterion for the simplicity of the Tractarian objects
in 1930:
What I once called ‘objects’, simples, were simply what I could refer to without running
the risk of their possible non-existence; i.e. that for which there is neither existence
nor non-existence, and that means: what we can speak about no matter what
may be the case.218
Objects are those parts of states of affairs that cannot but exist. Note that this
specification of objects is de dicto. It says that an object is anything that fulfills
a certain condition, viz. the condition that it necessarily exists.
Both the specifications of objects, i.e., that their composition is indifferent and
that they cannot cease to exist, are equivalent if non-existence of a thing means
its decomposition and separation into parts.
Even if its composition is indifferent, such an object nevertheless has internal
properties (an internal nature) which are its powers of combining with other objects
in a state of affairs. These properties are derived from the combinational
powers of the name that refers to this object. Here is Wittgenstein’s own exam-
ple:
This object is simple for me!
216
NB, p. 69.
217
NB, p. 69.
218
PR, p. 72.
69
If, e.g., I call some rod “A”, and a ball “B”, I can say that A is leaning against the
wall, but not B. Here the internal nature of A and B comes into view.219
The objects A and B function as simples here while the relation of leaning
against and the wall may be simple or complex. The internal natures of A and B
are such that A can stand in the relation of leaning against to the wall whereas
B cannot stand in that relation. Their internal properties are different with respect
to that relation. Obviously, they also share certain internal properties, e.g.,
the combinational power to stand in the relation of being near the wall. It may
now be argued220
that the internal nature of simple objects is nevertheless
grounded in the de re properties of these objects. B cannot be leaning against
the wall, because of its round or symmetrical shape. But this is again a de dicto
relationship between two concepts. Round objects in general cannot lean. This
is due to the relationship between roundness and the relation of leaning against.
The notion of simplicity is relative to a certain process of analysis and this is—
in my view—the true explanation for the shifting conception of simplicity as
mentioned above. What is (treated as) a simple on one occasion may appear to
be a complex on another occasion. This is tantamount to the claim that the
composition of an object is sometimes indifferent and sometimes not. If the
composition matters, then a description of such a complex has to be eliminated
in favor of names referring to simple objects. Wittgenstein supplements his fundamental
thought about these simple objects with the following claim: “even if
the name ‘N’ vanishes on further analysis, still it indicates a single common
thing [Ein Gemeinsames].”221
‘Gemeinsames’ can also be translated as ‘conjoint’
or even ‘combined’. ‘N’ would refer to a complex. In the same vein, the
sentence is ambiguous because ‘Ein Gemeinsames’ may apply to a thing (as the
English translation suggests); it might, however, also apply to the name itself.
On this reading, ‘N’ is then a name of multiple things; it is an indefinite description
in Russell’s sense.

219
NB, p. 70.
220
See Bradley, 1992, p. 89.
221
NB, p. 60.
70
I argued in §7.1 that the signs for relations should be eliminated in favor of internal
relations within the signifying state of affairs in a logically adequate language.
Now we see that signs for complexes should be eliminated too. All that
remains after the final analysis are signs for simple objects, i.e., their names.
Wittgenstein says, however, in several places that relations and predicates are
objects too.222
How does the account given here mesh with these claims?
My line of answer lies in the following consideration: we have to distinguish
relations (and predicates) both before analysis and after analysis. At 3.1432
Wittgenstein considers the complex expression ‘aRb’ and explicitly takes R to
be a relation. This is, however, a sign for a relation in the language to be analyzed.
But why is R a relation at all? What criterion is available to Wittgenstein
here? In general, R is a relation because of the logico-syntactical grammar of
this language (no semantics is involved here). As this language is not yet analyzed,
logical categories of linguistic entities have to be determined a priori.
This means that at this stage we can only employ preconceived criteria of categorization.
What a relation is is characterized essentially as its role in the combinational
system of entities. We can employ here a Fregean criterion (that relations
are unsaturated) or a Russellian criterion (that relations unify complex-
es).223
On both accounts, a sign for a relation is a complex sign that has to be eliminated
in favor of internal relations within a signifying fact. Wittgenstein says now
that even “if the analysis were completely carried out, its result would have to
be a proposition which once more contained names, relations, etc.”224
Relations
are, hence, among the constituents of elementary propositions in a completely
analyzed language. Because the constituents of elementary propositions are
names, some names express relations (and of course, some do not). There are,
as it were, at least two logical kinds of names. And since names act as proxies
for simple objects, there are also at least two logical kinds of simple objects—
particulars and universals—in the logically adequate language.
222
There is a notebook entry from 1915 “Relations and properties, etc. are objects too.”
(NB, p. 61) and a report by Desmond Lee from 1930 or 1931 (LWL, p. 120).
223
Cf. Johnston, 2009, p. 150.
224
NB, p. 61.
71
The key question at this point is what resources we have to distinguish between
these two classes of objects. We can rely only on the essential, which means the
internal, properties of these objects. We can distinguish between classes of objects
according to their combinational powers. Let me explain this with one example.
Suppose we have a class I of objects that cannot be combined with each
other:
I ≡ {a, b, c, d, …}(16)
Accordingly, the combinations of these objects like ab, ac, bd are not admissible
states of affairs. But there are admissible states of affairs consisting only of
a single object from I. However, let us suppose there is another class P of objects
that cannot be combined with each other but can nevertheless be combined
with objects from I.
P ≡ {p, q, r, s, …}(17)
Now, therefore, combinations like pq, qs are not admissible. There are, however,
admissible states of affairs such as pb, qa, sd. Finally, let us suppose that a
class R of objects can be combined only with two or more objects from I.
R ≡ {α, β, γ, δ, …}(18)
Inadmissible combinations are now, for example, αβ, αp, αd, while admissible
combinations can be αcd, βdb, γbcd, δabcd. Let me use bracketing to make this
logically adequate language more surveyable, which is not vital for its logical
function. Admissible states of affairs are now: a, c, p(b), q(a), s(d), α(c,d),
β(d,b), γ(b,c,d), δ(a,b,c,d).
I have only divided these simple objects into several classes according to their
combinational properties. Based on these internal properties, we can treat objects
from I as individuals, objects from P as predicates and objects from R as
relations.225
We can thus define what counts as an individual, a predicate (a
property, a one-place relation) or a relation. An individual is an object such that
it can make up a state of affairs by itself. A predicate is an object such that it can
be combined with one other object in a state of affairs. A relation is an object
225
The structure of classes might be more fine-grained. We might distinguish among relations
with different multiplicities or introduce higher-order predicates and relations.
72
such that it has to be combined with two or more other objects. Although some
of these objects are predicates or relations, these objects are nevertheless still
simple.
To conclude, the account given here is able to mesh with the following two
claims by Wittgenstein: that relations have to be eliminated in the course of logical
analysis, and that there can still be relations in fully analyzed sentences
(i.e., there are relations among simple objects). What a relation in a language to
be analyzed is has to be determined a priori. Such relations, then, are complex
signs that have to be eliminated in favor of internal relations within states of
affairs. What a relation in an analyzed language is is thus determined a posteriori
by internal properties of simple objects.226
226
An a posteriori account of logical categories is advanced by Johnston (2009) who attributes
this account back to Ramsey. Johnston argues, however, that the idea of the elimination
of relations (based on TLP 3.1432) is an anachronism in the Tractatus. I hope to show that
the idea of the elimination of relations can be reconciled with the claim that there are relations
among simple objects in a wholly analyzed language.
73
9. The picture theory
Wittgenstein famously writes in the Tractatus:
A gramophone record, the musical idea, the written notes, and the sound waves, all
stand to one another in the same internal relation of depicting that holds between
language and the world.227
It is obvious that the comparison between language and pictures lies at the heart
of the Tractatus. What is less obvious, though, is what the aim of this comparison
is. What is the point of involving pictures in the Tractatus? Does Wittgenstein
advance a picture theory of representation or meaning, or does he introduce
a mere analogy between pictures and language? Resolute readers of the
Tractatus tend to follow Wittgenstein’s later retrospective comments and conceive
of the involvement of pictures as a mere analogy.228
Other scholars, however,
ascribe to Wittgenstein some substantial claims about the essence of the
proposition (and by implication of language and the world) “which depicts the
facts that it describes”229
.
If internal relations occurred only in the context of a misunderstanding230
and if
they served only to overcome metaphysical claims about language and its relation
to the world and thereby play only a purely transitional role, then it would
be a mistake to ascribe to Wittgenstein the picture theory of meaning. In short:
if internal relations do not possess any positive role, then neither does the picture
theory, because it is articulated essentially in terms of an internal relation.
So, for example, McManus says that “[a]scribing a ‘picture theory’ of representation
to the Tractatus is […] a mistake anyway”231
. Diamond similarly says:
the so-called ‘picture-theory’ of the Tractatus is not merely something that fails to
fulfill some supposed narrow Tractatus criterion of sensefulness. The picture-theory
is a story about the relation between propositions and reality; it dissolves from within,
since we cannot attempt to grasp what it supposedly conveys without using the re-
227
TLP, 4.014, my italics.
228
Cf. Kuusela, 2011a, pp. 614f. for a discussion.
229
TLP, 4.016.
230
McManus, 2006, Ch. 7.3; cf. §3 above.
231
McManus, 2006, p. 65.
74
lational forms of our language, but what we are being shown is that, used in the way
we try to use them, they are empty.232
How does such an elucidation come about? Resolute readers take the picture
theory as setting a limitation233
on the combinational possibilities of names and
elementary propositions, i.e., on their internal properties and relations. Let us
restrict ourselves for now to names. Names are, supposedly, independent of
each other in the sense that no combination is excluded. The picture theory sets
a limitation which separates logical combinations of names from illogical ones.
The logical form of the world determines that some combinations of names are
logical and permissible while others are illogical and impermissible. But this
view is, according to the resolute reading, mistaken. We cannot consistently say
that there are any impossible combinations, but this is exactly what the idea of
the internal relation of depicting between language and the world is trying to
maintain. The role of the picture analogy is different. It helps us to see that according
to the context principle, we failed to give meaning to some of the elements
of the proposition in question.
Thus, the picture analogy helps us to see that ‘illogical combinations’ […] are problematic
just because we haven’t given those combinations any sense yet, any role to
play in our systems of representation.234
This is fair point, but this does not imply that the picture theory is a mistake. An
occurrence of ‘illogical combinations’ of signs indicates that the symbols they
stand for do not have the same combinatorial powers as the objects they repre-
sent.235
The picture theory says, inter alia, that the combinatorial powers of the
corresponding elements must be the same. Obviously, we can arbitrarily combine
signs as orthographical units. The point of introducing a logical syntax is
so that if we know what a sign symbolizes, i.e., if we treat signs as symbols, il-
232
Diamond, 2002, p. 273.
233
A limitation as opposed to a limit.
234
McManus, 2006, p. 70.
235
If we take names as signs regardless of their symbolizing role, we can combine them in
any way. Signs are mere orthographic units. But even thinking like this is misleading. Treating
something as a name amounts to conceiving of a sign as belonging to a logical category
of names. Hence, we are already considering its symbolizing role—viz. names symbolize
simples.
75
logical combinations are then excluded or, rather, cannot occur.236
This means
that illogical combinations of symbols cannot occur in a logically adequate language.
The situation is, however, different in a (possibly logically inadequate)
language that is to be analyzed. The point of introducing the picture theory of
representation and hence the internal relation of depicting is, on my understanding,
to improve the analyzed language in the direction of a logically adequate
language. Properly speaking, the combinational powers of objects and names
should thus be harmonized. In this sense, the internal relation of depicting then
serves as a practical command, as an imperative.
9.1. Levels of the pictorial relationship
The pictorial relationship has several distinct levels. Wittgenstein begins with
the lowest level of names and objects: “A name means an object.”237
The next
level consists of elementary propositions and states of affairs: “The simplest
kind of proposition, an elementary proposition, asserts the existence of a state
of affairs.”238
Possibly complex propositions, in turn, are pictures of reality.239
On the uppermost level, there is the pictorial internal relation between language
and the world.
The levels of each side of the pictorial relationship are bound by whole-to-part
relations, which are, of course, internal. Here is the language side: “An elementary
proposition consists of names.”240
“A [possibly complex] proposition is a
truth-function of elementary propositions.”241
Language is the totality of all
propositions.242
The world side is as follows: “A state of affairs (a state of
things) is a combination of objects (things).”243
“What is the case—a fact—is
the existence of states of affairs.”244
Reality, a fact, is “the existence and non-
236
Cf. TLP 3.334.
237
TLP 3.202.
238
TLP 4.21.
239
Cf. TLP 4.01.
240
TLP 4.22.
241
TLP 5.
242
TLP 4.001.
243
TLP 2.01.
244
TLP 2.
76
existence of states of affairs”245
. And finally, “[t]he world is the totality of
facts”246
. Not all these levels are of the same modal status. Names are internally
related to actual objects. Wittgenstein does not distinguish actual and possible
objects. Objects are those entities that we may refer to without any risk of their
possible non-existence.247
Hence, there are no possible objects or names that
can possibly refer to objects, i.e., no names that can be empty. On the other
hand, elementary propositions depict possible states of affairs and complex
propositions depict possible facts. There is a slight terminological problem with
the uppermost level. Language should depict the totality of all possible states of
affairs. But Wittgenstein defines the world as the “totality of existing states of
affairs”248
. ‘Existing’ here means ‘actual’. We should thus say rather that language
depicts an imagined world [gedachte Welt],249
for language can contain
propositions that are not actually true, i.e., that depict a non-obtaining fact.
This hierarchy begs the question: which level is the most fundamental? Complex
propositions are functionally dependent on elementary propositions, and
therefore the level of complex propositions and their corresponding facts cannot
be fundamental. The world is the mere totality of all mutually independent
states of affairs; hence this level is also not fundamental. Only the level of
names and objects or the level of elementary propositions and states of affairs
can be considered to be fundamental. And indeed, we find both of these suggestions
among commentators. Those commentators who tend towards the de re
account of simple objects, as discussed above,250
would consider the object level
as the most fundamental one, whereas advocates of the de dicto account
would prefer the level of elementary propositions and states of affairs.
The view that the level of names and objects is the most fundamental one finds
some support in Wittgenstein’s 1915 Notebooks: “And if the general description
of the world is like a stencil of the world, the names pin it to the world so that
245
TLP 2.06.
246
TLP 1.1.
247
Cf. PR, p. 72 quoted in §8.2 above.
248
TLP 2.04, my italics.
249
Cf. TLP 2.022.
250
See §8.1.
77
the world is wholly covered by it.”251
Accordingly, Raymond Bradley claims
that:
The logically noncontingent features [of states of affairs] are functions of the internal
(de re necessary, formal) properties of simple objects (how they can combine in possible
states of affairs). The logically contingent features are functions of their external
properties (how they do combine in possible states of affairs).252
In short, the internal properties of simple objects and their correlation with
names determine the whole picturing relation. However, this view faces several
objections. First, this account is inconsistent with the context principle which
Wittgenstein endorses.253
Names have meaning only in the context of a proposition.
But if the level of names and objects were fundamental, i.e., if their connection
determined the picturing relation, then names would possess meaning
prior to their involvement in propositions. A name’s meaning is the depicted ob-
ject.254
The second objection is that it is wholly mysterious how a connection
between a name and an object can be established. There would have to be some
kind of mechanism that injects meaning into signs and makes them into names.
One might think here of something like baptism, or of a mental, though not
psychological, mechanism as Hacker states:
That such configurations, in thought or language, actually represent […] is a function
of the will, of the metaphysical self […]. It is a mental act (albeit of a transcendental
self, not of the self that is studied by psychology) that injects meaning or significance
into signs, whether in thought or in language.255
251
NB, p. 53.
252
Bradley, 1992, p. 130. Peter Hacker expresses a similar view. Cf. McGinn, 2006, p. 89.
253
TLP 3.3. There is, however, another remark in the Tractatus that advocates of the namesobject
level fundamentality may invoke to support their case: “It [a proposition] is understood
by anyone who understands its constituents.” (4.024) There seems to be a tension
among these remarks. I take 4.024 as expressing a sufficient condition of understanding a
proposition without touching upon the issue of fundamentality (unlike the context principle
at 3.3).
254
TLP 3.203: “A name means an object. The object is its meaning.”
255
Hacker, 1986, p. 75.
78
The point is that if the connection between a name and an object is supposed to
be an internal relation256
, this connection cannot depend on any fact that results
from this kind of injecting act. If there is the question of how the connection
between a name and an object is established, then it is thinkable that they are
not connected. If it is thinkable that two entities are not related, the relation in
question is thus external. The upshot is, then, that if the level of names and objects
were taken as fundamental, the picturing relationship would be an external
relation.

As already said, those who prefer the de dicto account of simple objects would
tend to prioritize the level of elementary propositions and states of affairs. But
the de dicto view derives our knowledge of objects from our knowledge of
names. And furthermore, names have meaning only in the context of a proposition.
This implies that the preference for the level of elementary propositions
and states of affairs leads to a further preference for the side of elementary
propositions. Hence, the whole picturing relationship would be determined once
we knew what the elementary propositions signify.
The picturing relation obtains between the propositional signs and possible
states of affairs or, more generally, possible situations.257
A propositional sign is
projected onto a possible situation. Wittgenstein speaks of the method of projection
in this context.258
The whole consisting of a propositional sign and the
method of projection both belong to a proposition.259
A propositional sign
(which is an existent state of affairs or even a complex situation) depicts a possible
situation. Hence, the picturing relation obtains between two situations.
256
We are speaking of symbols here. Symbols are signs with sense. The relation between
name and object is internal, because for a given name (taken as a symbol) it is unthinkable
that it would have some other sense. The connection between a sign and its sense is arbitrary
and thus external.
257
Wittgenstein sometimes speaks of propositions without distinguishing whether they are
elementary or complex. Similarly, he sometimes speaks of situations without distinguishing
whether they are elementary states of affairs or complex facts. We can merge the levels of
elementary and complex propositions, because the picturing function of complex propositions
is fully determined by the picturing function of atomic propositions.
258
TLP 3.11.
259
TLP 3.13.
79
There is no mysterious gap between language and the world. In short, one situation
depicts another situation.
Let us look at Wittgenstein’s analogy of gramophone records, musical notes,
and waves of sound. The method of projection amounts to this: there is a certain
general rule for transforming written signs into waves of sound and another rule
for transforming grooves on a gramophone record into a musical score and so
on. These laws of projection are, Wittgenstein advises us, like the rules of translation
from one language into another. They belong to thoughts, which are
propositions with sense, in other words, these laws are essentially constitutive
of symbols. Consequently, a musical score does not show the slightest similarity
with waves of sound if we do not understand it. And to understand a score
amounts to knowing or possessing an appropriate method of projection. But we
can derive the resulting waves of sound from the musical score by means of the
projection method.260
This derivation is akin to a mathematical operation. If we
possess the score and the method of projection, we possess or, rather, we could
imagine the waves of sound (and there are many different methods of projection—i.e.,
many different ways of interpreting a score). This shows that the relation
of depicting is an internal relation.261
260
Wittgenstein gives another example of such a projection: “In order to understand the essential
nature of a proposition, we should consider hieroglyphic script, which depicts the
facts that it describes.” (TLP 4.016) This explanation suggests that the projection is something
natural. From the visual characteristics of a hieroglyphic sign for, e.g., a house, we
could determine that the sign represents a house without any prior knowledge. This is, however,
not how Egyptian hieroglyphs represent. Most hieroglyphic signs are (like English
letters) phonetic in nature, meaning that the sign is read independently of its visual characteristics.
The phonological value of a sign is derived from the name of the represented thing
by means of the so-called rebus principle according to which the representation of, for example,
a duck ‘son’ *ziɜ can stand for ‘hard ground’ *sɜt.w due to their similar phonetic
structures (analogously, the numeral ‘2’ can stand for the preposition ‘to’ in English). Cf.
Loprieno, 1995, pp. 12f. We can, thus, take the rebus principle as a peculiar method of pro-
jection.
261
My present interpretation follows the main features of Marie McGinn’s antimetaphysical
reading of the Tractatus. Cf. “The link between the items [the musical score,
the musical idea, the music, and the gramophone record] is internal insofar as it is made, in
each case, via a rule of projection that enables us, given any one of them, to construct the
others from it.” (McGinn, 2006, p. 80)
80
What, then, is constitutive of a possible (meaning: possibly non-existent in reality)
state of affairs that is represented by a proposition? All we need is this same
proposition. “In a proposition a situation is, as it were, constructed by way of
experiment [probeweise zusammengestellt].”262
All thinkable facts are, thus,
constructed from propositions. But if the logic of propositions “deals with every
possibility and all possibilities are its fact”263
, all possible facts are thus constructed
by propositions. What is thinkable is possible and what is possible is
thinkable. 264
There are no possible facts that are not thinkable, i.e., that are not
constructed by propositions. Possible facts are in a certain sense contained in
propositions. Hence, there are no possible facts outside propositions. It is clear
now why the relation of depicting is internal. A represented fact is nothing but a
projected propositional sign.
Two problems or questions emerge at this point: The first is how to be sure that
what can be represented by a proposition is also possible in reality. The second
problem is how a represented fact attains reality. The first problem is this: if the
level of elementary propositions is fundamental, how do we harmonize the level
of names and objects? More precisely, how do we harmonize the combinatorial
powers of names and objects? But it boils down to the question of how to analyze
the elementary propositions (after we have analyzed complex propositions
as truth-functions of logically independent elementary propositions). As discussed
in §7.2, if names correspond to simple objects, elementary propositions
must be logically independent of each other. If two seemingly atomic propositions
happen not to be independent, some of their parts do not refer to simple
objects (they may refer to complexes). This means that the combinatorial powers
(i.e., internal properties) of these parts do not correspond to the combinatorial
powers of simple objects. This consideration gives us a mere indication that
the combinational powers of names do not conform to the combinational powers
of simple objects.
The main indication that there is some kind of nonconformity between (the internal
properties of) names and objects is given by the fact that we are able to
express these internal properties and relations at all. By expressing an internal
262
TLP 4.031.
263
TLP 2.0121.
264
Cf. TLP 3.02; for a detailed discussion of the notion of thinkability see §6.2.
81
relation we are trying to express something that cannot be expressed in a logically
adequate language. A language in which it is possible to express an internal
relation is, thus, not logically adequate. §3 presents the general perspective
that expressing an internal relation may be taken as an imperative to improve
our language. Such an imperative invites us to improve the logic of our language.
In this case, it invites us to coordinate the combinatorial powers of
names and objects. It works like this: in an expression of an internal property, a
formal concept must be employed that is combined with other names. In a logically
adequate language, these names cannot be combined with formal concepts
and all formal concepts have to be substituted by propositional variables. In
Wittgenstein’s words:
The expression for a formal property is a feature of certain symbols.
So the sign for the characteristics of a formal concept is a distinctive feature of all
symbols whose meanings fall under the concept.
So the expression for a formal concept is a propositional variable in which this distinctive
feature alone is constant.265
Wittgenstein also proposes that the distinctive feature should be embodied in a
propositional variable. Such a variable is a structured variable where some parts
vary and some parts are constant. The distinctive feature corresponds, of course,
to the constant part. Let me illustrate this point with an example of an internal
relation between two shades of color:
Midnight-blue is darker than sky-blue.266
(19)
This internal relation asserts a certain relation Φ between two properties Ω1 and
Ω2. The expression
Ω1ΦΩ2(20)
should be analyzed into:
[Ω1x. Ω2y⊃xΦy].(21)
265
TLP 4.126.
266
This example is also discussed in §§6.2, 10.2, & 10.5.
82
This form is not, though, the final analysis. (21) is a propositional variable that
ranges over propositions like
Ω1a. Ω2b⊃aΦb,(22)
Ω1b. Ω2c⊃bΦc,
Ω1a. Ω2c⊃aΦc,
…
where a, b, c are names. This analysis is, however, a neat illustration of adjusting
the combinational powers of names by means of converting formal properties
into propositional variables. Signs for the properties (Ω1, Ω2) can be followed
only by signs for the objects (a, b, c). A sign for a relation Φ can occur
only between the signs for objects, but not between the signs for properties as in
(20). It is noteworthy to say that the sign Φ expresses an internal relation in
(20), but an external relation in (21) and (22). In an unanalyzed language, the
sign Φ may be ambiguous between expressing an internal and external relation.
The aim of the analysis presented here is to cope with this ambiguity. The combinatorial
powers of Φ are thus adjusted in such a way that the expressing of
(20) is impossible.
The second problem concerning how a fact that is represented by a proposition
can reach reality is the subject of the next section.
9.2. Language measures reality
Wittgenstein liked the analogy of comparing a proposition with a situation and
measuring an object against a yardstick.267
He writes: “Proposition and situation
are related to one another like the yardstick and the length to be measured.”
Later in the Notebooks, he writes: “The proposition is a measure of the
267
Wittgenstein’s interest in measuring may initially originate in his reading of Plato’s Theaetetus
where Protagoras’ famous fragment “Man is the measure of all things” is discussed.
Another possible source is Otto Weininger’s book Über die letzten Dinge [On Last Things].
The notebook entry “I have to judge the world, to measure things.” (NB, p. 82) is influenced
by Weininger, who advanced the idea that man lays down measures of things in their epistemological
and ethical aspects. The absolute measure is, for Weininger and for the Wittgenstein
of the Tractatus, logic. Such measures, Weininger argues, are given by or originate in
the biological nature of man. See my (Mácha, 2012a) for further details.
83
world.”268
The key claim from the Tractatus is: “[A picture] is laid against reality
like a measure.”269
What does this analogy say? A situation that is represented by a proposition is
compared to an actual situation in order to find out whether they match. If they
do, the proposition is true but otherwise it is false. These situations need270
not
be identical, though; they have to match at certain points. It is essential for a
yardstick to have graduating lines. In the analogy, pictures have graduating lines
too: “In a picture the elements of the picture are the representatives of ob-
jects.”271
What we measure are, in fact, the relations between these graduating
lines: “The fact that the elements of a picture are related to one another in a determinate
way represents that things are related to one another in the same
way.”272
What lies between these lines is not significant: “Only the end-points
of the graduating lines [Teilstriche] actually touch the object that is to be meas-
ured.”273
It is as if the yardstick only has two graduating lines and the outcome
of a measurement is bipolar. Either the thing measured is one yard long or it is
not. By analogy, a proposition is either true or false.274
The graduating lines correspond to names in this analogy. In the concrete act of
measurement, these graduating lines are coupled with some points of the measured
thing. There is no prior agreement about what is supposed to correspond to
the graduating lines while measuring. They can be paired with virtually anything.
If the analogy is sound, names do not need to correspond to objects prior
to putting a proposition (and the represented situation) up against reality. The
correspondence may be established ad hoc. In the Philosophical Remarks Wittgenstein
extends the analogy:
268
NB, pp. 32 & 41.
269
TLP 2.1512.
270
See §9.3.
271
TLP 2.131.
272
TLP 2.15.
273
TLP 2.15121.
274
There is an important disanalogy here. A yardstick is one-dimensional and thus has only
two end-points. A picture (and a proposition) may have higher multiplicity and more end-
points.
84
By application I understand what makes the combination of sounds or marks into a
language at all. In the sense that it is the application which makes the rod with marks
into a measuring rod: putting language up against reality.275
The analogy is extended as follows. We have to be able to recognize particular
lines as graduating lines (marks) on a rod in order to use it as a yardstick. The
pictorial relationship allows us to recognize a combination of sounds or marks
on a piece of paper as a represented situation. This is the case because we possess
a method of projection. The same method enables us to recognize some elements
of the measured situation as objects that are paired with objects in a represented
situation and—on that account—with names that occur in the proposi-
tion.
What else do the represented situation and the real situation have in common?
Wittgenstein’s answer is: their logical form.276
But what exactly makes up a logical
form? As already indicated above, what we are looking into when measuring
is whether the elements of the represented situation are related to each other
in the same way as the corresponding elements of the measured situation. In
other words, we are trying to find out whether these two situations exhibit the
same structure. If so, Wittgenstein can therefore say that “a proposition describes
reality by its internal properties.”277
Let us now go back to the analogy to the yardstick. The only elements that matter
are its two end-points (more precisely, the graduating lines on the yardstick).
The only structure is that these points are one yard (0.9144 meters) distant from
each other. A thing is one yard long if it exhibits the same structure of having its
end-points in the same relation. We can consider these points as being in Euclidean
space, with the end-points of the yardstick marking a unit vector.
275
PR, p.85.
276
TLP 2.2.
277
TLP 4.023.
85
Figure 1
The internal properties of the yardstick and the measured thing are marked by
the dashed lines, and the internal relation between them is marked by the dotted
line. The end-points A and B of the yardstick stand in the internal relation of
having the length of one yard. If the end-points C and D of the measured thing
stand in the same relation, then the thing is one yard long. This Euclidean space
is analogous to the logical space that Wittgenstein is invoking. In comparing the
represented situation with a real situation, we have to read off the internal relations
within both situations and compare them.
Wittgenstein finds this overall analogy problematic in his later philosophy. It
needs to be clear that the represented situation is an imaginary one that has to
be compared with a real situation. It is as if we had an imaginary yardstick that
we laid down when measuring. The comparison takes place in Euclidean space
where we compare two vectors. This is a case of a private comparison. In §16
“The standard meter”, I am going to argue that this analogy can be rectified by
getting rid of the abstract Euclidean space.
A[0,0] Yardstick
D[1,1]
B[1,0]
Measured object
y
x
C[0,1]
86
9.3. Frege’s objection
Frege, without having available the text of the Tractatus, but with Wittgenstein
in mind,278
raised the objection to the picture theory of meaning that it is unable
to give a satisfactory account of the correspondence of a picture with reality:
It would only be possible to compare a picture [Vorstellung] with a thing if the thing
were a picture too. And then, if the first did correspond perfectly with the second,
they would coincide. But this is not at all what is wanted when truth is defined as the
correspondence of a picture with something real. For it is absolutely essential that the
reality be distinct from the picture. But then there can be no complete correspondence,
no complete truth. So nothing at all would be true; for what is only half true is
untrue. Truth cannot tolerate a more or less. But yet? Can it not be laid down that
truth exists when there is correspondence in a certain respect? But in which? For
what would we then have to do to decide whether something were true? We should
have to inquire whether it were true that a picture and a reality, perhaps, corresponded
in the laid-down respect. And then we should be confronted by a question of the
same kind and the game could begin again. So the attempt to explain truth as correspondence
collapses. And every other attempt to define truth collapses too. For in a
definition certain characteristics would have to be stated. And in application to any
particular case the question would always arise whether it were true that the characteristics
were present. So one goes round in a circle. Consequently, it is probable that
the content of the word “true” is unique and indefinable.279
In Wittgenstein’s terms, this objection focuses on the comparison of a picture
representing a situation with a real situation. Frege’s argument proceeds in two
steps: (1) A real situation, in order to be comparable with a picture, has to be a
picture too. (2) These two pictures cannot be identical. They have to coincide
only in a certain respect. To specify such a respect, one needs to apply the picture
theory again. Therefore, any picture theory of meaning is circular in defining
the correspondence between language and the world.
278
Cf. Sluga 2002, p. 89. Frege must have been responding to conversations they had had
before the war, or possibly to lost letters from Wittgenstein to Frege during the war. Thanks
to Jim Klagge for this clarification.
279
Frege, 1956, p. 291. I have altered the translation of the German ‘Vorstellung’ from ‘idea’
into ‘picture’. From the context of Frege’s text it is clear that he means by ‘Vorstellung’ a
mental picture or image.
87
Frege’s first point leads to the question as to what is actually compared. Two
mental pictures or two facts? Wittgenstein says explicitly that a “picture is a
fact.”280
But what we have to compare with reality is thoughts, i.e., propositions
with sense, not mere propositional signs. Only “a thought contains the possibility
of the situation of which it is the thought.”281
By this we can understand that
a represented situation must be something imaginary.282
What Frege is insisting
on is that if we want to compare two entities, they have to be in the same space.
This is something that Wittgenstein fully realized in the early 1930s.283
I think,
however, that Frege’s requirement is met even within the Tractarian framework.
The pictorial relationship which is embodied in the method of projection
transforms a sign into a represented situation. Because this relationship is internal,
the represented situation is already there. It is in a certain respect identical
with the picture, i.e., with the propositional sign.284
If the relation were external,
some further account of bringing the represented and the real situation into the
same space would be needed.
The second point is much more serious and in the end fatal for the Tractarian
picture theory of meaning. The problem is this: how do we define what is significant
in a picture? In other words, how do we determine what makes up the
logical form of a picture—or of a sign? Employing again the analogy with the
yardstick, one could ask how we recognize its graduating lines. The method of
projection cannot be presupposed at this stage because it is a part of the sense of
the proposition that we are seeking to define.285
The present problem of giving
meaning to a sign is an instance of the more general problem of introducing the
internal/external distinction.
280
TLP 2.141.
281
TLP 3.02.
282
If a represented situation were real, it would automatically make the proposition true.
283
Cf. PR, pp. 48f.: “It’s easy to understand that a ruler is and must be in the same space as
the object measured by it. But in what sense are words in the same space as an object whose
length is described in words, or, in the same space as a colour, etc.? It sounds absurd.”
284
The pictorial relationship cannot wholly determine the depicted situation, because it is
not a perfect identity. The picture and the depicted situation are always identical in a certain
respect. This fact does not disturb the internality of this relationship.
285
Cf. TLP 3.11: “The method of projection is to think of the sense of the proposition.”
88
Frege considers and right away dismisses the alternative of perfect correspondence.
Perfect correspondence would mean that the picture and the situation it
represents would be identical. Everything in the picture would make up its logical
form. The picture would only represent itself. This is the case of a reflexive
internal relation, which is something inherently nonsensical. As Frege puts it,
“it is absolutely essential that the [represented] reality be distinct from the pic-
ture.”286
Let me illustrate the point at issue with the map-territory relationship. The expression
“The map is not the territory” is ascribed to the Polish philosopher Alfred
Korzybski. We can try to imagine a map (which is a kind of picture) that is
identical with the territory it is supposed to represent. Lewis Carroll, in his book
Sylvie and Bruno Concluded, makes this point in a humorous dialogue. “So we
now use the country itself, as its own map, and I assure you it does nearly as
well.”287
Jorge Luis Borges, in his very short story On Exactitude in Science,
depicts the unfortunate fate of a one-to-one map:
In that Empire, the Art of Cartography attained such Perfection that the map of a single
Province occupied the entirety of a City, and the map of the Empire, the entirety
of a Province. In time, those Unconscionable Maps no longer satisfied, and the Cartographers
Guilds struck a Map of the Empire whose size was that of the Empire, and
which coincided point for point with it. The following Generations, who were not so
fond of the Study of Cartography as their Forebears had been, saw that that vast map
was Useless, and not without some Pitilessness was it, that they delivered it up to the
Inclemencies of Sun and Winters. In the Deserts of the West, still today, there are Tattered
Ruins of that Map, inhabited by Animals and Beggars; in all the Land there is
no other Relic of the Disciplines of Geography.288
Maps or pictures in general are always pictures of something. A picture identical
with the represented thing is no picture at all. Reflexive cases of the internal
relation of representing make no sense. The maxim of no reflexive uses of internal
relations will be further addressed in subsequent chapters.289
286
Frege, 1956, p. 291.
287
Carroll, 1893, p. 169.
288
Borges, 1998, p. 325.
289
We have no evidence that Wittgenstein was familiar with Korzybski’s, Carroll’s, or Borges’
writings.
89

Let us move on now to Frege’s second alternative. A picture has to coincide
with the represented situation only in a certain respect. This means that some
features of the picture represent and some do not. The features that represent
make up the logical form of the picture. Wittgenstein gives an account of how
to determine a logical form in the Tractatus at 3.31s where he introduces the
notion of a propositional variable. A proposition (which is a picture) represents
a class of situations. An expression (‘Ausdruck’) “is the common characteristic
mark of a class of propositions.”290
To get an expression we need to turn all the
representing parts291
of the proposition into variables “whose values are the
propositions that contain the expression.”292
Then we can say that an expression
is represented by means of a propositional variable. Such a propositional variable
represents a class of propositions which in turn represents a class of situations.
What remains in the propositional variable is the logical form of the
proposition293
which is identical with the logical form of the represented situations.
These representing relations (from an expression to a class of propositions
and then to a class of situations), thus preserve the logical form of the
proposition and of the situations it represents.
What remains to be specified is the range of values that a propositional variable
may take. These values have to be stipulated. It is, however, important that they
have to be stipulated a priori, and not derived from any particular case:
We portray the thing, the relation, the property, by means of variables and so shew
that we do not derive these ideas from particular cases that occur to us, but possess
them somehow a priori.294
But when it comes to the exact specification of how such a stipulation should be
carried out, Wittgenstein say merely that “To stipulate values for a propositional
variable is to give the propositions whose common characteristic the variable
290
TLP 3.311.
291
Except logical constants. See NB, p. 114.
292
TLP 3.313.
293
Cf. TLP 3.315.
294
NB, p. 65.
90
is.”295
It seems therefore that the stipulation amounts to an enumeration of the
represented situations. Wittgenstein’s account of how to specify the situations
that are represented by a proposition comes thus to this: these situations must
simply be enumerated one by one. They cannot be specified by any general
form, because this would make the whole procedure circular.
This account does face several problems though. The most striking one is that
propositions would be able to represent only a finite number of situations that
have been given a priori. However, this is not how language works. Wittgenstein
became unsatisfied with this in his later philosophy. His reflections about
the ostensive definition can be seen as a response to this problem. Frege’s worry
is that the expression determining the respect in which a picture and a situation
coincide has to have a meaning assigned in advance. Wittgenstein’s worry about
the ostensive definition is that the expressions determining a genus (e.g., ‘number’,
‘color’, ‘length’, and so on) need to be defined in advance as well.296
The analogy goes as follows: let us take a complex P that we want to use as a
picture. But P cannot coincide with represented things perfectly, but only in a
certain respect. — In which respect? — In color. Hence P represents or is a picture
of all other objects that have the same color. In other words, there is an internal
relation between the colors of P and all other objects of the same color.
One may ask further, however: what do you mean by the same color? Do you
demand pinpoint accuracy? — Yes, I demand the highest distinguishable accuracy.
— One may go on asking endlessly. An ostensive definition faces an analogous
problem. Suppose one says: “That is called ‘sepia’” — “In which respect?”
— “I mean this color is called ‘sepia’” — “Do you mean this color exactly
or is there any degree of fuzziness?” — And so on…
It seems that giving meaning to a picture is a hopelessly never-ending process,
like ostensive definition. It is difficult to imagine how this problem can be tackled
in the Tractatus. There is, however, a solution available in Wittgenstein’s
later thinking. As he puts it: “Explanations come to an end somewhere.”297
“The
295
TLP 3.317.
296
Cf., for example, PI §29.
297
PI §1.
91
chain of reasons has an end.”298
There is a last definition like there is a last
house in this road.299
The last definition or the last respect that has to be specified
is when agreement is reached. It is not, however, any prior agreement in
definitions but, in the end, in practice and in form of life.300
A definition or an
act of giving meaning to a picture is successful only when subsequent praxis
shows that there is public agreement here. Only subsequent praxis can show
that people use the same method of projection. It is obvious that this explanation
by means of public agreement is not available in the Tractarian framework.
298
BBB, p. 143 & PI §326. See also my discussion of the Principle of Sufficient Reason in
§12.1.
299
See PI §29.
300
Cf. PI §§241f.
IV. Wittgenstein’s later writings
10. Definitions of the internal/external distinction in the later
writings
In his later writings (from 1929 onwards), what Wittgenstein says is for the
most part consistent with his earlier account of internal relations. What changes
is his focus. Internal relations can be exhibited not only in tautologies, but also
in grammatical propositions in general. Internal relations are relations that hold
in virtue of grammar.301
Grammatical propositions are either explicit statements
of the grammar of a language-game or also—in Wittgenstein’s final texts—
implicit descriptions of our human form of life. The criterion of (un)thinkability
is very soon replaced by the criterion of temporality: internal relations are expressed
in propositions that are timeless as opposed to temporal propositions
which express external relations. Wittgenstein now insists resolutely that internal
relations hold only between concepts and any talk of internal relations between
objects has to be understood as referring to internal relations between
concepts describing these objects.302
10.1. Disappearance of the saying/showing distinction
In Wittgenstein’s later writings, we can find only a few remarks mentioning the
saying/showing distinction in connection with the external/internal distinction.
The main idea is that internal relations hold between complexes. They cannot
be described, but only shown in descriptions between these complexes.303
The
reason why the relata of internal relations must be complexes is that they must
be capable of being described. These complex relata do not need to be restricted
to facts; they can also be complex acts, practices, or types of behavior. Let us
consider two complexes: one is described by ‘There are three white circles’,
301
VW, p. 237.
302
LFM, p. 73.
303
Ms 109, p. 291, 1.2.1931; Ts 211, p. 147.
94
whereas the other is described by ‘There are two black circles’. These descriptions
show several internal relations: between the colors (white is lighter than
black), the numbers (three is more than two) and the identity of the shapes.
These relations are internal because when these two complexes (with their descriptions)
are given, their internal relations are also given. If one fails to see
these internal relations in the complexes, one would see, in a sense, different
complexes. And the sense here is that in such a case they would not be able to
describe the complexes by these descriptions, but only by other ones. These
complexes would be different yet unchanged at the same time. This paradoxical
appearance will later be developed in Wittgenstein’s account of aspectperception:
“The expression of a change of aspect is the expression of a new
perception and at the same time of the perception’s being unchanged.”304
It would, however, be a mistake to believe that Wittgenstein rejected or repudiated
the saying/showing distinction. Rather, it has been transformed into the difference
between what language expresses and what is shown by grammar.305
Accordingly, internal relations are relations within grammar or grammatical relations.
This characteristic is thus turned into the essential feature of internal
relations: “Internal relations (internal properties) are nothing other than what is
described in grammar.”306
This definition of internal relations can be taken as a
generalization of the pre-Tractarian definition that Wittgenstein dictated to
Moore.307
Internal relations are still shown in the notation and can be expressed
not only in tautologies, but also in grammatical propositions generally. Undoubtedly,
logical tautologies are grammatical propositions.
We have the following recording of a conversation between Wittgenstein and
Thouless: the claim that a grammatical proposition states an internal relationship
is further explained by this subsequent definition: “an internal property is
one without which the X possessing it would no longer be called X, that is, it is
one which seems to state the essence of X”308
. Here I see an echo of the Tractar-
304
PI II, p. 196.
305
VW, pp. 130–2.
306
VW, p. 237.
307
NB, pp. 116f.
308
PPO, pp. 389f.
95
ian definition in terms of unthinkability.309
For an object X and its internal property
p, it is unthinkable that it could not possess the property p. It is unthinkable
to call an object X and, at the same time, refuse it the property p. This means
that there is a grammatical proposition (a grammatical rule) stating that X implies
p. This grammatical proposition also states that there is an internal relation
between the concepts X and p. Any mention of an essence of things is, thus, understood
as internal relations being expressed by grammatical propositions. In
short: “Essence is expressed by grammar.”310
Internal relations are thereby not
reified as independent things.
The Tractarian definition of an internal relation in terms of unthinkability311
is
thus altered into a mere consequence of the new definition: if an internal relation
is expressed in a grammatical proposition, it is unthinkable that its relata
would not be related by this relation, i.e., it is unthinkable that a grammatical
proposition could not be true. However, this conception of unthinkability later
becomes relativized to a language-game. It is possible to conceive of a grammatical
proposition becoming a genuine empirical proposition in another lan-
guage-game.312
This conception of unthinkability becomes, at the very least,
ambiguous in Wittgenstein’s later texts, which might be the reason why he
ceased to define internal relations in terms of unthinkability.
This ambiguity of the notion of unthinkability can be characterized as follows:
for a given true proposition p (p is for instance ‘1+1 = 2’) we can ask whether it
is thinkable that p is not true. We are invited to imagine a situation in which p
does not hold. Clearly, we have to come up with some non-actual situation, for
in the actual situation p is true. But then it is unclear to what extent we are allowed
to change (in imagination) the actual situation. It is plausible that we can
change everything factual, but we have to preserve conceptual matters. In other
words, we cannot change the meaning of p. If we nevertheless could think that
p means something different than it actually means, the criterion of
(un)thinkability would break down. We could always imagine that p means “It
309
TLP 4.123.
310
PI §371.
311
Wittgenstein occasionally mentions unthinkability in his conversations: VW, p. 236;
WWK, p. 54.
312
See OC §§95ff.
96
is raining” together with the fact that it is actually not raining. Hence we can
think that p is not true for every p.
The criterion of thinkability therefore rests on the distinction between conceptual
and factual facts. But the distinction between the conceptual and the factual
is obscured in ordinary language and in the distinction between internal and external
relations.313
However, Wittgenstein also offers other the following con-
sideration:
It might be imagined that some propositions, of the form of empirical propositions,
were hardened and functioned as channels for such empirical propositions as were
not hardened but fluid; and that this relation altered with time, in that fluid propositions
hardened, and hard ones became fluid. […] [T]he same proposition may get
treated at one time as something to test by experience, at another as a rule of test-
ing.314
The criterion of thinkability is of no help here. Wittgenstein eventually came up
with another criterion that can help us to distinguish between sentences that express
internal and external relations. This is the criterion of temporality, which
is going to be addressed in §10.4.
10.2. Concepts and objects
What is going to be discussed in this section is the central characteristic of the
distinction between internal and external relations in Wittgenstein’s later writings.
Its centrality is backed up by numerous remarks from Wittgenstein’s manuscripts
spread over his entire later philosophical career.
As already elaborated in the chapter on the internal/external distinction in Wittgenstein’s
early texts (§6), internal relations hold between concepts (or properties,
qualities, universals) while external relations hold between objects (or particulars).
The problem is, however, that the same verbal manifestation can be
313
This obscurity is obvious in Wittgenstein’s later remarks. In On Certainty he writes: “But
there is no sharp boundary between methodological propositions and propositions within a
method. […] The lack of sharpness is that of the boundary between rule and empirical
proposition.” (OC, §§318–9)
314
OC, §§96–8.
97
used for describing one and the same state of affairs, which at times can express
an internal relation, but at other times express an external relation. One of the
tasks of philosophical analysis is to make clear whether an internal relation or
an external relation is involved. Moreover, for the sake of this book, the uncovering
of this distinction is regarded as the leading methodological principle of
Wittgenstein’s thinking.
Wittgenstein repeatedly presents various clusters of examples in order to illustrate
how this confusion comes about and how it can be eliminated. The examples
which Wittgenstein used most frequently are the relation longer than and
the relation darker than.
Let us turn first to the relation of being longer than.315
The state of affairs (or
simply: the situation) is as follows: we have two line segments drawn on a piece
of paper.316
Figure 2. Two lines
The line segment a is 3 m long and the line segment b is 2 m long. We can describe
this situation as follows:
Line a is 3 m long.(23)
Line b is 2 m long.(24)
315
See WWK, pp. 54–5; Ms 108, p. 299; WV, p. 238; ROC §1 for various uses of this ex-
ample.
316
It is not essential that we use a geometrical constellation in this example. We could have
taken material objects like two sticks (see ROC §1) or two distances (‘Strecken’, see WV, p.
238).
a b
98
These propositions express the external properties of these two line segments.
In addition to that, they also express an internal relation of being longer than
between these two propositions. This relation can be expressed as
a is longer than b.(25)
But this expression is ambiguous. It might mean that
Line segment a is longer than line segment b.(26)
or
The length 3 m is longer than the length 2 m.(27)
The former proposition expresses an external relation holding between two line
segments. The latter proposition expresses an internal relation holding between
two numbers. This proposition expresses that 3 > 2. It might be used as a (partial)
definition of the relation longer than. Thus, here, it is a grammatical propo-
sition.
This example is in accord with what has been expounded so far. It is thinkable/possible
that the line segment a could become smaller and cease to be longer
than the line segment b. It is, however, unthinkable/impossible that 3 could
not be greater than 2.
We can now, in turn, proceed to the relation of being darker than.317
As we have
already seen, in the Tractatus Wittgenstein uses this relation as an example of
an internal relation.318
A certain shade of blue is darker than some other shade
of blue. This is an internal relation. A linguistic expression of this relation may,
however, be confusing. Suppose we have two material objects (suits, bodies,
plates, rectangles, etc.) a and b. Body a is black and body b is white. Now consider
the following expression:
a is darker than b.(28)
By this sentence we can express an external relation between these two objects
on the one hand. It might be used in the sense that
317
Wittgenstein uses this example in the following places: WWK, pp. 54–5; WV, p. 238;
RFM I, §104, p. 75; ROC I, §1.
318
TLP 4.123.
99
Suit a is darker than suit b.(29)
By (28) one can also express, on the other hand, an internal relation between the
two colors:
The color of a is darker than the color of b.(30)
This proposition is, in the context of our example, tantamount to
Black is darker than white.(31)
At this point, I have to refer back to our discussion of the Tractarian example of
an internal relation between two shades of blue:
Midnight-blue is darker than sky-blue.(32)
I then argued that this proposition is true because of the fact that sky-blue has a
brightness of 92% and midnight-blue has a brightness of 44% in the HSV color
space. Within this color space (32) expresses, thus, an internal relation which
may be put as 44 < 92. In the same color space, black has a brightness of 0%
and white one of 100%. Hence the propositions (30) and (31) both express the
internal relation 0 < 100. These grammatical propositions are, however, not
eternal truths. One could imagine a color space (albeit an uncanny one) in
which sky-blue would be darker than midnight-blue or even white darker than
black.
These two relations are by far the most extensively discussed examples in connection
with the external/internal distinction in Wittgenstein’s writings. The
next three examples appear far less often. The relation of being earlier can relate
two events or two dates. Wittgenstein uses the following sentence:
Caesar [was born] before Augustus.319
(33)
If two persons or their dates of birth were related here, the relation of being earlier
than would be external. But a relation of being earlier than between two historical
dates (here 100 BC and 63 BC) is internal.
The penultimate example is the relation of having the same number of elements.
We can say, for instance, that
319
WWK, p. 55.
100
This sack has the same number of potatoes as that sack.320
(34)
This sentence may express an external relation between these two sacks or an
internal relation between the numbers of potatoes in these sacks. Wittgenstein
compares this sentence with the following one:
The hand has the same number of strokes as the pentagram has points.321
(35)
This sentence is not ambiguous between expressing an internal and an external
relation. It expresses an internal relation between the (human) hand and the pentagram.
The fact that the pentagram has five points lies in the definition of the
pentagram. One could argue that a human hand does not necessary have five
strokes, because one of its fingers may be cut off. In this case, four strokes
would not depict a human hand, but a human hand without one finger. The internal
relation expressed in (35) holds not between actual depictions, but between
abstract geometrical figures: between the pentagram and the schema figure
of the human hand.
320
LFM, p. 73.
321
LFM, p. 73.
101
Figure 3. Pentagram Figure 4. Hand
The relation of having the same number of elements makes sense even if it were
a unary relation, i.e., the property of having a certain number of elements. Here
is Wittgenstein’s example:
Exercises: Number of notes–the internal property of a tune; number of leaves–the external
property of a tree.322
For the sake of this example let us concentrate on a single musical motif as a
part of a tune. The central opening motif in Beethoven’s Fifth Symphony is a
four-note figure:
Figure 5. The opening motif in Beethoven’s Fifth Symphony323
It is essential to this motif that it consists of four notes. If it had a different
number of notes, it would not be the opening motif of Beethoven’s Fifth Sym-
phony.
For a particular tree though, it is not essential that it has a definite number of
leaves. A tree can lose some or all of its leaves or new leaves can grow but the
tree still remains the same in the sense of numerical identity. If the matter that
322
RFM I, §77.
323
Symphony No. 5 (Beethoven) on Wikipedia. Retrieved March 1, 2013, from:
http://en.wikipedia.org/wiki/File:Beethoven_symphony_5_opening.svg.
102
concerns us is the concept of a tree, it is not essential either how many leaves
have objects (i.e., particular trees) that fall under this concept.
The last example that I am going to discuss in this section is the difference between
external and internal similarity324
which is prominent in Wittgenstein’s
writings on the philosophy of psychology. And here again, there is an ambiguous
sentence at the outset:
The two faces are similar.(36)
Using this sentence, one may mean “the faces of those men that interest me, or
it may be these facial forms, wherever I encounter them.”325
The similarity between
those men is external, while the similarity between their facial forms is
internal. A similarity between two men is external because they may change
their appearances and cease to be similar. A similarity between two facial forms
is an internal one, if and to the extent to which these forms can be characterized
by similar descriptions. So, for example, a round form is similar to an elliptical
form but dissimilar to an angular form. The question of whether the two forms
are similar is a geometrical one.326
One may say, for example, that the round
form and the elliptical form are both continuous curves while the rectangular
form is not.
10.3. Crossing different language-games
What all these examples have in common is that a sentence can be used to describe
one and the same state of affairs in two different ways. On the one hand,
one may be interested in particular objects in the state of affairs, in their properties,
and in the relations between them. Or one may be interested, on the other
hand, in the properties of and the relation between the concepts used to describe
the state of affairs. In the former, external relations are involved, while in the
latter internal relations are involved.
But why does this distinction matter? Why and in which cases is it important to
distinguish whether one intends to speak about objects and their relations, or
324
LWPP I, §156.
325
LWPP I, §155.
326
LWPP I, §158.
103
about concepts and their relations?327
Why is it sometimes important to be clear
about whether one is using a proper proposition or a grammatical proposition?
Consider the following sentences mentioning two objects, labeled a and b:
a has the same length as b.(37)
a has the same color as b.(38)
a was created at the same time as b.(39)
a has the same number of parts as b.(40)
a is similar to b.(41)
a and b are similarly beautiful.(42)
These sentences may all be used in order to assert this or that similarity between
the objects in question. That these two objects are the same or similar in this or
that way is an accidental fact. Object a can have the same length or color or
number of parts as object b, but it does not need to. In this sense these sentences
express the external relations between the objects a and b.
One may intend by (37)–(39) to point out the identity of two lengths, two colors,
or two points in time. The objects a and b serve only as examples here. One
may use (37) to explain what is meant by two objects’ having the same length
(and in the case of any doubts, they would have to be placed alongside one another
and compared).328
Sentence (38) may be used to illustrate what counts as
the same color (e.g., that tiny subtleties do not matter and, hence, do not disturb
their identity329
). ‘At the same time’ can mean at the same second or in the same
century. Sentence (39) may be helpful in clarifying this ambiguity. Things can
usually be decomposed into their parts in many different ways. Sentence (40)
can be used to explain what counts as a part here. Sentence (41) can be used to
focus on the similarity in question. And finally, (42) may be used to elucidate
some aesthetic feature or rule.330
The important thing here is that in the case of
confusion, other objects could be used for the same purpose, which is to express
the same internal relation.
327
See LWPP I, §§157–162.
328
See §16 on the standard meter.
329
See §15.2 for a more detailed discussion.
330
Cf. LWPP I, §161 for a variation of this example.
104
The point of all these examples is to show that these sentences could “fall ‘between
several games’”331
where they can be used to serve different purposes.
They can be used factually for expressing external relations, or conceptually for
expressing internal relations.332

The notion of falling between different language-games deserves more of an indepth
scrutiny. Wittgenstein delimits the role of language-games by claiming
that they “are rather set up as objects of comparison which are meant to throw
light on the facts of our language by way not only of similarities, but also of
dissimilarities.”333
Accordingly, language-games are objects which are compared
with (some parts of) our language.334
Language-games are invented or
fictional, which means that it is not important whether such an activity actually
takes place. What kind of objects are they? In most of the examples Wittgenstein
gives us they are simplified or, rather, schematized descriptions of language
use together with other relevant extra-linguistic activities. Such schemata
are compared with (the descriptions of) actual language use.
What language and language-games have in common is that they are ruleguided
activities. Provided that we are interested in the rules of grammar, then
language-games can be individuated by their rules. Here is how this individuation
works: provided that we are able to split a given description of a linguistic
activity into two parts so that (at least partly) different grammatical rules are
active in these parts, then we can take these two parts as different languagegames.
One can also, however, take these two language-games as one game if
one has a reason to do so.
Language-games are also primarily (although not exclusively) the conceptual
tools employed in philosophical analysis for surmounting philosophical confusions.
335
I would like now to focus on one form such confusions can take which
331
LWPP I, §761.
332
For the opposition between the factual and the conceptual, see RPP II, §5.
333
PI §130.
334
Wittgenstein occasionally conceives of language-games as parts of actual practices. See,
for instance, PU §654ff.
335
I take language-games primarily to be what Glock calls fictional language-games (Glock,
1996, pp. 194–6). Wittgenstein, however, uses the notion of a language-game in other mean-
105
is central to Wittgenstein’s later writings. A philosophical confusion (but also a
simple problem of understanding) may arise if we are not able to assign to a
given linguistic expression its appropriate context of use. If the meaning of a
word is its use in language, we have to be able to put this word into its context
in order to understand the word at all. It is obvious that we do not need to imagine
the whole context in all its details; a schematic description of an appropriate
language-game is usually enough. Thus, a philosophical confusion may now
arise if we do not provide a language-game, or if we provide the wrong one.
Wittgenstein labels such a diagnosis as the “crossing of different languagegames”
or “fall[ing] between several games”.
We shall now consider some sentences that are prone to such a confusion—
sentences that may express a genuine proposition in one language-game and a
grammatical proposition in another. If language-games were individuated by
their rules, such sentences would also fall within multiple language-games. The
risk of confusion would be bolstered if these two different language-games differed
only in this one sentence. I will focus precisely on such sentences (sometimes
called Doppelgänger336
) and on language-games. Suppose the following
general scenario: one has to teach or learn a certain rule in order to apply it afterwards.
We can distinguish between two stages of this process: the languagegame
of teaching and the language-game of applying the rule. These languagegames
are different, for what is a rule in the latter is not a rule. Although it is
not necessary to mention the would-be rule explicitly during the process of
training, in many cases it is anyway. Such mentioning would have a declarative
character (using Searle’s terminology) and we can treat such sentences then as
imperatives. Hence, one can easy imagine that one and the same sentence expresses
a genuine proposition in the language-game of teaching and a grammatings
too. Language games can also be actual linguistic activities or even sub-languages of
certain communities (like, for instance, the language of science or religion). What counts as
a language-game here is determined by our actual practices. The present inquiry is, however,
focused on Wittgenstein’s method of analysis. In this context, the notion of a languagegame
is a methodological tool. This is not to say that certain language-games cannot (accidentally)
coincide with actual practices.
336
Cf. Moyal-Sharrock, 2004, p. 66.
106
ical proposition in the subsequent language-game of applying a rule.337
In other
words: the form of the sentence is the same, but it expresses an external property
in the former language-game and an internal property in the latter one. The
language-game of applying a rule logically presupposes the language-game of
teaching. There is, thus, a vertical relation338
between these two language-
games.339
10.4. The criterion of temporality
We know from the previous section that internal relations only hold between
concepts while external relations can hold between concepts, between concepts
and objects, or between objects. This difference might be confused in the surface
grammar, for an internal as well as an external relation can be expressed by
one and the same sentence in different language-games. So, given the context of
a language-game, one could wonder how to find out whether an internal or an
external relation is being expressed.
For this purpose Wittgenstein developed what I shall call the criterion of temporality.
The main idea behind this is that external relations are expressed by temporal
sentences while internal relations are expressed by timeless sentences.340
The temporal character of sentences is something that was of particular interest
to Wittgenstein throughout his later philosophical writings. One of the leading
337
This scenario has not escaped the attention of commentators. See, for example, Hintikka,
1982 or Baker & Hacker, 2005b, p. 62: “The training activity antecedent to the languagegame
of §2 [of the PI] is itself a language-game.” There are even different kinds or stages of
training which Wittgenstein subsumes under the family-concept “general training” (BBB, p.
98). Some of these stages may involve testing the application of a rule.
338
The expression “vertical relation” is from ter Hark (1990, p. 34). The failure to consider
vertical relations between language-games is called the “ground-floor fallacy”, for example,
naming and describing within the same language-game.
339
These thoughts draw upon my own paper Mácha, 2013; and see also Smith, 2013.
340
See ROC §1 for the most explicit expression of the connection of temporality and the
internal/external distinction. See also RFM I, §104 & LFM, p. 166.
107
questions he poses in the Philosophical Grammar reads: “Is time essential to
sentences?”341
Temporal sentences are such that their truth or falsity depends on time. More
exactly, their truth or falsity depends on accidental circumstances which are, of
course, time-dependent. Timeless (or non-temporal) sentences are, by contrast,
independent of external circumstances. Whether a sentence is temporal or timeless
does not need to be manifested in its mere appearance. Not all temporal
sentences contain an explicit adverbial of time. A deeper reflection on the sentence’s
use is now required.342
The reflection is this: a use of a sentence is regarded as temporal if this is indicated
grammatically or if it would be plausible to insert an explicit adverbial of
time or a time clause into the sentence in the same context of the languagegame.
In other words, this criterion consists in considering making the grammatical
form of the sentence temporal. At this point we must clarify what
counts as a plausible modification. Wittgenstein has left us with numerous examples
here as well. For example:
341
PG, p. 215. The German original reads: “Ist die Zeit den Sätzen wesentlich?” (Ms 212, p.
377) I have changed the translation of German ‘Sätzen’ into ‘sentences’ instead of ‘propositions’.
It would be more plausible to translate this sentence by “Is tense essential to sentences?”
in order to highlight the grammatical character of this consideration. I am grateful to
Deirdre Smith for emphasizing this possibility to me.
342
Jason Stanley (2000) recently proposed the so-called binding argument in order to detect
covert variables in the logical form of sentences. If a sentence contains a covert variable, it
is possible to bind the variable by an explicit quantifier. So if the sentence
(1) It rains.
contains a hidden temporal variable, so the sentence
(2) Every time John lights a cigarette, it rains.
will bind the presumed variable contained in (1), and the sentence
(3) For every time t at which John lights a cigarette, it rains at t at the location in
which John lights a cigarette at t.
will be a natural interpretation of (2) (Stanley 2000, pp. 415–416).
108
“The 100 apples in this box consist of 50 and 50”—here the non-temporal character
of ‘consist’ is important. For it doesn’t mean that now, or just for a time, they consist
of 50 and 50.343
We are asked to consider whether it would be plausible to insert ‘now’ into (43):
The 100 apples in this box consist of 50 and 50 now.(43)
Although such a variant of the sentence is syntactically correct, it does not contribute
to its meaning. Let us take the arithmetical statement:
100 is equal to 50 + 50.(44)
The grammar of English does not prevent us from inserting an adverbial of time
into this sentence:
100 is equal to 50 + 50 now.(45)
Or other modifications:
100 has been equal to 50 + 50 since arithmetic was invented.(46)
Every time I try to count it 100 is equal to 50 + 50 at that point in time.(47)
Sentences (45)–(47) are not meaningless; they even follow from (44). But they
do not contribute to the meaning of (44); they do not make (44) more accurate.
This is why it is not plausible to explicitly impose temporality on the sen-
tence.344
Let us proceed to some less trivial cases. The sentence
‘Dædalus’ contains seven sounds.345
(48)
343
RFM I, §101.
344
Cappelen and Lepore (2005, p. 74) have proposed the application of Stanley’s Binding
Argument to mathematical propositions like (44). In so doing we get (47). Although there is
obviously no hidden variable in (44), there is an explicit bound variable in (47). That is the
reason, they have argued, for the failure of the Binding Argument. The argument, however,
presupposes that (44) has the same logical form when it is uttered alone as when it is embedded
in (47). The point in Wittgenstein’s work is that the same sentence may have different
forms in different language-games. In certain language-games, (47) is not a plausible
elucidation of (44).
345
See RFM, p. 338 for Wittgenstein’s treatment of this example.
109
could be taken phonetically as a report of an experiment in counting. In this
case, it would be more accurate to say
As pronounced now ‘Dædalus’ contains seven phones.(49)
This is an empirical proposition. The word ‘Dædalus’ could be pronounced differently.
Then it is thinkable or possible that the result could have been different
from what it was. Sentence (48) could be understood phonologically, however,
as a grammatical proposition in the sense
The sound-pattern ‘Dædalus’ has seven phonemes.(50)
In this case, it is not possible to insert any adverbial of time in (48). To be clear,
(50) may be a result of a particular method of counting the sounds given particular
rules of pronunciation, but what is expressed here is that one must have
decided on this particular method of counting. We could also have decided on
another method: for example, we could have taken into account the consonants
or vowels. Then the sound-pattern ‘Dædalus’ would have contained four consonants
or three vowels. Other methods are also thinkable or even possible. These
considerations do not detract from the timeless character of (48) and (50). What
is expressed here is the grammatical proposition attributing to the word ‘Dædalus’
the internal property of having seven sounds.
Let us apply the criterion of temporality to the sentences (37)–(42) discussed in
the previous section. One may argue that the internal/external distinction does
not exhaust the ambiguity of these sentences. We can reformulate (37) as claiming
the equivalence of two definite descriptions:
The length of a is equal to the length of b.(51)
Then we can adopt Keith Donnellan’s distinction between attributive and referential
uses of definite descriptions.346
These definite descriptions can be used
either attributively:
The lengths of a and b, whatever they are, are equivalent.(52)
Or, we can use these definite descriptions referentially:
The length of a is A, and the length of b is B, and A=B.(53)
346
See Donnellan, 1966.
110
I claim now that on both these readings, these sentences express external relations.
(52) expresses an external relation because it makes sense to say that
The lengths of a and b, whatever they are, are equivalent now.(54)
Objects a and b may have the same length, but it is conceivable that they do
not. Object a can shrink, for example. It makes sense to add ‘now’ into (52).
(53) also expresses an external relation because we can insert a temporal adverb
into the first and the second clause:
The length of a is now A, and the length of b is now B, and A=B.(55)
I am going to argue in §16.4 below that we cannot separate the description ‘the
length of a’ from the actual length of object a. This means that we cannot take
the description ‘the length of a’ to rigidly refer to A (which is a numerical value).
If we could rigidly refer to the lengths of a and b, inserting a time clause
into (37) and (53) would make no sense. Then these sentences might express an
internal relation that is, however, empirical, which would undermine Wittgenstein’s
analysis. But adding a time clause does make sense here.347

The criterion of temporality is thus that if a sentence is or can be made temporal,
then it expresses an external relation. In short, temporality implies externality.
The reverse claim—that is, if a sentence expresses an external relation,
then it is temporal—is less plausible. Laws of nature may be counterexamples
here. They are contingent and express external relations.348
But they are expressed
by timeless sentences and it makes no sense to insert any adverbial of
time or a time clause into them. Let us take, for example, Newton’s Second law:
347
The distinction between attributive and referential uses of a definite description applies
only to descriptions used de re and hence expressing external relations. De dicto descriptions,
which express internal relations, are used neither attributively nor referentially.
348
This claim is contestable. Necessitarians argue that laws of nature are necessary. But
there is no room for monological necessity in Wittgenstein’s later philosophy. In a Wittgensteinian
framework, one could argue, however, that laws of nature are still contingent but
propositions that express them are expressions of internal relations. A law of nature, once
discovered empirically, can be hardened into a rule. (See OC, §96)
111
The relationship between an object’s mass m, its acceleration a, and the(56)
applied force F is F = ma.
This sentence is timeless and it makes no sense to convert it into a temporal
one. Laws of nature are those laws which apply everywhere and at all times
throughout the universe. Laws of nature are external relations expressed by
timeless sentences.349
Returning now to the discussion of the Tractarian definition of the internal relation
in terms of unthinkability in §10.1, it becomes clear what is wrong with this
notion. There are two kinds of thinkability mentioned in the last example. For
an empirical proposition like (49) it is thinkable that it does not hold in this language-game.
For a grammatical proposition like (50) it is unthinkable that it
does not hold in that language-game, but it is thinkable that it does not hold in
some other language-game. The notion of thinkability is or may be confused.
The criterion of temporality is at hand to deal with this confusion. The temporality
of a sentence is always being considered within a language-game where
the sentence is used. This criterion gives us a tool or method for determining
whether a given sentence expresses an internal or an external relation in a given
language-game.
10.5. Tertium quid
The last characteristic is that the terms of an internal relation are related directly,
without any mediation, to each other. Internal relations relate their terms
through the terms, and not through other things or rules. This characteristic was
already present in the early Tractarian account of internal relations: the identification
of the one term of an internal relation is, eo ipso, the identification of the
other term.
The existence of terms, which as we know now are concepts, is the necessary
condition of the holding of an internal relation between them. “An internal relation
[…] exists only if its components are present.”350
If one of the terms did not
exist, there would be no internal relation. This means that there is no internal
349
Here I am indebted to Peter Baumann.
350
BT, p. 77e. See also LFM, p. 75.
112
relation that does not actually relate its terms. There could not be the relation of
being darker without there being entities that are supposed to be related, namely
its color shades.351
That the terms are related by an internal relation lies within these very terms.
“An internal relation holds by virtue of the terms being what they are.”352
There
are, of course, concepts that are not internally related. But if the concepts are
related, their mere existence guarantees their internal relatedness. There is, thus,
no third thing353
that would relate the concepts. Internal relations, so to say,
emerge when concepts are given. In other words, internal relations are supervenient
to its terms or “supervenience is written into our understanding of the
concepts.”354
Having said that, I now want to expound two interrelated points about this characteristic
of internal relations. The first one is that from knowing one term and
knowing that this concept is internally related to another concept (or concepts),
it is possible to infer (logically) what the other concept is (or the other concepts
are). As in the Tractarian account, we can conceive of internal relations as sorts
of operations which can be applied to a term in order to compute the other term.
This analogy between internal relations and mathematical operations is, however,
not very close: the outcome of a mathematical operation is always unique,
but applying an internal relation to a concept can lead to a multiplicity of other
concepts that are related in this way. By applying the relation of being darker to
sky-blue, we get as an outcome not only midnight-blue, but all the other color
shades that have lower brightness than sky-blue, e.g., sapphire-blue. One can
make the outcome unique if one makes the relation more precise. For instance,
351
But there do not need to be things that actually have these colors or shades of color.
352
LWL, p. 57. See also LWL, p. 9; Ms 112, p. 63r; LFM, p. 85.
353
There is no third thing in the case of binary relations. There is no fourth thing in the case
of ternary relations and so on.
354
McGinn, 2000, p. 85. Colin McGinn provides an argument that modal properties are epiphenomenal
to causal powers. Whether some of an object’s properties are contingent or
necessary makes no difference to its causal powers. “Only the actual can be causal. […] The
modal is not part of the causal order” (p. 86). The same is true of internal properties and relations
that can be said to be epiphenomenal to external properties and relations. Thanks to
Modesto Gómez Alonso for this reference.
113
by applying the relation of being 40 percent darker to sky-blue, one gets as the
outcome midnight-blue exactly.
To make this clear, I would like to add a mathematical example. Consider the
numbers 1597, 2584, and 4181. Although these numbers do not seem to be related
at first sight, they are in fact a succession of the 17th, 18th, and 19th Fibonacci
numbers. They are internally related by the ternary relation of being in
the Fibonacci sequence. There is a unique way of computing the next element
out of the previous ones. The next element is such that
Fn = Fn-1 + Fn-2(57)
One might now have the impression that these three numbers are internally related
because they make up a Fibonacci sequence. So there is a third thing after
all: the rule of how to compute the next member of the series out of the previous
ones. The numbers 1597, 2584, and 4181 are, in fact, internally related in
infinitely many ways, because there are an infinite number of series that have
these numbers as their members. It would be a deep misunderstanding to think
that the rule of computing the next member of a series is a third item relating
the preceding members to the next one. This is something which I am going to
focus on now.355
The second point is that the internal relations hold only in virtue of their terms.
It would be misleading or even false to say that they hold in virtue of other rules
or other grammatical propositions. It seems now that grammatical rules are precisely
those that internally relate the terms. For instance, one might assume that
midnight-blue and sky-blue are internally related because of the rules in the
HSV color space. Or that ‘Dædalus’ has seven sound patterns because of the
rule that assigns the number of sounds in a given word.
These concepts are individuated by their roles in language-games. If two (or
more) concepts are internally related, they are so in virtue of the rules that define
these very concepts. Accordingly, the internal properties of a concept are
made up by the rules defining the concept. The relation that is not constitutive
355
In the Critique of Pure Reason (B15) Kant expresses the opposite view. He claims that
from the mere conception of a sum of five and seven we never obtain the conception of
twelve. In order to perform the synthesis, there has to be a third thing, which for Kant is the
power of intuition.
114
of its terms is an external relation. If we say that an internal relation does not
hold in virtue of a third thing, it means that it does hold exclusively in virtue of
the grammatical rules that define or govern its terms. The holding of an internal
relation can be deduced or computed out of these rules; so that no other rules
are necessary.
Let me illustrate this point with one of our familiar examples. That midnightblue
and sky-blue are internally related by the relation of ‘being darker’ can be
read off from the definitions that give their internal properties: midnight-blue
has a brightness of 44% and sky-blue one of 92%. No other rule is required
here. I can imagine, for instance, the relation of being more important (for me)
and say that sky-blue is more important for me than midnight-blue. This is an
external relation because such a rule is a third thing that cannot be deduced
from the internal properties of the related terms. That the sound-pattern ‘Dædalus’
has seven sounds must be evident from the pattern itself and from the rule
or method for counting sounds.
The example with the Fibonacci numbers is slightly more complicated, but only
at first sight. The matter of concern here is, in fact, a simple equation like 3 + 2
= 5. Wittgenstein asks himself “How can one calculate that 3 + 2 = 5?!”356
There is no internal relation between ‘5’ and ‘3 + 2’, one (here, a Kantian philosopher)
would have to say. Wittgenstein wants us to imagine a fact like this:
‘|||||’ correlates with ‘|| + |||’.357
It is evident now that there is a one-to-one correlation
between these complexes. Hence the relation must be internal. No other
rule is necessary except the rules for numbers and the rule for addition. The rule
for a number is simply to make a certain number of strokes and the rule for addition
is to put the strokes together.
356
Ms 110, p. 289, my trans. (“Wie kann man kalkulieren daß 3+2=5 ist?!”)
357
Cf. BT, p. 67 and my further discussion of numbers in §14.2.
115
10.6. No exceptions
The point that internal relations do not allow for any exceptions has often been
neglected among commentators. Wittgenstein writes that internal relation are
exact358
or that
they persist always, unalterably, in the whole that they constitute; as it were independently
of any outside happenings. As the construction of a machine on paper does
not break when the machine itself succumbs to external forces.—Or again, I should
like to say that they are not subject to wind and weather like physical things; rather
[they are] unassailable, like schemas.359
Since internal relations are the expressions of grammar in a language-game,
they hold whatever happens. The only way to change them is to change the
grammar, which is tantamount to changing the language-game. There is no possibility
of any failure with respect to the holding of an internal relation as opposed
to the ever-present possibility of a real machine failing. An internal relation
holds exclusively between concepts and in virtue only of these very concepts.
In this sense, internal relations can be expressed only by analytic truths.

Several commentators have already advanced the view that Wittgenstein, in his
later philosophy, allowed that internal relations could be expressed not only by
analytic truths, but also by synthetic ones. In their influential interpretation,
Baker and Hacker write: “Wittgenstein repudiated the implication that any expression
of an internal relation must be a necessary truth or tautology.”360
Baker
and Hacker argue that internal relations that are not expressed by analytical sentences
are, inter alia, relations between psychological concepts and their criteria.
There is, for instance, an internal relation between the concept of pain and
instances of pain-behavior (such as whining or moaning). This relation is internal,
because it is formulated in grammar. But the sentence ‘If someone is moaning,
they are in pain’ is not a tautology. The criteria for some words are defeasi-
358
Ms 111, p. 111.
359
RFM I, p. 74. I have changed the last word to ‘schemas’ as ‘Schemen’ appears in the
original typescripts Ts 221, p. 157 & Ts 222, p. 75.
360
Baker & Hacker, 1984, p. 109; see also Hymers, 1996, p. 598f.
116
ble. Someone may fake pain-behavior without really being in pain, and they
may suppress pain-behavior despite being in pain. The inner state (of pain) is
not necessarily related to its outward criteria since there are exceptions to the
rule. From this fact, Baker and Hacker conclude that “Though not tautology, the
relation of pain-behavior to pain is an internal relation.”361
Internal relations hold solely among concepts. A sentence asserting an internal
relation between objects (or events) has to be read as a sentence expressing an
internal relation between concepts.362
The relation of pain-behavior to pain is in
fact the relation between the concept of pain and the descriptions of painbehavior.
The relation is, however, not that pain-behavior is pain. This meaning
would be a behavioristic diminution of the concept of pain. The relation is that
pain-behavior is a criterion of being in pain.
The concept of a criterion carries the (albeit rare) possibility of deception on its
shoulders. To be a criterion of something is a different kind of internal relation
than, for instance, in ‘thunder and lightning mean a thunderstorm’. There can be
more distinct criteria of psychological states. That only one of these criteria actually
holds does not imply that the other ones have ceased being criteria for
now. There is no exception or temporary suspension of validity.
The criterial definition of a psychological concept can eventually be replaced by
a more rigid definition. The scientist may define pain as a stimulation of nociceptors.
This is, however, a conceptual change. This new concept of pain is not
identical with the old one based on criteria. These criteria thus turn into symptoms
of the new concept.363
So we have the following relations:
Pain-behavior is a criterion of painold.(58)
A stimulation of nociceptors means painnew.(59)
Pain-behavior is a symptom of painnew.(60)
361
Baker & Hacker, 1984, p. 110.
362
See LFM, p. 73.
363
See PI §§79 and 354. See also Glock, 1996, pp. 95–97.
117
The first two sentences express internal relations; the last one expresses an external
relation. Pain-behavior is only a concomitant phenomenon of pain if pain
is defined by (59).364
The conceptual stipulation is a very rare phenomenon; it is in a sense “a trivial
way”365
of introducing conceptual change. Our conceptual usage may naturally
evolve through time. It may be that such a change rests on “imponderable evidence”
that cannot be predicted a priori:
And now the question remains whether we would give up our language-game which
rests on ‘imponderable evidence’ and frequently leads to uncertainty, if it were possible
to exchange it for a more exact one which by and large would have similar con-
sequences.366
In his last writings, Wittgenstein thus seems to concede that language-games
with their grammar may evolve due to empirical evidence.367
This would in effect
undermine the distinction between grammatical propositions and empirical
propositions and by implication undermine the distinction between internal relations
and external relations.
I think that we can keep the distinction between internal and external relations
valid if we consistently insist that this distinction is relative to a given language-game.
Nobody will deny that our concepts evolve through time. If a
change of a concept is small, we tend to think that it is the same concept. If a
change is bigger or more substantial, we tend to the view that one concept has
evolved into another one, though these two concepts may have the same verbal
expression. It is of little importance what the cause of the conceptual change
was. It is also of little importance whether the change was due to a one-off stipulation
or a gradual evolution. But if this change leads to conceptual confusion,
we should conceive it as a case of the crossing of different language-games.
364
See Klagge, 2011, §7 for a detailed discussion.
365
PGL, p. 292.
366
LWPP II, p. 94.
367
Klagge (forthcoming) writes that “the factors that will or might lead to such a change are
unknown to us now.”
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10.7. Summary
In Wittgenstein’s later texts, we can therefore find the following characteristics
of internal and external relations:
(i) Internal relations hold only between concepts while external relations hold
between objects and concepts.
(ii) Internal relations can be exhibited in grammatical propositions which express
either rules of a language-game or general facts of our human form
of life.
(iii) Propositions that express internal relations are timeless, whereas propositions
that express external relations are temporal.
(iv) Internal relations relate their terms only in virtue of these very terms, not in
virtue of other things or rules.
(v) Internal relations allow no exceptions.
119
11. Intentionality
Wittgenstein’s account of intentionality is a good example of his method of
analysis based on the distinction between internal and external relations. The
metaphysical mystery of human intentionality is dissolved by making clear the
grammar of language.
Apart from a few exceptions, Wittgenstein does not discuss intentionality under
this precise name, but he does discuss various kinds of intentionality. The most
common kind is the nature of expectation and how an expectation is related to
its fulfillment. Other kinds of intentionality Wittgenstein touches on are wishes,
fears, hopes, desires, orders, beliefs, hypotheses, and other intentional states. An
intentional state is always related to something that does not or might not exist.
It may refer to a state of affairs in the past or in the future, or it may be unknown
or uncertain whether the state of affairs is realized. Although an intentional
state may refer to the past, its fulfillment (satisfaction, verification, falsification,
etc.) is possible only in the future. In short, the intended state is not
available at the moment of intending.368
If something is to be expected, that expectation is directed to the future. The
expected state of affairs may or may not be realized. The same is valid for a
wish or a desire. A hypothesis, a belief, or a fear may refer to the past; but they
are also directed to the future because of their future verification or falsification.
Intentionality also includes a family of allied phenomena which do not need to
have any common feature. Two states of affairs are usually related to intentional
states: the present intentional act and a state of affairs that satisfies the intention.
These two states are separated in time. The mystery of intentionality consists
in the way in which an intentional act is supposed to reach its satisfaction.
The main problem concerning intentionality that Wittgenstein faces is an epistemological
one. How does an agent come to recognize that a present state of
affairs satisfies their previous intention? How does one know that this is what
they wanted? How does one know that the present state of affairs verifies or fal-
368
The intended state may already be fulfilled without the agent’s being aware of this. Fulfillment
means realizing that the intended state is fulfilled (it may have been fulfilled before
as well). It makes no sense to expect that p or to have a fear of p and at the same time know
that p is the case. I am thankful to Peter Baumann for bringing this point to my attention.
120
sifies their previous hypothesis? Or to restate the problem in a more Wittgensteinian
fashion: how does one know that the present act is in accord with a rule
that was previously stated? In spite its own peculiarities (further discussed in
this chapter), the problem of the recognition of a previous intention belongs to
the family of rule-following problems.369
11.1. Russell’s causal theory
At the beginning of the 1930s, Wittgenstein approaches intentionality against
the background of the causal accounts of intentionality given by Russell and by
Ogden and Richards in the 1920s. The difference between these approaches and
Wittgenstein’s picture conception of intention is the same as that between external
and internal relations. Wittgenstein writes accordingly:
But the essential difference between the picture conception and the conception of
Russell, Ogden and Richards, is that it regards recognition as seeing an internal relation,
whereas in their view this is an external relation.370
Before turning to Wittgenstein’s own conception I shall sketch Russell’s account
of intention from The Analysis of Mind371
and the way in which this is
based on external relations.
For Russell, a prototypical type of intentional act is desire. The thing that is distinctive
about desire is that it is accompanied by a feeling of discomfort. This
feeling causes an action which should likely bring about quiescence. If this action
is successful, it will bring pleasure. The intentional object (of a desire) is
the final state of affairs which removes the discomfort and brings pleasure. Russell
calls the whole process from desire to pleasure a ‘behaviour cycle’.
The ontological status of a behavior cycle is now clear: the connection between
desire and fulfillment is causal. But there are two epistemological problems or
questions which emerge here. Firstly, when one has a particular desire, how
does one know what exactly will satisfy it so that one can choose a suitable ac-
369
Cf. “Any ‘reasonable’ expectation is an expectation that a rule we have observed up to
now will continue to hold.” (PR, p. 294 & PG, p. 231)
370
PR, p. 63.
371
Russell, 1921, §§III & XIII.
121
tion to achieve this satisfaction? Secondly, when one experiences a particular
pleasure, how does one know that the present state of affairs is exactly that
which one has desired? The first question arises at the beginning of a behavior
cycle, the second at its end.
As to the first question, one could infer from past experience (from the fact that
similar desires have been satisfied by certain states of affairs) that the actual desire
will most likely be satisfied by a certain state of affairs. As to the second
question, the end of a behavior cycle is recognized by the end of the dissatisfaction
following the pleasure. These mental events are dependent on the desire
and on the action that is supposed to satisfy this desire. If one knows that the
actual pleasure is exactly that which they desired by recognition, then the
pleasure is causally related to the previous desire. This epistemological problem
nevertheless remains when one behavior cycle is interrupted by another, as Russell
agreed.372
If so, when one experiences the pleasure, it is difficult to decide
which cycle is to be terminated.
The relation between desire and its object is an external one because there must
be a third element, namely the feeling of pleasure. The desire is not matched
with its satisfaction until there is a feeling of pleasure.373
An analogous external
relation holds between an expectation and the thing that has been expected:
We have first an expectation, then a sensation with the feeling of expectedness related
to memory of the expectation. This whole experience, when it occurs, may be defined
as verification, and as constituting the truth of the expectation.374
The relation between an expectation and the expected state becomes external,
since the truth of its realization depends on the feeling of confirmation. One has
to remember and confirm that the actual state of affairs is exactly that which
one has expected.

372
Russell, 1921, p. 65.
373
Cf. PR, p. 63.
374
Russell, 1921, p. 270.
122
Wittgenstein’s objections to Russell are focused on the second epistemological
problem, as discussed above. How do we get to know that the present state of
affairs is exactly that which we have desired? Wittgenstein wrote that:
I believe Russell’s theory amounts to the following: if I give someone an order and I
am happy with what he then does, then he has carried out my order.
(If I wanted to eat an apple, and someone punched me in the stomach, taking away
my appetite, then it was this punch that I originally wanted.)375
Wittgenstein’s objection to Russell’s account is that whatever brings pleasure
during the course of a behavior cycle is that which was originally intended (desired,
ordered). This fact makes intentional acts indeterminate, for one cannot
know in advance all the things that might bring satisfaction. Although there
may be some kinds of intentional acts that are indeterminate in this way, most
of them are determinate. An order is a determinate affair. Its objective may happen
to be irrelevant before the order has been carried out or it may be accomplished
by some unexpected event. But if this happens, the original intention of
the order remains unaffected.
The example with the apple seems to be unfair to Russell. A punch in the stomach
usually does not cause any pleasure. The behavior cycle is only interrupted
or overlaid by another behavior cycle. Then, however, subsequent pleasure may
terminate one of these cycles, or maybe both. This empirical indeterminacy is
brought about by the conceptual indeterminacy in Russell’s theory, which does
not make clear what counts as a behavior cycle and what counts as an interruption
or termination of a cycle. In any case, by introducing additional behavior
cycles, the indeterminacy cannot be avoided.376
The crux of the matter lies in the fact that one cannot directly compare a supposed
satisfaction with the original intention. “I cannot confront the previous
expectation with what happens.”377
The intentional act lies in the past; that is
why such a comparison in Russell’s theory can be made only indirectly, based
on a possibly unreliable memory. If the causal chain within a behavior cycle
375
PR, p. 64.
376
See Hymers, 1996, p. 596 for an extended discussion.
377
PR, p. 67.
123
were broken, we would have no criterion for recognizing the original intention
at all.
11.2. Internal relation between expectation and fulfillment
According to Wittgenstein, this indeterminacy can be overcome if we take the
relation between an (expression of) expectation and its satisfaction as an internal
one: “The causal connection between speech and action is an external relation,
whereas we need an internal one.”378
Wittgenstein’s account of intentionality can be summed up by the note: “It is in
language that an expectation and its fulfillment make contact.”379
This note on
intentionality is, however, like all of the discussion of intentionality in the Philosophical
Investigations, extremely compressed. Wittgenstein’s earlier texts
from the beginning of the 1930s present (save for minor exceptions) the same
theory in a more intelligible way.
Wittgenstein calls his early 1930s account of intentionality the ‘picture conception’.
This, of course, echoes his Tractarian picture theory of meaning. The picture
conception of intentionality can now be summed up as follows: “What is
essential to intention is the picture: the picture of what is intended.”380
This
claim must not be interpreted in a literal sense. Since the Tractarian proposition
is a picture of a fact, so an expectation is (or contains) a picture of what is expected.
The picture view of meaning amounts to an internal relation between a
proposition and a fact. This view is now analogous to the picture conception of
intention: there must be an internal relation between an intention and its fulfillment.
Or: the relation between an expectation and its fulfillment is internal. Although
Wittgenstein does not subsequently use the label ‘picture conception’,
this conception of intentionality is to be found in the Philosophical Investigations
as well.381
378
PR, p. 64.
379
PI §445 & PG, p. 140.
380
PR, p. 63.
381
See §11.3 for more about the picture conception (of intentionality).
124
We know from §10.2 that internal relations are realized among concepts only.
The terms of such internal relations are, thus, not mental acts but descriptions or
expressions of these acts. Given that expectation is, prima facie, an expression
of expectation, and its fulfillment is an expression of fulfillment,382
to describe
an expectation means to describe the conditions of its fulfillment. A description
of an expectation can differ from a description of its fulfillment. There must be
a grammatical internal relation that transforms the former description into the
latter. The relation between an expectation and its fulfillment is essentially a
grammatical relation. Here are some examples:383
‘The expectation that p’ = ‘The expectation which will be fulfilled if p’(61)
‘The wish for it to be the case that p’ = ‘The wish that is satisfied by the(62)
event p’
‘The hypothesis p’ = ‘The hypothesis that is verified by the fact p’(63)
‘The proposition p’ = ‘The proposition that the fact p makes true’(64)
The proposition on the left-hand side does not need to have the same form as
the proposition on the right-hand side. Consider, for instance, the order ‘Would
you like to close the window?’ and its fulfillment ‘The window is closed’.384
The descriptions may differ in their grammatical forms, mood, or tense. This
consideration only underlines the fact that the relation between an intention and
its fulfillment has a grammatical nature.
Wittgenstein likens this internal relation to an arithmetical calculation: “From
expectation to fulfilment is a step in a calculation.”385
Consider the process of
the calculation 25 × 25 = 625. There are certain formal steps that must be carried
out in order to calculate the result from the input. The conclusion is that the
same is valid for the relation between an expectation and its fulfillment. Moreover,
both relations can be calculated with the same degree of certainty.

Now we can return to the two epistemological problems which we discussed
above. On having a particular expectation, how does one know what its satis-
382
Cf. Ms 109, p. 172; BT, p. 265e.
383
See PG, p. 161f.
384
See BT, p. 275 for other examples.
385
PG, p. 160.
125
faction will be? How does one know whether the actual state of affairs satisfies
the previous expectation? If we admit that there is an internal relation between
the descriptions of these events, then the problems will disappear. Either a single
description will hold for both events; or one description can be computed
from the other one. The trouble lies in the questions themselves, because they
suggest an external view of intentionality: “And if expectation is the thought ‘I
am expecting it to happen that p’ it is senseless to say that I won’t perhaps know
until later what I expected.”386
In order to recognize a past intention, we do not need to remember our mental
state at that moment of intention. The only thing that is remembered is the description
of the intention. If we expect rain, we have to remember this description
and apply it in the actual situation. The problem of recognition of my past
intention is, hence, the problem of applying the description in a future situation.
To put it in other terms: in order to recognize my past intention p, I have to apply
the rule for p. The internal relationship between expectation and its fulfillment
is a special case of the internal relationship between a rule and its application
(further argued in §13).

As argued in §10.5, the terms of internal relations are connected directly without
any need for any mediation by a third thing. If one term is present, the other
one must be too. Now, the fact that there is an internal relation between an expectation
and its fulfillment might suggest that if there is an expectation, there
must also be its fulfillment. This claim implies that expectations are not automatically
fulfilled, which is obviously absurd.
The internal relation in question is, of course, between the descriptions of these
events. If there is a description of an expectation, there will be a description of
its fulfillment. No mediating thing is acting that could be described independently
of the expression “the fulfillment of the expectation”:
The fulfilment of expectation doesn’t consist in the occurrence of some third thing
that, in addition to being described as “the fulfilment of the expectation”, could also
386
PG, p. 140.
126
be described as something else, i.e. as a feeling of satisfaction, for instance, or of joy
or whatever.387
This account does not rule out that there may be some feeling of satisfaction.
This feeling is, however, not a necessary part of expectation. We have to distinguish
between knowing what the possible fulfillment of an expectation would
be, and knowing that the expectation has, in fact, been fulfilled. This is analogous
to the distinction between knowing the meaning of a proposition and
knowing that a proposition is true or false. There might be a feeling of satisfaction,
if a certain proposition is true, but this feeling is not regarded as essential
to the proposition.
This account of intentionality was rejected in T. Crane’s “Wittgenstein on Intentionality
and Mental Representation”. Crane agrees with Wittgenstein that there
must be a grammatical rule that ‘the expectation that p’ is ‘the expectation that
is fulfilled by the fact that p’. But for Crane, this observation cannot suffice to
cover the entire account of how intentionality works. The expectation is that p
can be fulfilled in addition by an event e that is not grammatically related to
p.388
Crane’s example is the following story: I expect that the postman will
bring my mail in the morning. There are, unknown to me, two postmen: Mr.
Jones and Mr. Smith. My expectation is fulfilled by the postman’s bringing my
mail. It is only true on Monday that my expectation that the postman will bring
my mail is the expectation that is fulfilled by Mr. Jones’ bringing my mail. But I
have not expected this, because I did not know that Mr. Jones is the postman.
Hence, my expectation was fulfilled by a fact that I had not expected. Crane
concludes:
What this shows is that you can describe what actually fulfills your expectation […]
in a way that is independent of the description of the expectation itself. Wittgenstein’s
point, by contrast, is that you can only describe the object of the expectation in the
way it is specified in the description of the expectation itself.389
It seems that Crane is wrong. Wittgenstein’s point is not that every description
of the fulfillment of the expectation p has to be internally related to p. The fact
387
BT, p. 284e; see also PG, p. 157.
388
Crane, 2011, p. 21.
389
Crane, 2011, pp. 21f.
127
that fulfills the expectation that p (like all facts) can be described in many different
ways. Some of these descriptions (or “actions under a description” to
borrow Anscombe’s term) are internally related to p, some are not (they are related
externally to p). In order to understand the expression ‘the expectation that
p’ it is not necessary to know that p is (or apparently is) e. This may be an accidental
fact.390
Curiously enough, Wittgenstein anticipated this objection in The
Big Typescript:
“I’m looking for my cane. – Here it is!” The latter is not an explanation of the expression
“my cane” that’s essential to understanding the first sentence, and that therefore
I couldn’t have given before my cane had been found. Rather, the sentence
“Here it is”, if it isn’t a repetition of a verbal explanation that could (also) have been
given earlier, must be a new synthetic proposition.391
By the same token: “I expect that the postman will bring my mail in the morning.
— There he goes, it’s Mr. Smith!” As argued in §10.6, internal relations can
be expressed only by analytic truths. The proposition that today’s postman is
Mr. Smith is synthetic and thus cannot express an internal property or relation.
Wittgenstein’s account, based on internal relations, aims at explaining the
meaning of the expression ‘the expectation that p’ regardless of whether the expectation
has been fulfilled. This story must be not confused “with a proposition
that asserts the existence, the being, of an object.”392
11.3. Yardstick and fitting
The inquiries about intentionality focus on two related metaphors or analogies
that Wittgenstein used. The first analogy is that expectation is like a yardstick or
a measuring rod [Maßstab] for measuring subsequent events to judge whether
they satisfy it. Expectation and its fulfillment come together like a yardstick
measures an object. The second analogy is that expectation and its fulfillment
fit [passen] together in pieces like a cylinder and a piston do. These analogies
390
Crane’s demand implies that in order to understand the meaning of a word one needs to
know all the connotations of the word (all the descriptions that accidentally refer to the
same object). This is an unrealistic account of understanding.
391
BT, pp. 272f.
392
BT, p. 273e.
128
aim to propose the same view, the aim of which is in the end to explain the relation
between language and reality:
You cannot compare a picture with reality, unless you can set it against it as a yard-
stick.
You must be able to fit the proposition on to reality.393
We must not forget that these remarks belong to the early 1930s context of the
picture conception of meaning and intentionality. An expectation is a picture of
its fulfillment. ‘Picture’ is used metaphorically here. This pictorial relationship
can also be likened to the relation between a yardstick and the object measured.
The core of the analogy is obviously the idea of comparing two things. A yardstick
is compared with an object in order to find out its length. Both the yardstick
and the measured objects are material things; expectation and its fulfillment
do not need to be.
The internal relation between expectation and fulfillment, or between a description
of an expectation and a description of its fulfillment, is analogous to the
internal relation between a yardstick and a measured object. And once again,
this relation holds between descriptions of these objects. In this sense, there is
the internal relation of having the same length between the length of the standard
meter and the length of all other objects that are one meter long. Wittgenstein
concludes from these considerations that a yardstick and the object measured
must have something in common. This implies that a single description
holds for both.394
The same can be true of expectation and its fulfillment. Let an expression of an
expectation be ‘I expect that p occurs’ and an expression of its fulfillment be ‘p
has occurred’. Then “p is—in the strictest sense—what is common with a yardstick
and the object measured.”395
This statement should not be understood in
the sense that there lies some third thing between an expectation and its fulfillment.
All that this analogy suggests is that the same expression p is involved in
both descriptions of these two events.
393
PR, p. 77.
394
Ms 108, pp. 211f.
395
Ms 109, p. 52, my trans. “p ist — im strengsten Sinne — das Gemeinsame zwischen
Maßstab und Gemessenem.”
129
I would like to point out a lingering possibility of misunderstanding here. If a
single description (like ‘to be one meter long’) held for the standard meter and
also for an object that is indeed one meter long, we could then say of the standard
meter that it is one meter long. This is something that Wittgenstein clearly
denies later in the Philosophical Investigations, §50. Although the same expression
(‘to be one meter long’) is involved in descriptions of these objects, it cannot
be applied to them in the same sense. ‘My table is one meter long’ is an empirical
proposition. ‘The standard meter is one meter long’ is the expression of a
rule. The confusion would be complete if one says
My table and the standard meter are both one meter long.(65)
Is this an empirical proposition, or an expression of a rule, or a muddle?396
Now we shall turn to the other term of the analogy: p is contained both in a description
of expectation and in a description of its fulfillment. One might think
that the fulfillment has to be somehow contained in the expectation, like a
shadow.397
Then, however, an expectation would be its own fulfillment; an expectation
would be fulfilled by the very act of expecting.398
This is absurd. We
cannot say that ‘the expectation that p’ is p. We can, however, say that ‘the fulfillment
of the expectation that p’ is p. This is an important asymmetry in the
internal relation between expectation and its fulfillment—or between an intention
and its object.
Wittgenstein says that this is the point at which the simile of a yardstick breaks
down.399
English grammar does not rule out sentences like (65). This sentence
is not prima facie meaningless, and one may imagine a context in which it
could be used without misunderstanding. But it is absurd to say that an expectation
already contains its fulfillment, and hence, it is fulfilled straight away. This
may be the reason why the analogy with the yardstick virtually vanished from
Wittgenstein’s writings after The Big Typescript, and why he afterwards spoke
of intentional phenomena mostly in terms of fitting.
396
See §16 for a detailed discussion of the standard meter.
397
PG, pp. 150f.
398
Ms 109, p. 61.
399
Ms 109, p. 61.
130
Let me attempt to generalize this point. Wittgenstein’s argumentative strategy in
analyzing intentionality is, first, to make clear the difference between conceptions
based on external relations and those based on internal relations and, second,
to assess the possible reflexive cases of internal relations. If an expectation
already contained its fulfillment and we could refer to both events by a single
description, it would be a case of the reflexive use of an internal relation. Such
reflexive cases are often nonsensical and indicate a wrong analysis of the phenomena
in question. Wittgenstein eventually realized this. His treatment of such
cases is the focus of §11.4.

The other analogy Wittgenstein employs in explaining intentionality is that an
expectation and its fulfillment fit together somehow. Unlike the previous one,
this analogy survived up until the Philosophical Investigations in almost the
same formulations.400
The initial consideration is that one might say that intentional
thoughts are something unsatisfied:
“A plan as such is something unsatisfied.” (Like a wish, an expectation, a suspicion,
and so on.)
By this I mean: expectation is unsatisfied, because it is the expectation of something;
belief, opinion, is unsatisfied, because it is the opinion that something is the case,
something real, something outside the process of believing.401
This metaphor of (non-)satisfaction resembles Frege’s account of functions as
being unsaturated. In what sense, then, might expectation be unsatisfied? Wittgenstein
considers two answers: the first one is based on external relations.
Hunger is a feeling of non-satisfaction which can be satisfied by providing
some food.402
Satisfaction of hunger brings quiescence or even pleasure. This
closes the behavior cycle. This view of satisfaction thus implies a causal, i.e., an
external account of intentionality. Wittgenstein is, thus, after some other sense
in which expectation is unsatisfied. The metaphor is not a causal or temporal
one, it is a spatial one:
400
The metaphor of fitting also plays an important role in Wittgenstein’s later reflections on
the philosophy of psychology. This role is investigated in §17.1.
401
PI §438.
402
PI §439.
131
What is our prototype of nonsatisfaction? Is it a hollow space? […] For example, if
we lay it down that we call a hollow cylinder an “unsatisfied cylinder” and the solid
cylinder that fills it “its satisfaction”.403
We can also have two cylinders, or a cylinder and a piston (which is, in fact, a
solid cylinder). We can speak of satisfaction if the piston fits into the cylinder.
Now we can restate the analogy: the expectation and its fulfillment fit together
like a cylinder and a corresponding piston.
There is, however, a certain danger of taking this analogy too literally. Cylinders
and pistons are material objects. The relation of fitting between them is an
external relation. Consider the following investigation into the grammar of ‘to
fit’:
Exercises: (1) When is a cylinder C said to fit into a hollow cylinder H? Only while C
is stuck into H? (2) Sometimes we say that C ceased to fit into H at such-and-such a
time. What criteria are used in such a case for its having happened at that time?404
Does the sentence ‘C fits into H’ express an internal relation or an external relation?
We can employ the criterion of temporality from §10.4. If we can say that
C ceased to fit into H, then the sentence ‘C fits into H’ expresses an external
relation. And as we already know, internal relations only hold between concepts.
Accordingly, we have to look for the appropriate descriptions of the objects,
if their fitting is supposed to be an internal relation. Obviously, if two cylinders
fit together, they must have—at least partially—the same shape. This
leads to the idea that “when one wants to describe these two one sees that, to
the extent that they fit, a single description holds for both.”405
Our analogy
yields, thus, that expectation and its fulfillment fit together insofar as a single
description holds for both. This is, however, misleading, for the distinction between
expectation and its fulfillment must also be preserved.
It is actually not quite correct that a single mathematical description holds for
both a cylinder and a piston with respect to whether they fit. If the cylinder can
be described by the convex function p, then the piston is described by the corre-
403
PI §439.
404
PI §182.
405
PG, p. 134 & Z §54. See also Ms 108, pp. 211f.
132
sponding concave function –p. Now we can apply this consideration to both expectation
and fulfillment:
The expectation of p and the occurrence of p correspond perhaps to the hollow shape
of a body and the solid shape. Here p corresponds to the shape of the volume, and the
different ways in which this shape is given correspond to the distinction between expectation
and occurrence.406
This remark restates the point made earlier: although the same expression p is
involved in the descriptions of both objects, it is not exactly the same description.
The expectation p is not the same state of affairs as p itself, which is its
fulfillment.

We can conclude this section by saying that both analogies aim to explain the
same insights into the nature of intentionality: expectation and its fulfillment
come together in language, by which we understand that a single expression is
employed in their descriptions. In this sense, they are internally related. They
are, however, not identical. It is important to point out that Wittgenstein employs
both these analogies in highlighting the internal relationships between
various other phenomena.
11.4. Intransitive intentionality
Wittgenstein offers a different, or rather a more complex, analysis of intentionality
in The Blue Book. As we saw in the previous sections, the picture conception
of intentionality was motivated by the Tractarian picture theory of meaning,
and further driven by epistemological questions concerning knowledge of
the intentional object. Wittgenstein’s focus shifted further in The Blue Book in
1934. Now he takes into consideration the use of words like ‘expecting’, ‘wishing’,
‘longing’, ‘fearing’, etc. It turns out that a single analysis would not be
sufficient for “These cases of expectation form a family; they have family likenesses
which are not clearly defined.”407
406
PR, p. 71.
407
BBB, p. 20.
133
Wittgenstein distinguishes between two kinds of use of expressions for intentional
phenomena here in The Blue Book. The analysis from the previous sections
suggests transitive uses, because intentional verbs are used transitively
here, i.e., they take a direct object: one expects something, is afraid of something,
wishes something, is longing for something, etc. In addition to this, Wittgenstein
considers the possibility that some of these verbs can be used intransitively,
i.e., without any direct object. Some cases are straightforward. One can
be afraid of something but one can also just have a feeling of fear without being
afraid of something in particular. The intransitive use of some of the other verbs
is harder to conceive of.408
Let us have a look at Wittgenstein’s most prominent
case of intentionality, namely: expectation.
Are there also cases of intransitive expectation? There are indeed: “There is a
totally different use of the word ‘expectation’ if we use it to mean a particular
sensation.”409
We can use the word ‘expectation’ to refer to a feeling of tension
or exultation without referring to what is expected. Such use can be made explicit
by the expression ‘the sensation of expectation’. Moreover, Wittgenstein
also considers the case of “the sensation [of] the expectation that B will
come”410
. Does this expression refer to the sensation of the expectation or to the
state of affairs that B will come? In this case, B is not an argument of the function
‘expecting that x will come’. It is rather an index411
that alludes to a certain
kind of expectation. The subordinate clause is a much closer specification of the
sensation. The ‘expectation that B will come’ is, thus, a characteristic of certain
sensations (e.g., expecting someone is associated with positive feelings). So although
the surface grammar suggests the transitive use, the expression ‘the sensation
of the expectation that B will come’ can be used intransitively.
This is clearly a concession to Russell’s analysis. If the phrase the ‘expectation
that B will come’ is used intransitively, then we cannot be sure what its fulfill-
408
It would be difficult to imagine a belief or a hypothesis without specifying its content.
409
BBB, p. 20.
410
BBB, p. 21.
411
For the difference between indexes and arguments, see TLP 5.02. Using an expression as
an index creates an opaque context. If we substitute the name ‘B’ with a different coreferring
name, the new clause does not need to characterize the same kind of sensations.
This is not the case if we use an expression as an application of a function to its argument.
See BBB, p. 21.
134
ment is until the behavior cycle has been closed. Wittgenstein adds, however,
that such intransitive uses of the word ‘expectation’ are not very common.412
I
want to point out here one important difference between Russell’s analysis of
intentionality and Wittgenstein’s intransitive cases. Russell’s account of desire
is based on external (or more exactly, on causal) relations. Wittgenstein’s analysis
of the intransitive uses of intentional expressions does not invoke any relation
at all. There is no semantic reference to anything in the future mediated by
a causal chain.
There might be a temptation to explain intransitive cases in a transitive way
with an unknown (or unconscious) intentional object. “Whenever before we
said ‘I have a sensation of fear’ (intransitively) we will now say ‘I am afraid of
something, but I don’t know of what’.”413
There are indeed such cases of ignorance.
What Wittgenstein is opposed to is accepting such postulating of an unknown
object for a general explanation for intransitive uses of intentional idioms.
The problem is actually that the verb ‘to know’ is used in a non-standard
way in expressions like ‘an unknown/unconscious object of fear’. It is difficult
to explain how one gets to know what the intentional object was. Suppose I had
an undirected feeling of fear and later on, I get to know what I was afraid of.
Then we are facing the second epistemological problem from §11.1: how does
one get to know that the present state of affairs is exactly that which one was
afraid of? This suggests that the postulation of an unknown intentional object
does not explain the cases of intransitive intentionality, because “the difference
which [we have] tried to explain away reappears when we carefully consider
the use of the word ‘to know’ in the two cases.”414
To make the point clear, the
relation between an intentional state and its unknown intentional object is external.
In order to establish the connection, there must be some third thing,
namely the recognition (“getting to know”415
) of a previously unknown object.

412
This is, of course, an empirical fact, and Wittgenstein offers only his guess that it is true.
The frequency of intransitive uses may vary for different kinds of intentional expressions.
413
BBB, p. 22.
414
BBB, p. 29.
415
BBB, p. 23.
135
Wittgenstein’s analysis of intentionality from The Blue Book is, in the sense described
above, more complex than his later ‘official’ account from the Philosophical
Investigations. Although he mentions intransitive or undirected uses of
intentional expressions in the Philosophical Investigations416
, there is a more
detailed discussion that appears in the Remarks on the Philosophy of Psychology:
here “‘expect’ can mean: to believe that this or that will happen—but also:
to occupy one’s time with thoughts and activities of expectation, i.e., [to] wait
for.”417
Undirected expressions are analyzed there in terms of experiencing
meaning.418
To recapitulate: Wittgenstein’s analysis consists of two main steps: The first
step is to distinguish between internal and external relations. Most of the cases
of intentional idioms can be analyzed by an internal relation between the intentional
act and its fulfillment. In the second step, we have to consider certain
singular cases where the intentional object is missing or unknown: First, we
have to resist the temptation to posit the intentional object in the intentional act
(that is, to say that expectation contains its fulfillment), for it would be a case of
a reflexive use of an internal relation. Second, there are cases in which intentional
idioms directly express a sensation. The intentional object is not the main
focus here. We have to be careful about explaining such cases by postulating an
unconscious intentional object. Such an analysis faces similar epistemological
problems to the one based on external relations.
416
PI §§574 & 586.
417
RPP II, §154.
418
RPP II, §§2–5.
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12. Reason, motive, and cause
In the previous chapter, Russell’s causal conception of intentionality was confronted
with Wittgenstein’s conception based on internal relations between an
intentional action and its object. We can now slightly modify the terminology
which allows us to put the problem discussed above into a broader context. We
can say of an intentional action that it has a reason or a motive; and in the same
vein, such an action must have a cause. Let us assume that I am about to perform
an action p in order to achieve q; that is to say, q was my reason for doing
p. Therefore, I am about to do p because of q; thus here, q was my motive for
doing p. So, for instance, an order to do p can be a reason for doing p; or my
fear of q is a motive for taking action in order to avoid q, etc. Independently of
this, one may ask whether q was the cause of p—or in fact what sort of causality
is operating in this example.
It is thus only a matter of terminology whether all species of intentionality can
be restated in this way—as a relation between an action and its motive.419
To be
on safer ground, one could say that the relation of being a reason for doing or a
motive for doing belongs to the family of intentional relations which Wittgenstein
aims to conceive as internal relations. As he stresses, the words ‘reason’,
‘motive’ or ‘cause’ can be used in very many different ways.420
The same is valid
for the related expressions ‘because’ or ‘why’, etc. The diagnosis is, then,
that the surface grammar of our everyday language confuses us about (or at
least does not fully distinguish between) internal and external relations.421
In
what follows I shall argue that the distinction between reasons and causes is another
instance of the distinction between internal and external relations.
To begin with, let us consider the following examples from Wittgenstein’s Lectures
and Conversations on Aesthetics:
‘Cause’ is used in very many different ways, e.g.
419
As Anscombe (1957) does.
420
LA, pp. 13 & 22; Ms 112, p. 112v; BBB, p. 15; VW, pp. 108–111. I think that Wittgenstein
uses the expressions ‘reason’ [Grund] and ‘motive’ [Motiv] interchangeably (Cf. BBB,
p. 15).
421
Cf. RPP II, §909: “We remain unconscious of the prodigious diversity of all of our everyday
language-games because the outward forms of our language make everything alike.”
137
(1) “What is the cause of unemployment?” “What is the cause of this expression?”
[Experiment and statistics]
(2) “What was the cause of your jumping?” “That noise.” [Reason]
(3) “What was the cause of that wheel going round?” You trace a mechanism.422
In order to avoid a misunderstanding Wittgenstein wants to reserve the expression
‘cause’ for a (relation of) mechanical causality between two events. A
cause in this sense can be found statistically or by tracing the underlying mechanism.
This is to say that what is the cause of a certain action is always a hypothesis
based on past experience. Such experience may include the knowledge
of certain physical processes in one’s brain which are typically not known to an
agent. An important consequence is that one cannot be absolutely sure what exactly
the cause of one’s action was. It should therefore be clear that causal relations
are external: they are realized between events, not concepts; they are expressed
in temporal propositions.423
The most striking difference between a cause and a reason/motive for Wittgenstein
is that an agent knows without any doubt the motive of their action: “we
can only conjecture the cause but we know the motive.”424
Wittgenstein takes
this statement to be a grammatical one. The motive for an action or the reason
for a belief is something constitutive of the very action or belief:
The causes of our belief in a proposition are indeed irrelevant to the question [of]
what we believe. Not so the grounds, which are grammatically related to the proposition,
and tell us what proposition it is.425
Now, I want to address two interrelated points: The first one concerns what
counts as a motive, or as a reason. A rational motive or a reason cannot be just
anything that an agent avows. The second point is the objection that a motive
can be unconscious, i.e., unknown to an agent. One may later forget the original
motive for one’s action or be self-deceived or insincere about it. Both these
422
LA, p. 13. The bracketed postscripts are by James Taylor.
423
See §§10.2 & 10.4.
424
BBB, p. 15. Cf. Ms 113, p. 57r; Ts 302, p. 19; VW, p. 109.
425
Z §437; see also BT, p. 209e. In an earlier form, this remark reads “internally related”
instead of “grammatically related”. See Ms 113, p. 43r & Ts 211, p. 591.
138
points threaten my claim that the relation of being a motive or a reason is inter-
nal.
Now to the first point: a motive for an action or a reason for a belief is not arbitrary.
If the relation between an action and its motive is not obvious, the agent
has then to indicate a rule that has led them—step by step426
—from the motive
to the action. The motive can itself be an expression of this rule. In the 1930s,
Wittgenstein pondered the idea that this rule must be a kind a calculation: “Giving
a reason is like giving a calculation by which you have arrived at a certain
result.”427
This statement means that between an action and its motive there is
the same kind of relation as between a mathematical equation and its result.
Note that the same kind of relation holds between both expectation and fulfill-
ment.428
As argued in §10.5429
this relation is internal.
A slightly different account of this relation is to be found in the second part of
the Philosophical Investigations, and in subsequent writings.430
The relation between
an action and its motive is established here in the language-game of the
judging of motives. All that is needed is a technique for the judging of a motive.
A judgment within this language-game may resemble a calculation, but it does
not need to. We can think of some simple instances of judgments and take these
as sorts of measuring rods in order to judge cases that are more complicated.431
This later account of the relation of being a motive is, thus, the generalization of
the calculation-model from the 1930s. What is important here is that the relation
between a motive and an action that an agent performed is an instance of the
relation between a rule and its application. This relation must be internal as I
shall argue in §13.
As to the second point: an agent might avow a different motive for their action
than the real one (it may be a case of ignorance or self-deception or a lie). As
argued above, knowledge of a cause is always hypothetical—as opposed to a
motive/reason. But it seems now that a motive can also be hypothetical in the
426
Cf. Ms 115, p. 136.
427
BBB, p. 15. See also BT, p. 296e & Ts 302, p. 19.
428
See p. 33.
429
See p. 24.
430
PI II, p. 224 & RPP I, §631.
431
RPP I, §633.
139
sense that it is determined by the agent’s sincere avowal.432
There is a certain
confusion lurking here, for ‘motive’ or ‘reason’ can be ambiguous here. A reason
may mean the actual reason or may mean any possible, hypothetical reason:
sometimes what we say acts as a justification, not as a report of what was done, e.g. I
remember the answer to a question; when asked why I give this answer, I gave a process
leading to it, though I didn’t go through this process.433
We have to distinguish between a report of an actual or past motive and a justification
of the action. The point of a report is that it should be sincere. When
someone is asked for their actual motive, they should report their motive truthfully
and the answer depends on their sincerity (and on their memory). But
something different goes on when the agent is asked for a justification. Then it
does not matter what the past motive was. All they need to give is a rule of
which the present action is an instance. It does not matter whether the agent had
really followed this rule.434
In §§10.2 & 10.4, there are several examples of sentences that are ambiguous
between expressing internal or external relations. The same is true of the following
kind of sentences:
p is the motive for doing q.(66)
p is the reason for believing q.(67)
If these sentences are reports of an actual motive or reason, they can be restated
as being explicitly temporal:
p was my motive for doing q.(68)
p is the reason why I now believe that q.(69)
According the criterion of temporality (§10.4), these sentences express external
relations. Asking for a justification is something different. In this case, (66) and
(67) are timeless and could be restated as:
p is a possible motive for doing q.(70)
432
See Glock, 1994, p. 76.
433
LA, p. 22.
434
See VW, p. 111: “the reason is what he specifies. He answers with a rule. He could have
also given this rule if he had not gone by it”.
140
p is a possible reason for believing q.(71)
Again, following the criterion of temporality, these sentences express internal
relations. I would like to elucidate this matter further by Wittgenstein’s analogy
with a route:
The question ‘Why do you believe that?’ can be compared with the question ‘How do
you come to be here?’.435
Wittgenstein says that this question allows two answers. There are, in fact,
however three answers to be found in Wittgenstein’s lecture notes. (1) The first
answer consists in giving the physical or psychological cause of one’s being located
here. This answer will have to describe various phenomena (e.g., stimuli,
reflexes, connections of pathways in one’s nervous system, etc.), the circumstances
in which they occurred, and the causal laws operating here. (2) The second
answer would be specifying the way I actually went here. (3) The third answer
is by giving any route that I could have got here by.436
The first answer
corresponds to giving the actual cause, the second one to a report of the actual
reason, and the third to a justification by giving a possible reason. The first answer
expresses an external relation and the last one expresses an internal relation.
In the second answer, there is expressed an external relation be means of
an internal one.
12.1. The Principle of Sufficient Reason
In this and in the following section, I am going to investigate the various philosophical
implications or applications of the method outlined above. One, and
for Wittgenstein maybe the most important implication, has already been discussed
in §11.1; namely, the implication for Russell’s causal theory of intentionality
and of meaning in general.437
I will, in turn, therefore discuss Wittgen-
435
VW, p. 47. See Ts 302, p. 19.
436
Wittgenstein employed this analogy several times. He considered the first and the second
answer at VW, p. 47 and the second and the third one at LA, p. 22.
437
For the critique of causal theories of meaning, see PG, p. 60: “Meaning, in our sense, is
embodied in the explanation of meaning. If, on the other hand, by the word ‘meaning’ we
141
stein’s denial of the Principle of Sufficient Reason and his (if not actual denial
then at least) doubts about the psycho-physical parallelism of mind. Finally, I
will look at Donald Davidson’s influential and explicitly anti-Wittgensteinian
account of subsuming rational relations under causal relations.
Wittgenstein mentioned the Principle of Sufficient Reason in his early manuscripts
and in a letter to Russell from 1914.438
Curiously enough, in this letter,
he equates the principle with the law of causality. In The Blue Book, Wittgenstein
considers the idea that there must be “a chain of reasons reaching back to
infinity.”439
This is, I claim, a formulation of the principle which is traditionally
conceived of in the following wording:
For every fact F, there must be a reason why F is the case.440
(72)
The history of the principle goes back to pre-Socratic philosophy. Its time of
glory came, however, in the 17th century. The principle became one of the central
ideas driving the metaphysical systems of Spinoza and Leibniz, who actually
coined the term “Principle of Sufficient Reason”. It is relevant to present
concerns that both these philosophers used the expression “causa sive ratio”
[cause or reason] in formulating their main principles, marking the fact that
they did not distinguish between cause and reason. Here are Spinoza’s formulations
of the principle:
Nothing exists of which it cannot be asked what is the cause (or reason)(73)
why it exists.
For each thing there must be assigned a cause, or reason, both for its(74)
existence and for its non-existence.441
These formulations explicitly do not distinguish between cause and reason. This
is deliberately so in Spinoza: because causal and rational relations have the
same root in substance, they are coextensive. Schopenhauer, who might have
mean a characteristic sensation connected with the use of a word, then the relation between
the explanation of a word and its meaning is rather that of cause to effect.”
438
NB, p. 130.
439
BBB, p. 14.
440
See Melamed & Lin, 2011. Their wording is slightly different.
441
Quoted after Melamed & Lin, 2011. The first formulation is from Spinoza’s exposition of
Descartes’ Principles of Philosophy, the second one from his Ethics.
142
inspired Wittgenstein here, in his dissertation On the Fourfold Root of the Principle
of Sufficient Reason accuses the philosophical tradition of confusing different
kinds of reasons.442
Before going into the details of Wittgenstein’s critique of the principle, I want
to pause now to extend this route analogy. The principle could amount to the
following claims:
“Wherever you are, you must have got there from somewhere else, and to(75)
that previous place from another place; and so on ad infinitum.”
“Wherever you are, you could have got there from another place ten yards(76)
away; and to that other place from a third, ten yards further away, and so
on ad infinitum.”443
(75) is analogous to the claim that the chain of actual reasons of an agent is infinite.
A chain of actual reasons is based on external relations. (75) is obviously
false. Nobody has gone on an infinite route. By the same token, (76) is analogous
to the claim that the chain of possible reasons is infinite. Even this claim is
for Wittgenstein problematic and in the end untenable.
Wittgenstein is also strongly opposed to the principle. Not all actions or beliefs
need to have a reason, but all events do have causes. The principle is the conclusion
of the following fallacious argument:
(P1) Causal chains are infinite.
(P2) Chains of reasons are causal chains.
(C) Chains of reasons are infinite.
Wittgenstein takes the first premise, i.e., the principle of causality, to be a
grammatical rule of the language-game of mechanics.444
The second premise is
442
Spinoza’s conception of causality is, however, more intricate. He maintains a kind of rudimentary
picture theory, namely the view that causal relations between facts presuppose
logical relations between propositions that describe these facts. Cause presupposes reason.
Facts are, however, not wholly independent for Spinoza; they are not contingently related
(in contrast to Wittgenstein’s atomic facts). There is an essential union of all events and
facts. These two claims imply that all relations are partly internal and partly external. Bradley
followed Spinoza in this respect. See §4.2. Thanks to Modesto Gómez Alonso for elucidating
Spinoza’s account of causality to me.
443
BBB, p. 14.
143
just an expression of the confusion of cause and reason. The language-game of
giving reasons is different from that of mechanics. “A reason can only be given
within a game. The links of the chain of reasons come to an end, at the boundary
of the game. (Reason and cause.)”445
There are propositions that cannot (or
do not need to) be justified within a given language-game: “And this again joins
on to the confusion between cause and reason. We need have no reason to follow
the rule as we do. The chain of reasons has an end.”446
The expressions of
rules belong to the propositions that we do not need to give reasons for—in a
given language-game. This is connected to the central question of the rulefollowing
discussion:
“How am I able to obey a rule?”—if this is not a question about causes, then it is
about the justification for my following the rule in the way I do. If I have exhausted
the justifications I have reached bedrock, and my spade is turned. Then I am inclined
to say: “This is simply what I do.”447
The main substance of this remark is the justification for following a rule. If
one aims to justify one’s rule-following, one has to give a reason for it. But following
a rule is constitutive for this very rule.448
In other words: a rule is constituted
by the praxis of its following. The reason why you follow this rule in this
way and not in that way is internally related to the reason why you follow this
rule and not that rule.
If a rule is the reason for an action or a belief, there is no reason for the rule itself.
Language-games with their rules are expressions of human praxis. This is
the bedrock; this is simply what we do. There is, nevertheless, a temptation to
transgress the praxis and ask further for a reason for the language-game itself.
Why exactly this or that language-game?
There are several ways to confront this temptation or answer this question. It
would be too easy to say that beyond the boundary of a language-game, i.e., beyond
the bounds of sense, there is simply nonsense. This attitude amounts to a
444
AWL, p. 16.
445
PG, p. 97.
446
BBB, p. 143.
447
PI §217.
448
See Ch. 13 for more details.
144
refutation of any answer to the question of why. I would like to offer three possible
answers.
Firstly, there is always a causal explanation for why we employ this or that language-game.
I have in mind historical, etymological, or evolutionary explanations,
etc. This kind of answer is based, of course, on an external relation with
all its disadvantages over an answer that is internally related to the languagegame
in question.
Secondly, by attempting to answer the question, i.e., by attempting to justify a
rule, one could step into another language-game. What is an expression of a rule
in one language-game can be an empirical proposition in another game. A certain
sentence can express an empirical proposition in the language-game of
rule-teaching; the same sentence will then express a rule as soon as the application
of the sentence has been mastered. Between these games there is a vertical
relation.449
It may happen that there is no other language-game where a rule can
be justified. If so, one can just invent or construct one. What else are metaphysical
systems if not invented attempts to justify our everyday praxis? The chain
of reasons may then only be infinite if we keep constructing additional language-games
in which the chain can continue.
The third way to justify a rule is in fact Wittgenstein’s own proposed answer
given in the quotation above: “This is simply what I do.” This is a reflexive
construction “which masks the beginning of the chain of reasons.”450
Similar
constructions are not uncommon in philosophy: consider, e.g., the Spinozian
concept of causa sui, or the biblical description of God: “I am that I am”451
. To
pick up on the general threads of this book, we have here again the reflexive use
of an internal relation. Such uses are not necessarily nonsensical. But we have
to be aware that they mark the end of a chain of reasons, i.e., a terminus ad
quem of a justification.
449
Cf. §10.3.
450
PG, p. 111.
451
Exodus 3:14.
145
12.2. Psycho-physical parallelism, super-mechanism and the mind
There are various kinds of relationships between mental and physical phenomena.
We can speak of their isomorphism (viz. they have the same structural properties),
parallelism (viz. they occur in tandem) or even identity (viz. the mind
and body are one and the same thing as claimed by Spinoza452
). Wittgenstein
offers no knockdown argument against these metaphysical conceptions of mind,
though. His aim is rather to show that these conceptions are unnatural and ultimately
unjustified, and that our inclination for their validity results from a primitive
conception of grammar which confuses cause and reason.
The language-games of giving reasons/motives (reasoning) and of giving causes
(mechanics) are distinct activities. It may, however, seem that they share the
same realm. This is to say that they are different ways of referring to the same
phenomena, which here are mental states. One difference might be the one between
the first-person and the third-person perspectives on mental states. Wittgenstein
is opposed to this idea:
A motive is not a cause “seen from within”! Here the simile of “inside and outside” is
totally misleading – as it so often is. – It is taken from the idea of the soul (of a living
being) in one’s head (imagined as a hollow space). But this idea has been mixed with
other incompatible ideas, like the mixed metaphors in the sentence: “The tooth of
time that heals all wounds, etc.”.453
Let me try to unravel this from the end. This metaphorical talk of time contains,
in fact, two metaphors: ‘time has teeth’454
and ‘time heals all wounds’. Now
Wittgenstein claims that these metaphorical ideas are incompatible. Time’s teeth
leave marks of disintegration and thus collapse. We can see such marks at castle
ruins. The metaphor of healing wounds suggests, however, that something broken
will be put back together again. The whole metaphor thus implies both integration
and disintegration. The view that a motive is a cause seen from within
also contains two incompatible ideas: the mind is a mechanism and the mind is
something inside the head. The former idea is of a mind that is a mechanism
452
Ethics, 3p2s.
453
BT, p. 296e. Cf. Ms 138, p. 23b.
454
The figurative German expression ‘der Zahn der Zeit’ (literally ‘the tooth of time’) is
usually translated non-figuratively as ‘the ravages of time’.
146
consisting of one’s brain (or the head or the whole body). The latter takes the
mind as hovering in a hollow space and perceiving the mechanism from within.
The mind cannot be, then, both a mechanism and located inside this same
mechanism. This, therefore, is how these ideas are incompatible with each other,
which completes the argument for the claim that a motive is also not a cause
as seen from within.

However, Wittgenstein gives us some independent arguments against these two
conceptions of mind. My present concern here is with Wittgenstein’s argument
against the mechanistic model of the mind. He has, in fact, a lot of arguments in
this respect and so I will focus on the one related to the main subject of this
book, namely the difference between internal and external relations.
Taking the mind as a mechanism or as a machine means conceiving of the language-game
of mechanics as superior to other language-games. The mechanistic
model does not need to imply directly any kind of identity theory as discussed
above. One has to explain, however, the way in which other languagegames
are based on (or grow out of) the language-game of mechanics. All these
language-games are expressions of human practices. We have to explain in particular
how it is possible that we are able to be involved in the practices or techniques
of reasoning, inferring, intending something, or calculating, all of which
are based on internal relations. In short, how is it possible to achieve the superrigidity
of internal relations out of the underlying causal, i.e., external relations?
Internal relations hold with logical necessity or super-rigidity, whereas external
relations do not. Moreover, internal relations hold only in virtue of their precise
terms.455
To know one term amounts to knowledge of the other terms of an internal
relation. If the mind were a mechanism, we would necessarily know at
least some future states of this mechanism:
The machine as symbolizing its action: the action of a machine—I might say at
first—seems to be there in it from the start. What does that mean?—If we know the
455
Cf. §10.7.
147
machine, everything else, that is its movement, seems to be already completely de-
termined.456
A present state of the machine and its future state are related externally by the
causal relation. We assume, however, that the machine could compute internal
relations. Then, however, its consecutive states have to be related internally as
well. We can denote these states of the machine M as s1 and s2 and consider the
following conditional description of M:
If M is in state s1, then its next state must be s2.(77)
As established in §10.6, internal relations hold without exceptions. Does (77)
hold without any exception? It is obvious that (77) holds unless the machine
malfunctions:
If M is in state s1, then its next state must be s2, if M is working correctly.(78)
If (77) has to be understood as (78), then no internal relation is expressed. But
the saving clause would not be necessary if the machine could not have a malfunction.
We have arrived, thus, at the idea of a machine that cannot have a malfunction—the
idea of a perfectly rigid mechanism,457
of a super-mechanism.458
If such a machine could not break down, it would have to contain all of its future
movements and states, i.e., they would have to “be really—in a mysterious
sense—already present.”459
Wittgenstein also argues that the idea of a supermechanism
is confused: “‘People say there is a super-mechanism, but there
isn’t.’ But no one knows what a super-mechanism is.”460
The upshot of this argument
is thus: we wanted to explain the mind by using something that we are
familiar with, but the picture has been mixed up by something unknown or even
mysterious.
We have to distinguish here between an actual machine (which is able to perform
a computation) and an ideal machine, e.g., a Turing machine computing a
recursive function. Actual machines are liable to malfunction, but so too are
456
PI §193.
457
LFM, p. 196.
458
LA, pp. 15ff.
459
PI §193.
460
LA, p. 15.
148
human minds. But if the mind were a machine, the notion of malfunction would
not make any sense. There would not be any authority that could detect a malfunction
and correct the actual machine. The correct way of functioning would
be defined by the actual working of the machine. The actual machine therefore
lacks normative force.461
If the mind were such a machine, it would lack normative
force too.
There is a close connection here to the problem of rule-following which is going
to be discussed in the next chapter. Suppose that a machine has to compute
the plus function, but the machine computes that ‘68 + 57’ is 5. What then? Or
who could be the judge here? How could we decide whether the machine is
computing the plus function, but malfunctioning, or computing another function,
say quus? Or it could be another machine that does the judging. But other
machines are liable to malfunction in the same way. Or is it agreement among
several machines? This question is analogous to the one of whether an appeal to
communal agreement can decide the correct application of a rule in a novel sit-
uation.
These problems lead us to a very simple question: what internal relation is exhibited
in a given machine and how do we know it? Frege pointed out that it is
impossible to ascribe a unique number to a pile of playing cards.462
This means
in our terms here that material objects such as a pile of cards have no internal
properties. Frege’s strategy of escaping from Russell’s paradox was to ascribe
numbers to concepts. Wittgenstein’s insistence that internal properties can only
be ascribed to concepts and that internal relations hold between concepts also
draws on this Fregean idea.
Now we can pursue this idea a little bit further. The so-called triviality arguments
against functionalism in the philosophy of mind argue that every physical
machine (of a certain complexity) implements every possible computation.463
Every series of movements of a machine can be interpreted as an implementation
of any internal relation. This means, however, that a machine on its own
cannot implement every internal relation unless “we can already presuppose a
461
Cf. Livingston, 2010, §IV.
462
Frege, 1980, §55.
463
See Searle, 1990; cf. Livingston, 2010, fn. 34.
149
distinction between the correct and incorrect functioning of the machine.”464
Either the mind-machine is a super-machine which cannot break down, or we
have to provide an independent account to explain its normativity. Thus, the
idea of the mind as a mechanism cannot be the whole account of the human
mind.

In the remaining part of this section I shall briefly focus on a very weak version
of psycho-physical parallelism. It is based on the idea that the actual processes
of reasoning must make a physical difference in the agent’s brain. Wittgenstein
is, however, not willing to accept this form of parallelism either:
903. No supposition seems to me more natural than that there is no process in the
brain correlated with associating or with thinking; so that it would be impossible to
read off thought-processes from brain-processes. I mean this: if I talk or write there
is, I assume, a system of impulses going out from my brain and correlated with my
spoken or written thoughts. […] The case would be like the following—certain kinds
of plants multiply by seed, so that a seed always produces a plant of the same kind as
that from which it was produced—but nothing in the seed corresponds to the plant
which comes from it; so that it is impossible to infer the properties or structure of the
plant from those of the seed that it comes out of—this can only be done from the history
of the seed. So an organism might come into being even out of something quite
amorphous, as it were causelessly; and there is no reason why this should not really
hold for our thoughts, and hence for our talking and writing.
904. It is thus perfectly possible that certain psychological phenomena cannot be investigated
physiologically, because physiologically nothing corresponds to them.465
What is Wittgenstein’s argument here and how is this argument related to the
distinction between cause and reason? Several commentators have pointed out
that Wittgenstein is arguing here inter alia against Wolfgang Köhler’s trace theory
of memory.466
I think we can take Wittgenstein to be arguing against the
more radical view which Klagge calls the denial of supervenience, i.e., the view
that “there could be a difference in memories, or resulting plants, without any
464
Livingston, 2010, fn. 34.
465
RPP I, §§903 & 904.
466
Cf. Hacker, 1996a, pp. 496–503. See Klagge, 2011, pp. 101ff. for an in-depth discussion.
This section draws mostly on §8 of Klagge’s book.
150
difference in brains, or seeds.”467
Let me first analyze the following claim about
plants:
If two plants are different, there must be a difference in their seeds.(79)
It is, however, perfectly possible that different plants grew up from the same
seeds. Their growth is influenced by their environment and these influences
may be responsible for their difference. Wittgenstein is considering here, in
fact, a stronger claim:
If two plants are different, there must be a difference in their histories.(80)
I take the history of a plant to be its growth from a seed to its present form.
Now, (80) seems to be analogous to:
If two memories are different, there must be a difference in the underlying(81)
brain states.
What kind of compulsory force is expressed by the ‘must’ in these sentences? Is
it a logical compulsion? These sentences are timeless and thus, according to the
criterion of temporality, they should express internal relations. As noted
above,468
Wittgenstein takes the law of causality to be a grammatical rule.
Something like this must be true of (80) and (81) too. The compulsion of the
‘must’ is also the compulsion of a grammatical rule. If (81) were a rule, then
which language-game would it belong to? Wittgenstein is reported to have
made the following statement which might be helpful here: “Now (today) we
have every reason to say there must be a difference.”469
We can thus take (81) to
mean:
If two memories are different, there is a reason to say there must be a(82)
difference in the underlying brain states.
Let me call this the principle of causal mediation. This wording suggests that
the rule belongs to the language-game of reasoning. On the other hand, the rule
states that causality between psychological phenomena must be mediated physiologically.
This is to say that the rule must belong to the language-game of me-
467
Klagge, 2011, p. 102.
468
§12.1.
469
UW, p. 411.
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chanics. There is an obvious conclusion: (82) is crossing over different language-games.
If so, if (82) were a rule of both these language-games (that is,
those of reasoning and of mechanics), we would have to give up the principle
that internal relations hold between concepts only.
The true status of the principle of causal mediation is that of an ideal: “There is
an ideal—a direction in which investigations are constantly pushed. ‘There
must be’ corresponds to this ideal.”470
This declaration has a certain Kantian
flavor and I would like to argue that Wittgenstein follows—maybe unknowingly—in
Kant’s footsteps. Wittgenstein sees the source of the temptation to follow
the principle of causal mediation in a primitive conception of grammar:
The prejudice in favour of psycho-physical parallelism is also a fruit of the primitive
conception of grammar. For when one admits a causality between psychological phenomena,
which is not mediated physiologically, one fancies that in doing so one is
making an admission of the existence of a soul alongside the body, a ghostly mental
nature.471
What is the primitive conception of grammar that leads to an admission of the
existence of a Cartesian soul detached from the body?472
I think that the primitive
conception of grammar amounts to taking grammar for something other
than a conceptual scheme expressing human practices, activities, and techniques.
It amounts to mistaking grammatical rules for metaphysical statements
answerable to the essence of reality. In short, the primitive conception of
grammar is such that it is not autonomous. Only if we took the grammatical
rules of the language-games of reasoning and mechanics for metaphysical principles
would we be tempted to postulate any unifying principles (like the principle
of causal mediation) underpinning these games and hence unifying the
realms of reality which correspond to them.

470
UW, p. 411.
471
RPP I, §906.
472
Wittgenstein might have had in mind Spinoza’s psycho-physical parallelism as a reaction
to Descartes’ dualism of soul and body.
152
This line of argument finds its predecessor in Kant’s Antinomy of the Judg-
ment.473
Kant considers this antinomy in the context of explaining living organisms
by purely mechanical laws. Can living organisms, and human beings especially,
be explained by causal laws only? Or do we need another kind of explanation?
He states this antinomy in two variants. The first variant is presented by
the following two maxims:
Thesis: All production of material things is possible in terms of merely mechanical
laws.
Antithesis: Some production of material things is not possible in terms of merely mechanical
laws.474
These theses obviously contradict each other, because the antithesis is a negation
of the thesis. They are irreconcilable because they are “constitutive principles
concerning the possibility of objects”475
. A principle is constitutive in this
sense if it is an a priori presupposition of our experience. It is a necessary condition
of possible experience. For the antinomy to be resolved, we have to convert
these principles into regulative maxims which then would read:
The first maxim of judgment is this thesis: All production of material things and their
forms must be judged to be possible in terms of merely mechanical laws.
The second maxim is this antithesis: Some products of material nature cannot be
judged to be possible in terms of merely mechanical laws. (Judging them requires a
quite different causal law – viz., that of final causes.)476
Taking the principles as regulative maxims of judgment does not solve this antinomy;
it is rather the first and preliminary step towards its resolution. As regulative
principles, these maxims do not discredit any mechanical explanation of
living organisms; they actually help us in our search for genuine mechanical
explanations.477
There must be, in the end, a mechanistic explanation of nature
(living organisms included) which can be called knowledge. Kant’s method in
473
Kant, 1987, §69ff. We possess, however, no evidence that Wittgenstein read Kant’s Critique
of Judgment.
474
Kant, 1987, §70.
475
Kant, 1987, §70.
476
Kant, 1987, §70.
477
See, for example, Breitenbach, 2011 for details.
153
his dialectics can be summed up as the process of converting metaphysical
principles into constitutive transcendental principles (which is done mostly in
the Critique of the Pure Reason) and converting these, in turn, into regulative
maxims in order to resolve the antinomy of judgment (in the Critique of Judgment).
This process is driven by our striving for the unity of all knowledge.
I think that Wittgenstein goes even further in questioning this striving for unity.
He clearly recognized the regulative character of the principle of causal mediation.
In addition to this, he does not see any need to provide a unification of
causal explanations and explanations based on reasons and motives. That there
must be a causal explanation is only an ideal that we may (but do not need to)
pursue.478
12.3. Davidson’s causal theory of action
I am going to conclude my discussion of the distinction between cause and reason
by looking at an influential critique of it by Donald Davidson. I will argue
that Davidson’s critique does not pose any serious problem for Wittgenstein’s
position and that Davidson is guilty of several confusions that Wittgenstein was
combating. This discussion will, I hope, provide us with some deeper assessment
of the distinction between cause and reason.
Before going into the details of Davidson’s argument, let me reiterate the crux
of Wittgenstein’s position. The question ‘Why did you do action X?’ allows for
basically three kinds of answers: (1) an actual cause of X, (2) the actual reason
for doing X, and (3) a possible reason for doing X. These answers express, successively,
an external relation, an external relation by means of an internal one,
and an internal relation. Moreover, as argued in §12.2, it is not necessary that
the relation of being the actual reason is accompanied by a causal relation.
Davidson’s position is that rationalization, i.e., the explanation of an action by
giving the agent’s reason, “is a species of causal explanation.”479
What Da-
478
I see in this one of the main traits of Wittgenstein’s philosophy, which can be summed up
by a quotation from King Lear: “I’ll teach you differences.” See §1.
479
Davidson, 1963, p. 3. Cf. however Davidson’s summary of his paper “Actions, Reasons,
and Causes” where he concedes a substantially weaker claim that rationalizations “can, and
154
vidson is after is also not merely a justification of the action (a possible reason
for it), but the actual reason why it was performed.
But then something essential has certainly been left out, for a person can have a reason
for an action, and perform the action, and yet this reason not be the reason why
he did it. Central to the relation between a reason and an action it explains is the idea
that the agent performed the action because he had the reason. Of course, we can include
this idea too in justification; but then the notion of justification becomes as
dark as the notion of reason until we can account for the force of that ‘because’.480
What makes a possible reason for the action into the real reason? Davidson
claims now—“failing a satisfactory alternative”481
—that the actual reason for
the action is such that there is a causal connection between the reason and the
action. Davidson’s (unlike Wittgenstein’s) linguistic intuition is that ‘because’
suggests a causal connection. But even if Davidson were right about the meaning
of ‘because’, it would not imply that the real reason for the action is also its
cause.
If the existence of a causal connection is not, according to Wittgenstein, the distinguishing
feature of the actual reason over a merely possible reason, how can
Wittgenstein then account for this distinction? The easiest way to find out the
agent’s actual reason for their action is to ask them: “The reason may be nothing
more than just the one he gives when asked.”482
There does not need to be
any further reason,483
because the chain of actual reasons always has an end
(§12.1). Such an avowal may be disturbed by the possibility of ignorance, selfdeception,
or a lie as already argued above.484
But then, if there is a suspicion,
we have to look at the context of the action. Sometimes there is evidence that
often must, invoke causal connections” (p. xiv). If this is Davidson’s position, then there
would be no disagreement between Davidson and Wittgenstein on this matter.
480
Davidson, 2001, p. 9.
481
Davidson, 2001, p. 11.
482
AWL, p. 5.
483
Cf. Davidson’s acknowledgement of this point. One may “answer the question ‘Why did
you do it?’ with, ‘For no reason’, meaning not that there is no reason but that there is no further
reason, no reason that cannot be inferred from the fact that the action was done intentionally;
no reason, in other words, besides wanting to do it.” (2001, p. 6)
484
See p. 46.
155
there is a reason to believe that the agent’s avowal did not reveal the actual reason.
Sometimes we can use the methods of psychoanalysis to reveal the agent’s
reason for their action. In such cases, however, a different language-game is being
played.485
This means that what counts as a reason is defined by the praxis
and techniques of asking for reasons. We can, but do not need to, include psychoanalytic
methods in this praxis.
To sum up: if Davidson’s claim is that an explanation of an action by giving the
agent’s reason is a species of causal explanation, then this claim is based on an
unfounded assumption which I have called the principle of causal mediation. If
Davidson’s argument is that the existence of a causal connection is the only account
for the distinction between an actual reason and a merely possible reason,
then we can find another account in Wittgenstein. And finally, if Davidson’s
claim is that the agent’s reason is perhaps (or even often is) causally connected
with their action, then there is no disagreement between Davidson and Wittgenstein
at all here.
485
“Here there are two motives—conscious and unconscious. The games played with the
two motives are utterly different.” (LA, p. 23)
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13. Rules and their applications
Language is internally related to the world. This is the core of the Tractarian
picture theory of meaning. Wittgenstein abandoned this view at the beginning
of the 1930s. However, we must not understand this to mean that the relation
between language and the world then became external. The internal relation between
language and the world had instead been transformed into the internal
relation between a rule and its applications.
The problem is that Wittgenstein was never wholly explicit on this matter. The
claim that rules are internally related to their applications is for the most part an
exegetical construction advanced primarily by Baker and Hacker in their response
to Kripke’s rule-following paradox.486
Wittgenstein’s most explicit statement concerning the relation between a rule
and its applications is his remark from 1931:
One thing is clear, if I have (correctly) followed a rule, the result will stand in an internal
relation to the source fact and to the expression of the rule. I cannot express
this internal relation except by restating those three complexes because […] everything
is already settled in this restatement.487
In the context of this remark, Wittgenstein argues that an application of a rule to
a certain source fact (Vorlage) that leads to a certain result is not a causal process.
Wittgenstein talks about three complexes, but we can take the expression
of the rule together with the source fact (Vorlage) as one complex and the result
of the application as another complex. The rule has to match the source fact in
order to transform it into the result. Thus understood, we can then say that there
is an internal relation between a rule and its application.
Put this way, however, this statement becomes rather misleading. We must not
forget that, strictly speaking, the expression of the rule applied in a certain sit-
486
See Baker & Hacker, 1984 & Kripke, 1982.
487
“Eines ist klar, daß wenn ich der Regel (richtig) gefolgt bin das Resultat zu der Vorlage
und dem Ausdruck der Regel in einer internen Beziehung stehen wird, die ich nicht anders
ausdrücken kann, als durch die Wiedergabe jener drei Komplexe, weil [...] in dieser Wiedergabe
allein schon alles bestimmt ist.” (Ms 109, p. 291; my trans)
157
uation stands in an internal relation to the result of this application. An example
may help here. Suppose we have a rule which orders us to do B, if A. The expression
of this rule (if A, do B) applied in the situation A stands in an internal
relation to B.
We know from §10.2 that internal relations hold solely between concepts. Accordingly,
then, the internal relation is not between a rule and an act that is in
accord with it but between the expression of the rule (applied in a situation) and
a description of the result. We have to insist here on the distinction between a
rule and an expression of this rule.
Taking the internal relation in question in this way, we can infer from knowing
a rule what its application is in a given case. In short: knowing a rule amounts
to knowing its application. Wittgenstein expresses this idea in a negative context:
“It seems so clear here: it is one thing ‘to understand a word’ and another
‘to apply it’.”488
Wittgenstein speaks here of knowing a word, but we take him
to be speaking about rules in general, for words are what they are due to their
grammatical rules.
13.1. The rule-following paradox
Baker and Hacker are generally right that “the concept of a rule and the concept
of what accords with it (what is a correct application of it) are internally relat-
ed”489
and that this view is implicit in Wittgenstein’s treatment of rules and rulefollowing.
Their argument (or one of their arguments) against Kripke and his
rule-skepticism is that they take this relation to be external. This means that
“the identity of the rule is divorced from its applications; what the rule is is one
thing, what its applications are is another, and only an agent’s independent interpretation
links the two.”490
This claim means, however, that from an expression
of a rule we cannot infer what a description of its application would be, let
us say, in a novel situation. This creates a paradox. One has mastered a rule
through a finite set of its applications; thus, one might be unable to apply it in a
novel situation.
488
Ms 116, p. 144, trans. by ter Hark, 1990, p. 47.
489
Baker & Hacker, 1984, p. 72.
490
Baker & Hacker, 1984, p. 100.
158
Let us take the following series: 2, 4, 6, 8, 10… Suppose that I have mastered
the rule of this series for numbers less than 1000. I am now asked to continue
the series above 1000. Ought I to continue the series in this way: 1000, 1002,
1004… or this way: 1000, 1020, 1040…? The point is that there is a rule for
every continuation of the series. Or Kripke’s most famous example: suppose
that I have never computed ‘68 + 57’. The correct result of this computation is,
of course, ‘125’. But there is also a non-standard interpretation of the sign ‘+’
according to which the result of this computation is then ‘5’. And Wittgenstein
now admits this general point: “This was our paradox: no course of action could
be determined by a rule, because every course of action can be made out to accord
with the rule.”491
Kripke distinguishes between a straight solution and a skeptical solution to this
paradox. A solution is straight “if it shows that on closer examination the scepticism
proves to be unwarranted”492
by rejecting one or more of the premises of
the paradox. A skeptical solution, however, accepts the paradox and tries to argue
that it does not disturb any of our ordinary practices or beliefs. First to
skeptical solutions: the paradox can be surmounted by an appeal to some kind
of third step that mediates between a rule and its applications.
This third step could be (i) an interpretation or a mental image. But, as Baker
and Hacker argue, an expression of the interpretation is, in fact, another formulation
of the rule. If it had been impossible to apply the rule in the former formulation,
it would also be impossible to do so in the latter formulation. Hence,
an interpretation of a rule is not a third step that could mediate between a rule
and its application. Another skeptical solution may appeal to (ii) human biological
nature (to the human form of life as it is understood biologically). It is obvious
that our language is conditioned by human nature or in Wittgenstein’s
terms by the “general facts of nature”. Such facts, being the basis of grammar,
lie outside grammar. They may cause us to have this or that rule, but they cannot
justify it or be a reason for it.493
(ii) Kripke’s solution to the paradox is to
appeal to community agreement. “The situation is very different if we widen
our gaze from consideration of the rule follower alone and allow ourselves to
491
PI §201.
492
Kripke, 1982, p. 66.
493
See PI, p. 230 for Wittgenstein’s argument. See also PI §142; RPP I, §48; LWPP I, §209.
159
consider him as interacting with a wider community.”494
Kripke’s solution is
based, thus, on what he calls the “public checkability” of applications of the
rule. If someone applied a rule wrongly, the community is then able to check
and possibly correct their action.
Internal relations do not allow for any third mediating term between its terms as
argued in §10.5. In order not to disturb the internal relation between a rule and
its applications, one would have to argue that the third term is internally related
both to the rule and to its applications. We saw that an interpretation of a rule is
internally related to a rule, but there is no independent reason why such an interpretation
should be internally related to an application of a rule.
Kripke’s solution is based on the idea that community agreement is an internal
property both of a rule and of its applications. In fact, agreement about a rule
would amount to agreement about its applications. The crucial question is: is
the agreement of the community an internal property of a rule? Kripke’s critics
like Baker and Hacker argue that community agreement is external to a rule:
Community agreement yields a sense of objectivity (or assertion-conditions for correctness)
only by severing the internal relation between the rule and what accords
with it. In place of that internal relation the community view substitutes the notion of
community agreement, which is not an internal property of the rule.495
A community can agree on a rule at time t0, which means agreeing on its future
applications. What guarantees, however, that agreement about the rule’s application
at time t1 matches the previous agreement at time t0? The majority of the
community may be mistaken about it. Baker and Hacker thus conclude:
There is no possibility of building consensus in behavior (or shared dispositions) into
the explanation of what ‘correct’ means except at the price of abandoning the insight
that a rule is internally connected to acts in accord with it.496
An agreement of the community or a communal agreement, to sum things up,
cannot then be the desired mediator between a rule and its applications. But fol-
494
Kripke, 1982, p. 89.
495
Baker & Hacker, 1984, p. 75.
496
Baker & Hacker, 1985, p. 172.
160
lowing a rule has an essentially social character; it is a practice, a habit, a custom,
an institution—to use Wittgenstein’s attributes. A kind of communal
agreement is, of course, among these practices. Using language presupposes
communal agreement about this or that language.497
Communal agreement is,
like the “general facts of nature”, a part of the framework which makes rulefollowing
possible.
At this point, Kripke or a rule-skeptic may admit that there has to be an internal
relation between a rule and its applications498
without having a communal
agreement as its internal part. They may insist, however, that the problem is an
epistemological one. The problem is: “How do I know that ‘plus’, as I use it,
denotes a function that, when applied to 68 and 57, yields 125?”499
In other
words, how do I know that, thus far, I have followed this or that rule under the
expression ‘plus’? How do I know which internal relation is the right one? The
role of communal agreement might be to decide about this epistemological
problem. Hence, an appeal to the internal relation between a rule and its applications
is void if one strives for a skeptical solution of the paradox.
A straight solution of the skeptical paradox will consist in showing that no third
mediating term between a rule and its application is necessary, because there is
neither an ontological nor an epistemological gap between a rule and its applications.
All the preceding attempts to give a skeptical solution miss out the fact
that internal relations are held among concepts. They focus on the application of
a rule as an act, as an event. What we have to focus on is, however, a description
of the application. In what follows, I am going to reiterate the argument
from §10.5 and adjust it to the present problem.
497
Cf. PI §241: “ ‘So you are saying that human agreement decides what is true and what is
false?’—It is what human beings say that is true and false; and they agree in the language
they use. That is not agreement in opinions but in form of life.”
498
Kripke has never wholly explicitly made such an admission. I think, however, that he
makes this admission in footnote 19 (1982, pp. 25–6). He attributes to Wittgenstein the
“view that the relation between the desire (expectation and so on) and its object must be ‘internal’,
not ‘external’”. This is parallel to the view that “The relation of meaning and intention
to future action is normative, not descriptive.” (1982, p. 37)
499
Kripke, 1982, p. 13.
161
We have to show that there is an internal relation between an expression of a
rule applied in a given situation and an expression of the result of this application.
In Kripke’s case, we need to show that there is an internal relation between
‘68 + 57’ and ‘125’. These concepts (the plus sign and the signs for the natural
numbers) are individuated by grammatical rules that define these concepts. All
we have to prove now is that these rules are not independent.500
Ironically, this
idea is formulated by Kripke himself:
One might even observe that, on the natural numbers, addition is the only function
that satisfies certain laws that I accept – the ‘recursion equations’ for +: (x) (x+0=x)
and (x) (y) (x+y′=(x+y)′) where the stroke or dash indicates successor; these equations
are sometimes called a ‘definition’ of addition.501
Kripke, however, rejects this idea on the following grounds:
The problem is that the other signs used in these laws (the universal quantifiers, the
equality sign) have been applied in only a finite number of instances, and they can be
given non-standard interpretations that will fit non-standard interpretations of ‘+’.
Thus for example ‘(x)’ might mean for every x<h, where h is some upper bound to
the instances where universal instantiation has hitherto been applied, and similarly
for equality.502
To cast skeptical doubt upon the rule for addition would involve doubting other
rules too. Kripke mentions the rule for the universal quantifier and for the
equality sign. Most importantly, these skeptical doubts would involve rules for
natural numbers too. We can recursively define addition as the only function on
natural numbers which satisfies the conditions mentioned above. On the other
hand we can define natural numbers by their algebraic properties with respect to
addition, e.g., associativity, commutativity, existence of the identify element,
etc. It is not important what defines what; whether addition is defined by employing
natural numbers or vice versa.
500
The role of a mathematical proof in general is to show that certain grammatical propositions
are not independent: “A mathematical proof connects a proposition with a system.”
LFM, p. 136. See §14.3.
501
Kripke, 1982, pp. 16f.
502
Kripke, 1982, p. 17.
162
The skeptic cannot then question the rule for addition while leaving all the other
grammatical rules (and especially those for natural numbers) untouched. The
rule for addition is not left hanging in the air. It is grammatically (internally)
connected to a myriad of other rules, e.g., to the rule for subtraction and, most
importantly, to the grammatical rules for natural numbers. If someone applied
the rule for addition wrongly, a community might be able to correct them. However,
this cannot be an arbitrary decision; the community has to point out the
other rules that are internally related to this rule. The community (or even a single
person) has to give a reason why the application of the rule was wrong.
To conclude: in order to give a straight solution to Kripke’s paradox, one has to
stress the internal relation between an expression of a rule applied in a certain
situation and an expression of the result of this application. What determines an
application of a rule in a novel situation is the rule itself together with the
grammatical rules for the description of this situation. These rules themselves
are then sufficient to compute the result.
13.2. Inexplicability of rules and the determinacy of sense
Towards the end of §12.1, I mentioned a possible way of explaining or justifying
a rule by the use of the reflexive constructions “This is simply what I do” or
“I do what I do”503
. This means that there is no independent reason for this rule
available. If there is an explanation, it must be internally related to this very
rule, which implies that the (expression of this) rule must already contain its
explanation. Wittgenstein writes in the early 1930s:
What I want to say is that a sign in some sense cannot be explained. It must speak for
itself in the rules for its use. It has to say everything, by giving every possible (clear)
explanation.504
Wittgenstein demands that rules, which explain or govern the meaning of signs,
have to speak for themselves. In the end, “the whole language must speak for
503
Ms 112, p. 113r.; PU §217.
504
Ms 109, p. 93. The English translation is taken from Krüger (forthcoming) and slightly
amended.
163
itself.”505
But neither signs nor rules are pictures or pictorial representations.
The determinacy of sense cannot be presupposed so easily. There must be the
possibility of misunderstanding, because language, in fact, sometimes fails to
speak for itself. We are now facing the following dilemma. On the one hand,
signs are being used in accordance with rules and rules aim to explain the
meaning of signs. They are hence already explanations. On the other hand, we
cannot determine rules by means of any expressions, i.e., by means of any
signs, because these signs have to be explained by other rules. In short: rules
determine signs and signs determine rules. 506
The problem stems from the fact that language is for the Wittgenstein of the
early 1930s a calculus, an internally related system of rules. All external relations
are forbidden here.507
Language as a calculus falls short of its practical
dimension. Language users play no role in this system.
Wittgenstein eventually realized that rules (or at least some rules) could be determined
by human practices and behavior. The question as to how to follow a
rule is pointless, because it suggests that there must be an answer, an explanation
of the rule. Some rules, however, cannot be explained by signs in certain
situations. If someone does not understand a rule, they may call for further explanations.
There must be, however, a final explanation. They follow this expression
of a rule without any further explanations. This consideration explains
why Wittgenstein says: “When I obey a rule, I do not choose. I obey the rule
blindly.”508
We are tempted to cut off any need for further explanations by the
reflexive construction “This is simply what I do”. To say that I obey the rule
blindly or “This is what I do” is, in fact, not a reply at all to the question concerning
the following of the rule.
If human praxis is constitutive for the rules of our language, then language users
must agree in this praxis. This agreement, however, does not encompass an-
505
Ms 109, p. 294.
506
The present problem is neatly summarized by Krüger (forthcoming, sec. I): “On the one
hand it is only a game if the signs are used in accordance with a rule; on the other hand it is
impossible to determine the use of signs by the expressions of their rules because expressions
– of any kind – are interpretable.”
507
Cf. Krüger, forthcoming, sec. III.
508
PI §219.
164
ything factual. Language users do not need to agree about what is true and what
is false. Rather, they must agree in the language they use:
“So you are saying that human agreement decides what is true and what is false?”—It
is what human beings say that is true and false; and they agree in the language they
use. That is not agreement in opinions but in form of life.509
Here, we have to look at language from an outside perspective which Wittgenstein
also calls “the ethnological approach.”510
This perspective is in contrast to
an inner perspective on language, which is a closed, internally interrelated system
here. What is important here is that these perspectives do not exclude each
other. Rules, on the one hand, make up a system; they are not independent of
each other. The rule for addition is not independent of the rules for natural
numbers. Each of these rules, on the other hand, is an expression of the actions
about which language users agree. The rules for natural numbers and for addition
are expressions of our practices of counting which do not need any further
justification. This is simply how we count, or—to mention other practices addressed
in this book—this is simply how we measure things, ascribe colors,
produce works of art, etc.
Internal relations are embodied in the rules of our language. We can look at internal
relations from the inner perspective and conceive of them as a system or
rather as systems. Complex algebraic structures like lattices, rings, monoids,
groups, groupoids, loops, or categories are examples of such systems where the
inner perspective is more important. Human practices with such formalized systems
are rather conservative.511
The outside perspective is, thus, less important.
This is not to say that human practices are of no relevance in mathematics. The
nature of its praxis is just such that the outside perspective does not reveal many
important insights in this field.
Internal relations are, however, also constituted by human practices. Shared
human practices are models or vehicles for conveying internal relations.512
The
internal relations between the paradigmatic samples for numbers, colors, or
509
PI §241.
510
Ms 162b, p. 67v.
511
Cf. PI §240.
512
Cf. Altieri (forthcoming, sec. IV)
165
units of length are constituted by our practices of counting, ascribing colors, or
measuring. It matters how we check that two sets have the same number of elements.
We have to pair their elements off. It matters how we ascribe color to a
thing. We look at its surface (not the inside) and if we are uncertain about the
exact color, we have to compare it with a sample in a sample book. Finally, it
matters how we measure the length of an object. We can use another object of a
known length and place them alongside each other. People used to use their
hands or feet as the objects of comparison. Not all hands or feet are, however,
the same length; and moreover, the lengths of hands and feet change as people
grow. It is better to use a uniform object of comparison whose length is more or
less constant. This includes using rulers and yardsticks. There has to be agreement
about these methods of counting, ascribing colors, or measuring things,
and not about the color or the length of a certain thing. All of these practices
and the internal relations that they constitute shall be addressed in the next
chapters.513
The tendency in Wittgenstein’s thinking to view internal relations as constituted
by our praxis culminates in his final writings, which were collected in the book
On Certainty. At least some internal relations are rooted not in this or that language-game;
they are rooted deeper in our pre-linguistic behavior such “that a
language-game is based on it, that it is the prototype of a way of thinking and
not the result of thought.”514
This behavior is not the result of any decision, any
training, or any act of baptizing. These internal relations are also not learned, at
least not explicitly: “I do not explicitly learn the propositions that stand fast for
me.”515
In conclusion, internal relations may be expressions of our human form
of life, of general facts about our human nature.
513
These paragraphs draw on a detailed analysis in Krüger (forthcoming).
514
Z §541.
515
OC §151.
166
14. Mathematics
We have seen in some of the previous chapters that mathematical statements are
paradigmatic cases of internal relations.516
And indeed, the core of Wittgenstein’s
conception of mathematics can be summed up in the motto that “arithmetical
rules are statements of internal relations.”517
This is not to say that this
taking of arithmetic (and in fact of all mathematics) as based on internal relations
is all Wittgenstein has to say about the philosophy of mathematics. On the
contrary, his contribution to the philosophy of mathematics is extremely diverse.
This is why in this chapter I have to restrict myself to the discussion of
topics directly related to the distinction between internal and external relations.
In particular, I am going to focus on Wittgenstein’s insistence on the pictorial
aspect of mathematical notation, which is, of course, his Tractarian heritage.
Mathematical notation must always be capable of depicting a state of affairs.
Here is a clear expression of this attitude:
There must always remain a clear way back to a picture-like representation of numbers
leading through all arithmetical symbols, abbreviations, signs for operations,
etc.518
Wittgenstein identifies numbers with numerals and this strategy also applies to
mathematical proofs: “A proof must of course have the character of a model.”519
Numbers and proofs are for Wittgenstein kinds of prototypes of certain activities—especially
the activities of counting and performing experiments. Numbers
or proofs are the yardsticks or measures of reality. Like in the Tractatus,
the pictorial relationship here is based on internal relations.
516
Cf. mathematical statements 3 > 2 (§10.2), 50 + 50 = 100 (§10.4), 1597 + 2548 = 4181
(§10.5), 25 × 25 = 625 (§11.2), 68 + 57 = 125 (§13).
517
PPO, p. 390. Cf. also: “All the errors that have been made in this chapter of the philosophy
of mathematics are based on the confusion between internal properties of a form (a rule
as one among a list of rules) and what we call ‘properties’ in everyday life (red as a property
of this book).” PG, pp. 476f.
518
WWK, p. 224.
519
RFM, p. 159.
167
This finitistic conception of mathematics is threatened, however, by general
arithmetical propositions. Do they picture some general characteristics of numbers?
There is an analogous question within the Tractarian framework. If a
proposition were a picture of reality,520
what would a generalized proposition
depict? Wittgenstein realized that his Tractarian account of generality was deficient.
In this chapter, I am going to argue that the Tractarian account of generality
fails because it confuses internal generality with external generality. Then I
will proceed, in turn, to Wittgenstein’s conceptions of numbers and proofs.
14.1. Generality
First let me turn to Wittgenstein’s conception of generality (or of general propositions)
from 1929. Here he criticized his Tractarian view that quantified propositions
are infinite conjunctions or disjunctions.521
The problem lies in an attempt
at quantifying over infinite domains. We can never capture such quantification
by enumeration. In order to understand a generalized proposition one has
to know all elements from the infinite domain, which is, of course, impossi-
ble.522
This shows that the word ‘all’ is ambiguous here. “There are as many different
‘alls’ as there are different ‘ones’.”523
And indeed, general propositions can
sometimes be analyzed as finite conjunctions. But then we have to provide an
account of quantifications over seemingly infinite domains. In order to provide
such an account Wittgenstein distinguishes between an internal generality and
external generality.524
Consider the following arithmetical statement:
(x) ((x + 1)2
= x2
+ 2x + 1)(83)
This statement is supposed to quantify over the infinite domain of (let us assume)
natural numbers in order to express something general. The Tractarian
account yields the following analysis of (83):
520
TLP 4.01.
521
PG, p. 268.
522
VW, p. 165.
523
PG, p. 269.
524
Ms 106, pp. 110 & 133.
168
((0 + 1)2
= 02
+ 2×0 + 1) ∧(84)
((1 + 1)2
= 12
+ 2×1 + 1) ∧
((2 + 1)2
= 22
+ 2×2 + 1) ∧
…
However, according to the later Wittgenstein this would be an inappropriate
analysis. The notation for generality525
(x)—as well as the notation for existence
(∃x)—indicates that an internal relation is expressed between the two sides of
the equation.526
The notation for generality, then, expresses that there is an
arithmetical operation transforming the one side of the equation into the other
one. This is a case of an internal (or provable) generality which is, in fact, no
generality at all. (83) is thus not about all numbers (natural, real, or whichever);
but it is about two expressions and their structural relation.
Internal generality is opposed to the external generality of non-arithmetical language.
Consider the following non-arithmetical general proposition:
All men die before they are 200 years old.527
(85)
This is an empirical proposition expressing an external property of all men. The
generality of this proposition is based on induction. If there were a man over
200, this proposition would be falsified. No evidence could, by contrast, falsify
(83). If one found a number that did not comply with (83), (83) would be evidence
of a miscalculation. Wittgenstein further argues that not even all cases of
external generality can be analyzed into a logical product. This is, however, not
our present concern. We are concerned with mathematical notation in this chap-
ter.
14.2. Numbers
There are different kinds of numbers in mathematics, e.g., the cardinal numbers,
the rational numbers, the real numbers, the complex numbers, etc. We are
tempted to think that there is some essential feature which all numbers have in
525
We usually use the sign  for the universal quantifier today.
526
If there were neither equation, nor any relational statement in (83), an internal property
would be expressed instead.
527
PG, p. 268.
169
common. There is no such thing according to Wittgenstein. It would also be
wrong to say that they make up a family resemblance class. We have to distinguish
sharply between numbers that have a finite representation and numbers
that have a potentially infinite representation. Only numbers that are capable of
a finite expression are for Wittgenstein numbers in a proper sense; they are
mathematical extensions. Clearly, only finite numerals can be identified with
numbers. He calls such numbers cardinal numbers, which was in conformity
with usage at the time. Numbers with a potentially infinite expansion—like irrational,
real, or complex numbers—are, in fact, not concepts but rules which
generate their infinite expansions.528
Wittgenstein also had problems with what was then the received definition of
cardinal numbers. Drawing on Cantor, Frege and Russell defined cardinal numbers
as cardinalities of equinumerous [gleichzahlig] sets. Cardinalities of finite
sets are natural numbers. There are, in addition to these, so-called transfinite
cardinal numbers, which describe the cardinalities of infinite sets. Wittgenstein,
however, rejected the view that there are different infinite cardinalities in his
finitistic conception of mathematics. But if we deprive cardinal numbers of the
hierarchy of transfinite numbers, what we get is precisely the natural numbers.
We can, thus, take Wittgenstein’s claims about cardinal numbers as claims
about natural numbers.
Now to Wittgenstein’s key definition of a cardinal number: “A cardinal number
is an internal property of a list.”529
Or more extensively:
The sign for the extension of a concept is a list. We might say, as an approximation,
that a number is an external property of a concept and an internal property of its extension
(the list of objects that fall under it).530
A number is an external property of a concept, for a concept does not determine
the number of elements of its extension.531
The concept of a book, for instance,
528
For instance, the number π has an infinite expansion which cannot be written down. It
can, however, be captured by an infinite converging series which can be described by rules,
for example, by the Leibniz formula for π.
529
PR, p. 140.
530
PG, p. 332.
531
This is true for material concepts, but not for formal concepts. Cf. Frascolla, 1994, p. 48.
170
does not determine how many books there are or must be. In other words, if a
concept is given by a defining property (by an intension), the number of elements
falling under this concept is not determined a priori.532
On the contrary, if we have a concrete extension of a concept, its number is its
internal property. The following list of strokes | | | has three elements and this is
its internal property. If we added one stroke to it, it would be a different list.
Lists of strokes like | | | are for Wittgenstein prototypes of (natural) numbers:
If 3 strokes on the paper are the sign for the number 3, then you can say the number 3
is to be applied in our language in the way in which the 3 strokes can be applied.533
There is, however, further clarification needed here. The list of strokes is not an
abstract list. It is a concrete list written down on the paper. We can write down
three strokes on a piece of paper and use this sheet of paper as a paradigm for
the number 3. We can furthermore store this sheet of paper in a mathematical
archive and use it whenever someone might be uncertain about the meaning of
the numeral 3. The list | | | serves in this sense as a yardstick. The numeral 3 is a
substitution or rather an abbreviation for the list | | |. Numerals are, thus, picturelike
representations of numbers.
We should not conceive of the list | | | as a set with three members or even as
any set of cardinality 3: “Here the strokes function as a symbol, not as a class.
Russell’s argument rests on a confusion of sign and symbol.”534
Wittgenstein’s
concrete and finitistic approach takes numerals as concrete objects as opposed
to Frege-Russell’s approach that is based on abstract sets. The decisive advantage
of Wittgenstein’s conception of numbers over Frege and Russell’s is
that numbers are rooted in our primitive activities535
like children’s fingercounting
or counting on an abacus. Moreover, Russell’s definition of numbers is
based on an actual correlation between equinumerous sets:
532
AWL, p. 205.
533
PR, p. 307. Cf. also: “But in what sense is | | | | | | | | | | | | the paradigm of a number? Consider
how it can be used as such.” RFM, p. 149.
534
WWK, p. 223. See also: “According to the Frege-Russell abstraction principle the number
3 is the class of all triples.” WWK, p. 221.
535
Cf. PI §1.
171
In Russell’s theory only an actual correlation can show the ‘similarity’ of two classes.
Not the possibility of correlation, for this consists precisely in the numerical equality.
Indeed, the possibility must be an internal relation between the extensions of the
concepts, but this internal relation is only given through the equality of the 2 num-
bers.536
This argument is, however, a little bit tricky. An actual one-to-one correlation
between two classes of things is an external relation. This correlation presupposes
that the two classes are numerically equivalent. Then, however, “the numerical
equivalence is not determined by the correspondence, but the numerical
equivalence makes the correspondence possible.”537
Two classes are equinumerous
if it is possible to correlate their elements one-to-one. The possibility of
correlation is an internal relation. We can say that
There are three books lying on the table.(86)
if it is possible to correlate them with the paradigmatic list | | |. We can reformulate
(86) by inserting the paradigm into it:
There are | | | books lying on the table.(87)
This means that there is an internal relation of possible correlation between the
paradigmatic list and those books lying on the table. And more appropriately:
there is an internal relation of a possible correlation between the number of
strokes on the paradigmatic list and the number of books.

Now, we have to make explicit the distinction between statements of numbers
in mathematics and statements of numbers outside mathematics:
536
PR, p. 140. Cf. also: “Dr. Wittgenstein made a very successful attempt. He began by
quoting and criticizing Russell’s definition of number, that is, ‘the class of classes similar to
a given class’, similarity being defined by means of a 1-1 correlation, and pointed out that
Russell confuses the existence of this correlation with the possibility of its existence.” PPO,
p. 373.
537
Waismann, 1951, p. 109.
172
Statements of number in mathematics (e.g. “The equation x2
1 has 2 roots”) are
therefore quite different in kind from statements of number outside mathematics
(“There are 2 apples lying on the table”).538
Consider the following equation (which is a statement in mathematics):
2 + 2 = 4.(88)
Inserting the paradigmatic lists into this statement yields:
| | + | | = | | | |.(89)
There is possibly a one-to-one correlation between both sides of the equation,
which means that there is an internal relation (of a possible correspondence).
The analysis of statements outside mathematics like (86) is a different one. First
of all, (86) is an experiential statement. The meaning of the numeral 3 here is
defined by a reference to the paradigmatic list. This internal relation holds between
the number of books and the number of strokes. It is not a relation within
the sentence like in (88) or (89), but is a relation to something else, namely to
the paradigmatic list.
Confusing these two uses of numerals would result in possibly nonsensical reflexive
uses of internal relations. Consider the following statement:
There are 3 strokes in | | |.(90)
What is the meaning of the numeral 3 in this sentence? Its meaning must be derived
from the very same paradigmatic list. If so, we get:
There are | | | strokes in | | |.(91)
This is, however, a very peculiar statement of identity aiming to express a reflexive
internal relation between | | | and | | |. (90) cannot be a definition either,
because “The form 3 can only be transposed, it cannot be defined.”539
Hence we
can take (90) as a substitution rule transposing the form | | | into the form 3. The
number 3 is an internal property of the list | | |. “It is nonsense to say of an ex-
538
BT, p. 410e.
539
WWK, p. 225.
173
tension that it has such and such a number, since the number is an internal
property of the extension.”540
To sum up: numerals outside mathematics are being used transitively, deriving
their meaning from paradigmatic samples (paradigmatic lists). Numerals within
mathematics express internal relations between different samples. But we cannot
ascribe a number to the very same paradigmatic list which defines this
number. This would then be a nonsensical reflexive use of an internal relation.
14.3. Proofs
The concept of a mathematical proof is—as one would expect in the Wittgensteinian
spirit—a family resemblance concept. There are logical differences
among different kinds of proofs. A recursive proof, for instance, is, in fact, a
guide to the construction of special proofs. Wittgenstein was critical of the notions
of an inductive, a logical (Russellian), or an existence proof inter alia. After
excluding these suspicious kinds of proofs, he nevertheless then tries to capture
something like the nature of proof.541
We can proceed from the assumption that mathematical propositions are statements
of internal relations. In this respect they are like grammatical proposi-
tions.542
A proof of a mathematical proposition aims to show or rather lay down
its internal relatedness to a system of other mathematical rules:
What is proved by a mathematical proof is set up as an internal relation and withdrawn
from doubt.543
Proof must shew the existence of an internal relation.544
A mathematical proof connects a proposition with a system.545
540
PR, p. 141.
541
Cf. RFM, p. 174.
542
“To say ‘mathematics has the function of grammar’ would be false. It has many other
functions. But mathematical propositions are of the same kind as grammatical propositions
even when they appear to be experiential propositions.” PPO, p. 389.
543
RFM, p. 363.
544
RFM, p. 434.
545
LFM, p. 136.
174
Consider again the proposition ‘68 + 57 = 125’ as discussed in §13. Its proof
must show that this proposition is compatible with the rule for addition and
with paradigmatic samples of the natural numbers.546
To prove a mathematical
proposition amounts to showing how to arrive at it from other (primitive) propositions
by means of formal operations. For Wittgenstein, mathematical proofs
must be constructive (hence his aversion to existence proofs that are not con-
structive).
A mathematical proposition which is proved is thus an internal part of the proof.
We may say that “the completely analysed mathematical proposition is its own
proof.”547
In other words, mathematical propositions get their meanings from
their proofs.548
This account threatens the existence of mathematical problems,
i.e., mathematical propositions that have not been proven yet. Proven and unproven
mathematical propositions are, however, not at the same level:
The proposition with its proof doesn’t belong to the same category as the proposition
without the proof. (Unproved mathematical propositions—signposts for mathematical
investigation, stimuli to mathematical constructions.)549
The very notion of an ‘unproven mathematical proposition’ is misleading, for it
suggests that they are also statements of internal relations that are not apparent
for the time being. The expression ‘mathematical conjecture’ would be more
appropriate here. The crucial question is whether a mathematical conjecture expresses
an internal relation or an external relation. Let us consider the famous
Goldbach’s Conjecture (an example Wittgenstein himself employed):
Every even number greater than 2 can be expressed as a sum of two(92)
primes.
Although we still do not possess any rigorous proof of Goldbach’s Conjec-
ture,550
we can understand it. We just do not know whether the conjecture is true
546
“What a proof really proves is the compatibility of the proposition with the propositions
from which one started, the primitive propositions, or rather the incompatibility of the opposite.”
LFM, pp. 73f.
547
PR, p. 192.
548
“[A] mathematical proposition only gets its meaning from the calculus in which it is embedded.”
LFM, p. 137.
549
PG, p. 371.
175
or false. Would this undermine Wittgenstein’s position that mathematical propositions
get their meaning from their proofs? Wittgenstein, however, calls our
understanding of this conjecture into question:
To believe Goldbach’s Conjecture, means to believe you have a proof of it, since I
can’t, as it were, believe it in extenso, because that doesn’t mean anything, and you
cannot imagine an induction corresponding to it until you have one.551
What we understand here is that for a given number n, we are able to find out
whether n can be expressed as the sum of two primes. But if (92) is supposed to
be a mathematical proposition, the general quantifier must express an internal
relation.552
We cannot, however, imagine such an internal relation until we
know it or are able to construct it. We can employ brute force techniques or statistical
considerations in order to give a heuristic justification of Goldbach’s
Conjecture. If so justified, we can hardly treat Goldbach’s Conjecture as expressing
an internal relation. Goldbach’s Conjecture justified heuristically has,
thus, a different meaning from (92) than if it were rigorously proven.553
We can
understand Goldbach’s Conjecture in a compositional way, i.e., we understand
all the concepts involved and the way in which they are combined. If Goldbach’s
Conjecture were rigorously proven, we would understand it by virtue of
its proof, i.e., by virtue of its internal relations to other mathematical proposi-
tions.
We can conclude from the previous discussion that a mathematical proposition
gets its meaning from its proof, which lays down an internal relation to other
mathematical propositions. We cannot understand a mathematical proposition
until we possess its proof.554
On the other hand, a mathematical conjecture is
550
As of May 2013.
551
PR, p. 204.
552
Cf. §14.1.
553
The same point could be made about the conjecture that the group ‘7777’ occurs in the
decimal expansion of π. See PI §§352 & 516; RFM, pp. 284 & 407f.
554
Contra Floyd: “Wittgenstein is not insisting that […] we do not understand a mathematical
proposition until we possess its proof” (2000, p. 244).
176
not meaningless.555
Although it could have the same surface form as the corresponding
mathematical proposition, it expresses an external relation and thus
has a different meaning from the mathematical proposition. The proof of a
mathematical conjecture “alters the grammar of a proposition”556
. The proof
changes a proposition that expresses an external relation into a proposition that
expresses an internal relation.

What now needs to be examined further is the pictorial aspect of mathematical
proofs. Wittgenstein is quite explicit in this respect:
When I say “a proof is a picture”—it can be thought of as a cinematographic picture.
[…]
Proof, one might say, must originally be a kind of experiment—but [it] is then taken
simply as a picture. […]
The proof must be our model, our picture, of how these operations have a result.557
A proof is also a picture—or rather a motion picture—of an experiment. What
kind of experiment? Consider a class of some already-proven mathematical
propositions. We can, as it were, experimentally try to transform them by applying
mathematical operations in order to yield the desired proposition which has
to be proven. The experiment consists in trying to construct the desired proposition
(which has the status of a conjecture for the time being) out of alreadyproven
mathematical propositions. There is no systematic way of choosing suitable
initial propositions and suitable operations. This may involve constructing
ancillary terminology and proving ancillary mathematical propositions, i.e.,
lemmas. These peculiarities are one of the main reasons why some mathematical
conjectures are so hard to prove.
There is, however, another sense in which a proof can be taken as a picture of
an experiment. We may transform every mathematical proposition that is con-
555
Contra Shanker: a mathematical conjecture is “a meaningless expression albeit one
which may exercise a heuristic influence on the construction of some new proof-system.”
(1988, p. 230).
556
PG, p. 367.
557
RFM, pp. 159–161.
177
tained in a certain proof into a statement outside mathematics as demonstrated
in the previous section. For instance, we may transform
1 + 1 = 2(93)
into
One apple and one apple on my table makes a total of two apples.(94)
Sentence (93) expresses an internal relation, whereas (94) expresses an external
relation. If we transform all the steps of a proof in this way, we get a description
of a real experiment. We arrive at something like this: if one starts with a certain
state of affairs and proceeds according to prescribed rules, then the resulting
state of affairs must be so-and-so. The ‘must’ in the preceding sentence is, however,
not a logical must. There needs to be a ceteris paribus clause added: ‘…
the resulting state of affairs must be so-and-so, if nothing goes wrong’. There
are thousands of ways that an experiment could go wrong. A description of an
experiment based on external relations is not normative here. If we want to insert
normativity into it, we have to add the ceteris paribus clause or—and this is
of the utmost significance—we have to take it as a picture “of how these operations
have a result”558
. This is how an experiment can be taken as a proof.
I have identified two ways in which a proof can be taken as a picture of an experiment.
The first one is an experimental attempt at transforming some mathematical
propositions in order to arrive at the proposition that has to be proven.
The second one is to take the mathematical propositions involved in a proof as
statements outside mathematics expressing external relations. Then a mathematical
proof can be taken as a picture of a real experiment. These two ways complement
rather than contradict each other.

Having said this, we are now in a position to investigate the next and final twist
of Wittgenstein’s considerations about mathematical proofs. As noted above,
mathematical proofs aim to integrate mathematical conjectures into the system
of already-proven mathematical propositions. A conjecture is turned into a
proposition by providing its proof, which is a picture of an experiment. But how
558
RFM, p. 161.
178
can we take a proof as a picture? There must be some act that elevates something
into a picture so that it will be considered as a picture of some other thing.
Wittgenstein employed the idea of depositing something in the archives in order
to explain how we can handle standards of colors like color swatches or standards
of length like the standard meter.559
These are particular objects that are
deposited into some prominent place and considered as paradigmatic cases (as
samples as opposed to examples) of particular properties. Wittgenstein now
imagines that we can also put some significant calculations and proofs into the
archives:
A calculation could always be laid down in the archive of measurements. It can be
regarded as a picture of an experiment. We deposit the picture in the archives, and
say, “This is now regarded as a standard of comparison by means of which we describe
future experiments.” It is now the paradigm with which we compare.560
A proof—not an abstract proof, but rather its visual shape impressed on a piece
of paper—is deposited in the (mathematical) archives and regarded as a paradigm
of future experiments. A proof on a piece of paper is a particular object
like the standard meter or standard sepia. There must be an internal relation between
the proof and an experiment. Although internal relations do not hold between
objects, they do hold between concepts. There is, however, no genuine
contradiction here. We have to focus on the visual shape of a proof. Some of its
features have to correspond to some features of an experiment. We can see this
as a generalization of Wittgenstein’s account of numbers which was discussed
in §14.2.561
Numbers are defined by paradigmatic lists. We may, of course, perform
experiments regardless of any proofs. But then the experiment would be
deprived of any normative force. We could not decide whether the experiment
went right or wrong and what its final outcome was.
559
I discuss these paradigms in §§15.2 & 16.
560
LFM, p. 104.
561
Cf. “The idea that the sequences of strokes in an arithmetical construction, like the figures
of a geometrical proof, [can] take the part of paradigms, symbols, or, in Tractarian
terms, variables, will develop, in later writings, into the conception of mathematical proof
as the picture of an experiment.” (Frascolla, 1994, p. 49)
179
Hence, in order to be able to read off the result of an experiment we need something
like a yardstick. “The proof is our model for a particular result’s being
yielded, which serves as an object of comparison (yardstick) for real chang-
es.”562
An experiment is a concrete process which results into a certain state of
affairs. This state has to be measured and “the proof serves as a measure”563
here.564

I opened my discussion of Wittgenstein’s thoughts about mathematics with his
insistence on the pictorial aspect of mathematical notation. Numerals like | | |
are picture-like representations of numbers; proofs and calculations are more
complicated cases of such picture-like representations. There is no fundamental
difference between them in this respect.
The final point I would like to discuss concerns the applicability of a picturelike
representation. Let us begin with some easy cases. How to apply the list of
strokes | | | to a particular state of affairs, e.g., to apples on my table? We have to
project the list onto the state of affairs. Wittgenstein insists that arithmetical
constructions can guarantee their applicability:
You could say arithmetic is a kind of geometry; i.e. what in geometry are constructions
on paper in arithmetic are calculations (on paper). You could say, it is a more
general kind of geometry. […]
The point of the remark that arithmetic is a kind of geometry is simply that arithmetical
constructions are autonomous like geometrical ones and hence so to speak themselves
guarantee their applicability.565
If arithmetical constructions are like geometrical constructions with respect to
their applicability, we have to ask what guarantees the applicability of geometrical
constructions. The question can be put in terms of the internal/external distinction
as pursued here: what guarantees that there is an internal relation between
the list | | | (deposited in the archives) and three apples on my table? A
562
RFM, p. 161.
563
RFM, p. 158.
564
See Diamond, 2001, pp. 123–125 for another focused discussion of the idea of depositing
something in the archives.
565
PR, pp. 306f.
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relation between these objects is an external one. The idea of depositing something
in the archives makes sense only if the deposited object guarantees its
own applicability, i.e., its own projection. This is essential to all mathematics:
“But I see the ‘mathematically essential’ thing about the process in the projection
too!”566
We can only deposit in the archives an object we know how to project. The object
has had to play some role in our activities and techniques before. We have
to know in advance that it is essential to the list | | | that it is a paradigm of a
number and not a paradigm of color or length. That is to say: in order to define
or rather transpose the number 3 using the list | | | we have to presuppose the
concept of number. This is a well-known idea from the beginning of the Philosophical
Investigations where Wittgenstein focuses on ostensive definition:
So one might say: the ostensive definition explains the use—the meaning—of the
word when the overall role of the word in language is clear. Thus if I know that
someone means to explain a colour-word to me the ostensive definition “That is
called ‘sepia’” will help me to understand the word.567
The list | | | (written down on a piece of paper and deposited in the archives) is
an instrument of language like the standard meter.568
The same is valid for calculations
and proofs as well. They do not have their meanings in isolation, but
rather within our practices.
566
RFM, p. 51.
567
PI §30.
568
RFM, pp. 167f.: “If I were to see the standard metre in Paris, but were not acquainted
with the institution of measuring and its connexion with the standard metre—could I say,
that I was acquainted with the concept of the standard metre?
A proof is an instrument—but why do I say ‘an instrument of language’?”
181
15. Colors
The theme of colors pervades Wittgenstein’s writings and this book as well. The
very first remark on colors in connection with internal relations is to be found in
an entry from the pre-Tractarian Notebooks from 1916:
That the colours are not properties is shewn by the analysis of physics, by the internal
relations in which physics displays the colours.569
I think that the point of this rather cryptic remark is to express a certain preference
for grammatical investigations over physical investigations of colors.
What the analysis of physics shows us is that colors are, at best, secondary
properties, i.e., they are not the (primary) properties of objects. What is distinctive—and
philosophically relevant—about colors is their mutual relations. What
colors are from the viewpoint of physics570
plays an inferior role in Wittgenstein’s
inquiries, or more precisely the physical viewpoint gradually fades away.
What he is after here is rather the logic, grammar, or geometry of colors (these
characteristics can be used here interchangeably). This is to say: Wittgenstein
investigates colors as they appear to us and as we speak about them.
This idea can be further developed by noting a certain analogy between color
systems and number systems:
We have a colour system as we have a number system.
Do the systems reside in our nature or in the nature of things? How are we to put
it?—Not in the nature of numbers or colours.571
Colors are neither properties of objects, nor Platonic qualities. Colors are nodes
in our color systems (the plural is important here). The relationship between
colors and mathematics is very close: mathematics has a certain pictorial aspect;
colors must be, in Wittgenstein’s view, connected to the way they appear—let
me call this the phenomenological aspect of colors. Furthermore,
569
NB, p. 82, 11.9.1916.
570
For example, colors are certain ranges of the electromagnetic spectrum interacting with
the light receptors in the eye.
571
Z §357. Cf. also ROC III, §3: “Here we have a sort of mathematics of colour.”
182
basic colors as well as natural numbers can be defined by paradigmatic samples,
i.e., by internal relations to these samples. Like the fact that not all numbers
are of the same sort,572
“not all colour concepts are logically of the same
sort”573
. This means that what we call ‘color’ in our everyday language may belong
to different color systems, to different language-games.
15.1. Lightness and darkness
For Wittgenstein, the most prominent example of drawing the distinction between
internal and external relations is the lightness relation. This example appears
throughout Wittgenstein’s works from the Notebooks to the last Remarks
on Colour. Here is the very first remark:
From the fact that […] one colour is darker than another, it seems to follow that it is
so; and if so, this can only be if there is an internal relation between the two; and we
might express this by saying that the form of the latter is part of the form of the for-
mer.574
The already quoted remark from the Tractatus reads: “This shade of blue and
that one stand, eo ipso, in the internal relation of lighter to darker.”575
This
thread is again picked up in Wittgenstein’s conversations at the beginning of the
1930s:
I can \certainly\ say: the one suit is lighter than the other one, but not: this grey is
lighter than that grey/ (if the word ‘grey’ is specified differently in both cases)/. That
is to say, ‘lighter’ and ‘darker’ are external properties of the materials but internal
ones of the colours.576
The theme recurs again in the Remarks on the Foundations of Mathematics577
and culminates in the very first paragraph of the Remarks on Colour:
572
As argued at the beginning of §14.2.
573
ROC I, §54.
574
NB, p. 118.
575
TLP 4.123.
576
VW, p. 239.
577
RFM, pp. 48, 75f.
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A language-game: Report whether a certain body is lighter or darker than another.—
But now there’s a related one: State the relationship between the lightness of certain
shades of colour. […] The form of the propositions in both language-games is the
same: “X is lighter than Y”. But in the first it is an external relation and the proposition
is temporal, in the second it is an internal relation and the proposition is time-
less.578
In previous sections of this book579
I have expressed an internal relation between
two color shades as an arithmetical statement within the HSV color
space. I chose exactly this color space (which was designed a long time after
Wittgenstein’s death) because it has brightness as one of its dimensions and,
thus, there is an easy way of expressing the lightness relation in a mathematical
statement. Most important here is the whole idea of color spaces as abstract
mathematical models which represent colors (and other visual properties) as
tuples of numbers and allow the translating of statements about colors into
arithmetical statements. In addition to this, they provide a surveyable representation
of the color space in two- or three-dimensional space.580
That is why we
can speak of the geometry of colors. Wittgenstein is one of the progenitors of
the idea of color spaces, which is now indispensable in computer graphics or
image processing.
The common point of the remarks quoted above is that the lightness relation is
an internal relation between color shades and an external relation between objects.
Internal relations hold within a color space. They constitute “the logic of
colour concepts”581
or rather a part of it. The lightness relation is not the only
relation that can be captured in a color model; one can think of other relations
such as saturation or colorfulness.
578
ROC I, §1.
579
§§6.2, 10.2, & 10.5.
580
The HSV color space represents colors in a three-dimensional cylindrical coordinate system.
There are four-dimensional color spaces which of course cannot be presented as spatial
objects.
581
ROC I, §22.
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15.2. Paradigmatic samples
We can think about color spaces on an abstract level without any reference to
actual colors as they are perceived. This may be useful in applied sciences, but
Wittgenstein insisted on the close connection between color spaces and the way
we perceive colors. We may ask the following question: how do we define color
concepts? Wittgenstein’s way of answering this question is to ask another question:
how could children learn to use color concepts? A reasonable answer may
be that we can give meaning to a color concept by giving an ostensive definition
by pointing at a paradigmatic sample. Wittgenstein initially considers a
very simple case:
The word “blue”, for example, is correlated with a certain colored patch which is a
sample.582
Suppose I pointed to a piece of paper and said to someone: “this colour I call ‘red’”.
Afterwards I give him the order: “now paint me a red patch”. I then ask him: “why, in
carrying out my order, did you paint just this colour?” His answer could then be:
“This colour (pointing to the sample which I have given him) was called red; and the
patch I have painted has, as you see, the colour of the sample”.583
According to these remarks, we can think of color concepts as substitutes for
samples. If someone is uncertain about the actual application of a color concept,
they can then compare the actual case with the sample. Color concepts are analogous
to the concepts for numbers which are also substitutes for paradigmatic
lists. And just as there is an analogous internal relation between a yardstick and
a measured object,584
so too is there an internal relation between the sample for
red and any red object. More precisely, there is an internal relation between the
color of the sample and the color of any red object.
Having said this, we now have to address several points that may complicate
this simple picture. First, it is not self-explanatory how a comparison of a sample
with an object should be carried out.
582
AWL, p. 143.
583
BBB, p. 14.
584
Cf. §§11.3 & 16 et passim.
185
There are even many different kinds of comparing and copying. There is rough and
exact copying, comparison of this green with other greens, comparison of two colors
by means of a color wheel with respect to the amount of yellow they contain.585
The sample and the object that is compared have to be of the same color. It is,
however, not clear what counts as the same color. Sometimes we rely on perceptual
evidence, at other times more precise methods are needed. Sometimes
we need exact agreement, while at other times we are satisfied with approximate
agreement. Sometimes we take environmental influences into account,
and at other times we do not. And so on. In a nutshell: “Everything depends on
the method of comparison.”586
Dealing with color samples and the comparing of
colored objects have to have their place in our practices before we define any
particular color concept. This means that we must know how to project the
sample onto other objects or use it in our practice.587
A second point is that children usually do not use a single paradigmatic sample
in order to apply color concepts. The schema above is to be taken as an idealization.
Children are usually taught or rather trained to use color concepts via a
series of examples (as opposed to samples).588
The normative aspect of the practice
is warranted in the first place by an authority, e.g., by their parents. But if
their parents were uncertain or in disagreement about the application, a reference
to a paradigmatic sample might be helpful. A child needs to be trained using
series of examples until they are able to apply the color concept in a novel
situation alone. This means that the child has to grasp a rule for the color concept.
What interests us here is, in fact, only the application of the rule.589
What a
child has to learn using a series of examples is the method of projection of the
sample, i.e., the method of comparison.
585
AWL, p. 88.
586
ROC III, §259.
587
Cf. §14.3 where we discuss presuppositions of depositing something in the archives.
588
This is, of course, an idealization of children’s learning. They may learn color concepts
in some other way than by observing their parents pointing at examples. Such speculative
ontogenetic considerations, however, are meant only to illustrate the philosophical point.
589
Cf. BBB, p. 14: “The rule which has been taught and is subsequently applied interests us
only so far as it is involved in the application.”
186
Suppose now that we have succeeded in defining some color concepts by paradigmatic
samples. The problem is, however, that the linguistic expressions for
these colors may be ambiguous with respect to the distinction between internal
and external properties. They may fall between several language-games as addressed
in §10.3. Let us focus on a simple language-game here (as introduced at
§§48–49 of the Philosophical Investigations). We have four colors: white,
black, red, and green. The syntax of this language-game is very simple. A sentence
is only a series of the initial letters of these colors. So for instance, the
sentence ‘RRG’ means that there are two red squares followed by one green
square. We may, however, hesitate here regarding what a sentence consisting of
one single letter, e.g., ‘R’, actually means. This expression could describe a
complex consisting of only one square on the one hand, or it could also be a
name for that very square on the other hand.
Wittgenstein considers, then, the possibility that the naming (of a square) is a
limiting case of the describing (of a complex of squares). This would, in effect,
dismiss the difference between describing and naming. To say that there is a
complex consisting of one red square would be tantamount to saying that the
square is red. This stance, though, could easily lead to confusion. We can use
the expression ‘R’ or the sentence ‘There is a red square’ in the course of ostensive
teaching to explain the meaning of ‘R’ or ‘red’. In short: we could use the
sentence ‘R’ either as a genuine proposition or as a rule (that is, as a grammatical
proposition). This situation fits exactly within our general scenario as presented
in §10.3.
To avoid any misunderstanding we are invited to take naming and describing as
different activities. In Wittgenstein’s words: “For naming and describing do not
stand on the same level: naming is a preparation for description. Naming is so
far not a move in the language-game.”590
Naming is thus not a move in the language-game
of describing colored squares. It is, however, a move in the language-game
of teaching the rule. In this preparatory language-game, one has to
learn what ‘R’ or rather ‘This is red’ means. It is crucial that in this languagegame,
the demonstrative ‘this’ refers to the color of the square, not to the square
itself. ‘This is red’ actually means ‘This color is called red’.
590
PI §49.
187
Mastering a paradigm changes the nature of the language-game: “By bringing
in a paradigm we have altered the game.”591
Hence, after the rule has been mastered
we can go over to the language-game of applying the rule, i.e., of describing
colored squares. Red is an internal property of the color shade that one was
pointing at in the preparatory language-game. ‘R’ or ‘This is red’ now means
that there is a complex consisting of one red square. ‘This’ refers now to the
complex one is pointing at, and ‘R’ or ‘This is red’ ascribes the external property
of containing a red square to this complex.592

The next point that needs to be addressed is whether we can meaningfully assert
that the paradigmatic sample of a color has this very color. Can we, for instance,
meaningfully assert the following sentence?
The paradigmatic sample of blue is blue.(95)
Wittgenstein gives a straightforward answer. We cannot assert sentences like
(95). “Samples such as this are part of our language; the patch is not one of the
applications of the word ‘blue’.”593
There is a more detailed discussion of this
issue in the Philosophical Investigations:
Let us imagine samples of colour being preserved in Paris like the standard metre.
We define: “sepia” means the colour of the standard sepia which is there kept hermetically
sealed. Then it will make no sense to say of this sample either that it is of this
colour or that it is not.594
There is, however, no prima facie reason why we could not assert that the sample
of blue is blue and the standard sepia is sepia-colored. Why could we never
assert sentences like (95)? Maybe we cannot compare the paradigmatic sample
with itself. But Wittgenstein does not use this argument. The remark quoted
above continues, however:
591
AWL, p. 143.
592
This discussion is taken from my own paper (Mácha, 2013).
593
AWL, p. 143.
594
PI §50.
188
We can put it like this: This sample is an instrument of the language used in ascriptions
of colour. In this language-game it is not something that is represented, but is a
means of representation.595
It makes no sense to assert (95), but only with the reservation ‘in this languagegame’.
Now we have to ask which language-game is meant and what Wittgenstein’s
argument is here. In the language-game of teaching the rule, we can as-
sert
This is blue.(96)
while pointing to the paradigmatic sample (which we are attempting to define),
because this sentence in fact means:
Let this color be called blue.(97)
We cannot assert (95) or (96) in the language-game of applying the rule, because
the sample “is a means of representation”. We can take this idea quite literally
and imagine a language without any color concepts. When one wants to
say that an object has a certain color, one will have to take a paradigmatic sample
and put it in front of the object. To express (96), one would then have to
take the paradigmatic sample of blue and attach it to the object like we attach
labels to exhibits in museums. In order to apply the sample to another object,
one would have to remove it from the previous object and attach it to the new
one. In this scenario, it would be impossible to express the assertion that the
paradigmatic sample of blue is blue, because we cannot attach this sample to
itself.
We can understand the claim that samples are means of representation in the
sense that they are, or at least can be, involved in grammatical rules governing
color concepts. In my previous discussion of colored squares, I pointed out that
‘R’ or ‘This is red’ can be used as a genuine proposition or as a rule (that is, as a
grammatical proposition). Hence, (95) and (96) are genuine propositions in the
language-game of teaching, but they are grammatical propositions in the language-game
of applying. This is why they cannot be asserted. This is, however,
not the reason why they are possibly nonsensical. We can use these sentences to
report a rule of an actual language-game. (96) is a typical example of a sentence
595
PI §50, my italics.
189
that is ambiguous as regards the distinction between internal and external rela-
tions.
The reason why they are possibly nonsensical lies deeper. As stated at the beginning
of §15.2, sentences like (96) express an internal relation between the
paradigmatic sample and the object pointed at. In the language-game of applying
the rule, sentence (96) expresses an external property of this object. But if
this object were the paradigmatic sample itself, (96) would express an internal
relation of the sample to itself. At the same time, it should express its external
property (that is, its color). This is therefore flawed, since blue is an internal
property of the paradigmatic sample of blue. The same surface form
x is blue(98)
is used to express an internal property of the paradigmatic sample (employing a
reflexive construction) and an external property of other objects within the same
language-game.
We have here another instance of Wittgenstein’s general argumentative schema,
which is the main focus of this book. The first step consists in drawing a clear
distinction between conceptions based on external and internal relations. The
second step is to assess possible reflexive cases of internal relations. First, we
have to consider whether sentences of the form (98) are used to express the internal
relation of color identity or an external property of an object. If the former
is the case, we have to look at possible reflexive cases of this internal relation,
i.e., an internal relation of the paradigmatic sample to itself.
15.3. Relations between colors
Definitions of colors by reference to paradigmatic samples do not take into account
the mutual relations between colors. Each color and each color shade has
its own paradigmatic sample which is entirely independent of the other samples.
There are human practices that use colors in this way. We have color sample
tools in graphics software; or we use a color sample if we want to instruct the
person pointing our house. We just pick whichever shade of color we want and
later we can compare the result with the sample. These practices are in fact an
exception. In a great deal of human practices, relations between colors are im-
190
portant and useful. Color samplers usually consist of a great number of paradigmatic
samples. If we take into account relations between colors, we can significantly
reduce the number of samples to a few basic ones. These are the socalled
basic (or simple, or pure) colors. Other shades of color, the so-called intermediate
or mixed colors, can be defined by internal relations to basic colors.
Realizing that ascriptions of colors are not mutually independent led Wittgenstein
to abandon the program of logical atomism. I am not going to focus on the
color exclusion problem, but on the related problem of the status of the proposition
ascribing intermediate colors. Consider the following statement:
A mixed colour (or better, an intermediate colour) made up of blue and red is a mixed
(intermediate) colour via an internal relationship to the structures of blue and red.
Expressed more precisely: What we call “an intermediate colour between blue and
red” (or “bluish-red”) is so called because of a relationship that shows in the grammar
of the words “blue”, “red” and “bluish-red”.596
This formulation suggests that an intermediate color like ‘bluish-red’597
shares a
part of the structure of red and a part of the structure of blue. We can think here
of surface structures that reflect light of certain wavelengths. The structure of
red (and of blue too) is that such a surface reflects light of 620–740 nm (450–
495 nm for blue). ‘Bluish-red’ would then be a mixture of small pigments reflecting
light of these wavelengths. However, according to Wittgenstein himself,
this analysis is wrong. The quotation from The Big Typescript continues:
(The very proposition that talks about an internal relationship between the structures
originates in an incorrect idea – in that idea that sees complicated structures in the
concepts “red”, “blue”, etc., structures whose inner construction must be shown by
analysis.)598
Wittgenstein repudiates the idea that simple color concepts have structures, and
not the idea that an intermediate color is defined by internal relations to simple
596
BT, p. 342e. Cf. also PR, p. 108.
597
Today, one would call such a color ‘purple’ or ‘magenta’. It is important to note that
these reflections are specific to German. ‘Purple’ has been established as a basic color concept
in English for far longer than in German.
598
BT, p. 342e.
191
colors. In Wittgenstein’s earliest manuscripts, internal relations were defined in
terms of structures.599
Here we can now notice a certain shift in Wittgenstein’s
thinking. The incorrect idea mentioned above is that internal relations are exclusively
relations between structures. Wittgenstein is thus heading towards a more
general account of internal relations as rules of grammar. Here is the final part
of the quotation from The Big Typescript:
But the relationship of pure colours to their intermediate colour is of an elementary
kind. That is to say, it doesn’t require that the proposition that ascribes the colour bluish-red
to an object is made up of the propositions that ascribe the colours blue and
red to it. And in a like manner the relationship among various shades of a reddishblue,
for instance, is an elementary relationship.600
Consider a proposition ascribing bluish-red to an object:
A is bluish-red.(99)
We cannot eliminate the expression ‘bluish-red’ by equating (99) with the product
of the following propositions:
A is partly red.(100)
A is partly blue.(101)
(99) implies neither (100), nor (101), nor their product. The relationship between
blue, red, and bluish-red is of a grammatical nature. In the HSV color
space, color shades are defined by their hue in terms of grades ranking from 0°
to 360°. Red has a hue of 0° or 360°, blue 240°, and bluish-red (or purple) 300°.
That bluish-red is an intermediate color between blue and red is expressed by
the following mathematical statement:
240° < 300° < 360°(102)
In the same manner, an internal relationship between various shades of reddishblue,
e.g., that violet is bluer that magenta, can be expressed as:
274° < 300°(103)
599
See §6.1.
600
BT, p. 342e.
192
Internal relations between color shades are not relations between the structures
of colors. They are grammatical relations within a color space meaning that
they are relations within structures used for a surveyable representation of col-
ors.601
“Here we have a sort of mathematics of colour.”602

In the previous discussion, I took blue and red to be simple colors. But let me
pause to look more closely at the issue of simplicity of colors. That red is simple
is a fact about our grammar. This is to say that to be simple is an internal
property of red, blue, or white inter alia. Here is a confirmation of this idea in
Geach’s lecture notes:
Jackson: “White is simple” is timeless.
Wittgenstein: “White is simple” is to “the room’s colour is simple” as “7 is prime” is
to “the number of papers in my pocket is prime”. It makes no sense to ask what white
would be like if it weren’t simple.603
As we know from §10.4, timeless sentences express internal properties and relations.
Now if grammar is autonomous, i.e., not answerable to extra-linguistic
reality, then what counts as a simple color is arbitrary. One community may
count green as a simple color, but another as an intermediate one. Furthermore,
is the question of simplicity of colors only a matter of choosing this or that color
space? The answer is obviously no, because we have to take into consideration
our experience of colors. We can therefore ask: is green/red/yellow a simple
color, or an intermediate color? Why is orange or violet an intermediate color?
What experience would be decisive in this matter?
The point of introducing simple or primary colors is to describe all other color
shades as combinations (mixtures) of these simples. What we are after, therefore,
is an experience of color mixtures as combinations of certain simple colors.
We have to select simple colors in such a way that given any color shade
we will be able to recognize what it is a mixture of. Let us begin with an unproblematic
example: “In any case, orange is a mixture of red and yellow in a
601
Here I am indebted to Ondřej Beran.
602
ROC III, §3.
603
PGL, p. 18.
193
sense in which yellow isn’t a mixture of red and green.”604
Perceiving an orange
spot, one can experience the resemblance between orange red and yellow. This
experience is not available in the case of yellow if one tries to see it as a mixture
of red and green. This kind of evidence should convince us that orange is
an intermediate color between yellow and red and that yellow is not an intermediate
color between red and green. The same is valid of green. We cannot
describe it as bluish yellow or yellowish blue and hence green is a simple col-
or.605
Here is another kind of experience:
It makes sense to say of a colour that it is not pure red, but contains a yellowish, or
bluish, whitish or blackish tinge; and it makes sense to say that it contains none of
these tinges but is pure red. In this sense one can speak of a pure blue, yellow, green,
white, black, but not of a pure orange, grey or reddish-blue. […] That is to say, the
colour circle has four special points. For it does make sense to say “This orange is
situated closer to red than that one (not on the plane of the colour circle, but within
colour space)”; but it’s not an equivalent expression to say “This orange is situated
closer to bluish-red than that one” or “This orange is situated closer to blue than that
one”.606
Of intermediate colors like orange or violet, it makes no sense to say that they
contain a reddish tinge. Simple colors are also those that can be experienced in
a pure as well as a tinged form. This is to say that the difference between pure
and tinged makes sense only when applied to simple colors. Intermediate colors
are, in contrast to simple colors, those (i) where the distinction between pure
and tinged makes no sense, and (ii) where we are able to recognize which simple
colors they are mixtures of. Employing these criteria, Wittgenstein identified
four simple colors in accordance with the classic color theory of Ewald Hering
(1872): red, green, blue, and yellow (white and black are also simple colors
that affect the lightness and darkness dimension).
In addition to the relations of being a mixture of and resemblance, there is an
opposite relation of the impossibility of mixture. Wittgenstein’s most discussed
example is the impossibility of mixing red and green:
604
BT, p. 343e.
605
Cf. ter Hark, 1990, p. 206.
606
BT, p. 342e.
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“There is no such thing as a reddish green” is akin to the sentences that we use as axioms
in mathematics.607
It is obvious at a glance that we aren’t willing to acknowledge anything as a colour
intermediate between red and green. (Nor does it matter whether this is always obvious,
or whether it takes experience and education to make it so.)608
The topology of our color space should also mark out the set of simple colors
and make explicit which simple colors can be mixed together and which cannot.
The point I am driving at is that internal relations within a color space are not
totally independent of our experience. Although these internal relations are not
answerable to any single fact or any single piece of experience, they must comply
with the whole of our experience and the way in which experience is interwoven
into our practices. With regard to color concepts, we can say that “It is as
if our concepts involved a scaffolding of facts.”609

Next, I shall connect the theme of internal relations within a color space with
themes in the later Wittgenstein, namely with aspect-seeing 610
and rulefollowing.
Experiencing orange as a mixture of red and yellow amounts to seeing
orange as reddish yellow or as yellowish red. The simple colors that an intermediate
color consists of are, as it were, its aspects. To see that a color shade
is reddish is a case of seeing it as red. As already stated, not all combinations of
simple colors are possible. It is impossible, for example, to mix red and green.
This means that we do not have the ability to see red as greenish—or we cannot
imagine what a greenish red would look like.
An important point in this discussion is that we do not need to experience an
intermediate color as a mixture of simple ones all the time. In the same vein, an
aspect may change and we can experience a change or dawning of an aspect.
Wittgenstein describes such an experience in the following words: “what I perceive
in the dawning of an aspect is not a property of the object, but an internal
relation between it and other objects.”611
What this cryptic remark could mean
607
Z §346.
608
Z §359.
609
PI §350.
610
See Lee, 1999, pp. 238f. Following paragraphs draw on Lee’s account.
611
PI II, p. 212.
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is discussed in Chapter 17. It is clear, however, from this remark that an aspect
may change; an aspect may remain unnoticed, or it may dawn suddenly, etc. We
can make analogous claims about colors and their tinges: a tinge may remain
unnoticed, or it may suddenly occur to us. Let us consider a colored object, e.g.,
an apple. One may describe the apple as having a red color. Later on, one may
notice that the color of the apple has a yellowish tinge. This experience can be
described as a dawning of an aspect and the perceiving of an internal relation
between the color of the apple and yellow. Hence, Wittgenstein can say that:
‘internal properties’ of a colour gradually occur to us, which we hadn’t thought of at
the outset. And that can show us the course of a philosophical investigation. We must
always be prepared to come across a new one, one that has not occurred to us earli-
er.612
The idea is that at the outset, we learn color concepts by reference to examples
and paradigmatic samples, and later on, in the course of language acquisition,
realize that our color concepts are interconnected in various ways. The perception
of these internal relations is, however, conditioned by the experience of the
dawning of an aspect. Such an experience does not need to occur or there might
be people incapable of it. This kind of incapability Wittgenstein calls “aspect-
blindness”.613
However, aspect-blindness is not achromacy, i.e., the inability to
distinguish any color from grey. Aspect-blind people are naturally able to distinguish
between different color shades. What they are lacking is the ability to
experience certain relations between different color shades. They may, for instance,
fail to observe that orange is yellowish red. What they are able to do,
however, is use color concepts defined by reference to paradigmatic samples.
For aspect-blind people, the set of simple colors must be greater. If someone
were incapable of experiencing orange as yellowish red, for them orange would
be a simple color.
We can now think of our incapacity to see reddish green as a kind of deficiency,
and we can imagine people that lack this ability. Wittgenstein was preoccupied
with such cases in his Remarks on Colour:
612
ROC III, §63.
613
The concept of aspect-blindness is broader, for it covers non-optical aspects too. Our present
concern is limited to aspects of colors.
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There could be people who didn’t understand our way of saying that orange is a rather
reddish-yellow, and who would only be inclined to say something like that in
cases where a transition from yellow through orange to red took place before their
eyes. And for such people the expression “reddish-green” need present no difficul-
ties.614
The idea is that there might be people with a certain kind of aspect-blindness,
but are able to experience aspects that we cannot.615
Is this idea coherent? How
do we recognize such people? Wittgenstein suggests the following thought-
experiment:
Someone who is familiar with reddish-green should be in a position to produce a colour
series which starts with red and ends with green and which perhaps even for us
constitutes a continuous transition between the two. We would then discover that at
the point where we always see the same shade, e.g. of brown, this person sometimes
sees brown and sometimes reddish-green. It may be, for example, that he can differentiate
between the colours of two chemical compounds that seem to us to be the
same colour and he calls one brown and the other reddish-green.616
Of course, there are continuous transitions from red to green. However, people
with normal eyesight are able to produce such transitions only through another
color, e.g., through yellow.617
People with the ability to recognize reddish green
are able to produce this transition directly without referencing to any other simple
color. Suppose now there were a single person claiming that they could differentiate
between brown and reddish green. Such a case would be an instance
of a private language, however. The case would be different if there were a
614
ROC I, §78.
615
Recent empirical research suggests that certain color-blind observers may have their color
space extended relative to the norm and are hence able to distinguish color shades that
most people cannot. See Mollon et al., 2005.
616
ROC I, §11.
617
In the geometrical model of Hering’s opponent theory, red and green are separated by
yellow. This fact was widely accepted in Wittgenstein’s times. However, recent investigations
of the so-called filling-in phenomena (see, for example, Livitz, 2011) have cast doubts
upon this fact. Reddish green nevertheless seems to be possible, which means that red and
green are not opposite colors. This evidence does not disprove Wittgenstein’s arguments.
The geometry of our color space is, however, different from the one Wittgenstein adhered
to.
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community of people who were able to distinguish between brown and reddish
green and they agreed in their judgments on this matter.
This brings us to the problem of rule-following. Consider a rule ‘add a slight
greenish tinge to a given color shade’ or in mathematical notation ‘+G’. As established
in Chapter 13, there must be an internal relation between a rule and its
application. Accordingly, there must be an internal relation between this rule
applied to a red sample and a sample of greenish red. Does this consideration
force us to admit that there might be people who “see colours which we do not
see”618
? The answer is no, for “[t]here is, after all, no commonly accepted criterion
for what is a colour, unless it is one of our colours.”619
The ability to distinguish
between greenish red and brown does not need to be on a par with the
ability to distinguish between yellowish red and pure red. There are other visually
discriminable qualities that are not properly called colors.620
We can speak
of something’s being ‘gold-colored’, but gold is not one of the shades of yellow.
“We speak of the ‘colour of gold’ and do not mean yellow. ‘Gold-coloured’ is
the property of a surface that shines or glitters.”621
Such visual qualities are the
subject of the next section.
15.4. Colors in the visual field, transparency
As already noted at the beginning of §15, not all color concepts are logically of
the same sort. This is, I think, the main insight pursued in Remarks on Colour.
Some color concepts stand for simple colors, some for intermediate ones. In this
case, we do not need to speak about different logical kinds, because what all
these color concepts have in common is that they can be defined by references
to paradigmatic samples. Such samples are typically two-dimensional plates
and reference is made to their surface colors. This is, however, an idealization,
for we perceive colors in our visual space. In this space, we are able to distinguish
qualities that cannot be defined by reference to paradigmatic samples.
The most salient quality that Wittgenstein discusses is color transparency or
618
ROC I, §14.
619
ROC I, §14.
620
Cf. Lee, 1999, p. 237.
621
ROC I, §33.
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opacity.622
Some colors (like green) can be transparent, some (like white) can-
not.
That white is an opaque color (or that we cannot imagine a white transparent
object) seems to be a grammatical rule. We could therefore think of incorporating
these characteristics of colors into our color space; we might add an additional
dimension into this space. And indeed, Wittgenstein ascribes this suggestion
to Philip Otto Runge:
Runge: “If we were to think of a bluish-orange, a reddish-green, or a yellowishviolet,
we would have the same feeling as in the case of a southwesterly north
wind.... Both white and black are opaque or solid.... White water which is pure is as
inconceivable as clear milk.”623
According to this suggestion, transparency and opacity would be internal properties
of colors, because the claim that white is opaque is timeless. Wittgenstein
says, however, that “[o]paqueness is not a property of the white colour. Any
more than transparency is a property of the green.”624
Why is opacity not an internal
property of the color white? Alan Lee considers the explanation that
“opacity and transparency are properties of objects, [but] independent of their
colours”625
. But even then we would be tempted to say that opacity is internal to
the concept of white. If opacity were an internal property of white, we would
neglect the fact that we have color concepts of different sorts. Opacity is not a
property of the color white if this color is defined by reference to a paradigmatic
sample. If anything, opacity is an internal property of the concept of white in
our visual field: “Transparency and reflections exist only in the dimension of
the depth of a visual image.”626
The aim of color spaces is twofold: to define concepts for simple colors by reference
to paradigmatic samples and to define internal relations between color
concepts. It now seems that if we apply color concepts in our visual field, they
622
Wittgenstein considers other qualities as well, for example, glitter, reflection, mirroring
of light, etc.
623
ROC I, §21, Wittgenstein’s ellipses.
624
ROC I, §45.
625
Lee, 1999, p. 232.
626
ROC I, §19.
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acquire additional internal properties and relations. If these relations are internal,
there must be grammatical rules vindicating these relations. Wittgenstein
considers the kind of rules that may fulfill this function. They are the rules of
appearance for transparent colored things.627
Applying such rules628
we should
be able to infer that there cannot be transparent white objects.
In the course of his reflections, Wittgenstein also questions this suggestion:
“Why can’t we imagine transparent-white glass,—even if there isn’t any in actuality?
Where does the analogy with transparent coloured glass go wrong?”629
These questions point out the deeper problem of the status of rules for appearance.
Are these rules for appearance grammatical rules? Or is our inability to
imagine transparent-white glass of a factual nature? This inability might be
rooted in our biological nature630
—similarly to our inability to imagine reddish
green. The last sentence quoted above is followed by this general remark:
Sentences are often used on the borderline between logic and the empirical, so that
their meaning changes back and forth and they count now as expressions of norms,
now as expressions of experience.631
This remark may be taken as Wittgenstein’s last word on this, his last methodological
hint, which says that one and the same sentence may be used to express
both an internal as well as an external relation. What is important, however, is
that this ambiguity cannot occur within one language-game. We must strive to
split the language-game in order to cope with this ambiguity.632
627
Cf. ROC I, §29.
628
The rule might be something like this: “Every coloured medium darkens that which is
seen through it, it swallows light.” (ROC I, §30)
629
ROC I, §31.
630
These are among the “very general facts of daily experience” (RFM, p. 82) or the “general
facts of nature” (PI §142). See fn. 493, p. 75 for further references.
631
ROC I, §32.
632
This might be implied by Alan Lee (1999, p. 239). He thinks that “[t]he problematic status
of white among the colours presents a problem of demarcation between internal and external
relations of the colour system.” Once we admit that the distinction is relative to a language-game,
the problem vanishes.
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16. The standard meter
The analogy between (the rules of) language and a measuring rod or yardstick
(Maßstab) plays an important role throughout Wittgenstein’s philosophy. The
analogy goes roughly like this: as a measuring rod is laid against an object in
order to measure it, so language is laid against reality. It is, however, employed
differently in different stages of Wittgenstein’s thinking. In the Tractatus, a picture,
actually a proposition, “is laid against reality like a measure”633
, whereas
later in the Philosophical Investigations, a language-game is an object of comparison,
a measuring rod.634
In this chapter, I am going to examine the most famous
case of a measuring rod, namely Wittgenstein’s discussion of the standard
meter in Paris635
and his focus on a reflexive case in this analogy, i.e., the case
in which the standard meter is supposed to measure itself.
This chapter is framed by the following questions: Is the standard meter one
meter long? Can we assert this proposition? Is it an empirical or grammatical
proposition? Is this sentence meaningful at all? My investigation is driven by
the simple idea that all these claims must be related to a context of a languagegame.
That a certain sentence can or cannot be asserted or that it is an empirical
or grammatical proposition depends on a language-game. One and the same
sentence can be an admissible move in one language-game, a grammatical
proposition in another and even nonsensical in a third. These language-games
can be interconnected in various ways. My aim is to reveal these connections. I
will focus, in particular, on the relations between the language-game in which
the standard meter is defined, and the language-game where it is used as a pro-
totype.
633
TLP 2.1512; see §9.2 above.
634
PI §131.
635
PI §50.
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16.1. The definitions
Wittgenstein says that the only thing of which one can say neither that it is one
meter long nor that it is not one meter long is the standard meter in Paris.636
Kripke says in reply that he must be wrong;637
the stick which is actually the
standard meter is (but might not have been) one meter long as a matter of fact.
Most commentators have sided with Wittgenstein.638
Whether one could meaningfully
assert that the standard meter is one meter long depends on how one
meter is defined. In this chapter, I would like to suggest three possible definitions.
Both Wittgenstein and Kripke would agree that the meter must be defined
via an ostensive definition (or an act of baptizing, to use Kripke’s term). All the
definitions make use of an arbitrary chosen rod which shall play the role of
standard meter henceforward. Let us call it S.
(i) The first definition states that everything that has the same length that
rod S had at time t0 is one meter long.
(ii) The second definition states that any object with end-points coinciding
with those of rod S at any time (if the object were placed alongside rod S)
is one meter long.
(iii) The third definition states that—in a literal sense—rod S alone is one
meter long.639
Other objects are one meter long—although in a metaphorical
(or, rather, a synecdochical) sense—if their end-points coincide with
S when placed alongside it.
636
By the expressions ‘the meter rod’, ‘the standard meter’, ‘the rod that is the standard meter’
or ‘the rod that is just the standard meter’ I refer de re to the rod in question. I will indicate
a de dicto reference explicitly by the expression ‘the role of the standard meter’.
637
Kripke, 1980, p. 54.
638
See, for example, Loomis, 1999; Pollock, 2004; Baker and Hacker 2005a, pp. 189–199;
Dolev, 2007; Avital, 2008.
639
As far as I know, this idea has been suggested only by Fogelin (2002, p. 127): “To begin
with, it may not seem obvious that we cannot say of the standard meter that it is a meter
long; indeed, we may be inclined to say the opposite, that it is the only thing that really is
one meter long” (italics in the original). Fogelin, however, does not attribute this view to
Wittgenstein. This idea should not be confused with the related idea that the standard meter
is exactly one meter long and all other objects might be one meter long only approximately
or with some degree of tolerance (Pollock, 2004, p. 151), (Avital, 2008, p. 337).
202
The first definition is to be found in Kripke, for he uses the following wording:
“one meter is to be the length of S at a fixed time t0.”640
The second definition
has often been attributed to Wittgenstein,641
while the last definition is the proposal
I would like to advocate. I shall argue that the third definition can be attributed
to Wittgenstein as well. Moreover, I want to show that Wittgenstein’s
analysis will be better understood if we employ the third definition.
For Kripke, the statement ‘stick S is one meter long’ is a priori but only a contingent
proposition. Rod S may vary its length over time (and in counterfactual
situations as well) and, in this case, will cease to be one meter long. Following
the second definition, one cannot say that S is (or is not) one meter long, for a
measuring instrument cannot be applied to itself. That S is one meter long is not
an empirical proposition but a grammatical proposition which is actually—in
contrast to Kripke’s definition—timeless.642
The third definition is the reverse of
the second one. It starts by fixing the length of the standard meter and then applies
this unit of measurement to all other spatial objects in a derived sense.643
The second definition starts by defining the length of all objects; however, it
cannot proceed to the standard meter, because it is the blind spot of the system.
16.2. The longitudinal variability
The crucial point here is, of course, the longitudinal variability of all spatial objects.
The standard meter is no exception. It does not matter what the cause of
the variability is (e.g., temperature, pressure, or even human action).644
How do
640
Kripke, 1980, p. 54.
641
See Loomis 1999, p. 304; Pollock, 2004, p. 153; Baker and Hacker, 2005a, p. 172.
642
The fact that Wittgenstein’s definition of the standard meter is timeless will be important
later in this chapter where I will discuss the standard meter in connection with the distinction
between internal and external relations. See my discussion of the criterion of temporality
in §10.4.
643
I do not want to stipulate the counterintuitive claim that no object except the standard
meter is one meter long. By endorsing the third definition, I want to point out that the sense
in which the standard meter could be said to be one meter long is different from the sense in
which other objects could be said to be one meter long.
644
One could raise the objection that the length of the meter cannot change, since the current
definition of the meter is fixed by the speed of light. The meter is the length of the path
travelled by light in a vacuum during the time interval of 1⁄299 792 458 of a second. One
203
we detect that an object’s length has changed? The answer is simple: by means
of measurement using the standard meter (or a measuring instrument, e.g., a
ruler which is derived from the standard meter). But how do we detect whether
the length of the standard meter has itself changed? Consider the following
thought-experiment that is central to my argument. If all objects including all
particular measuring instruments except the standard meter were suddenly doubled
in size, then we would have a good reason to suppose that the standard meter
had been halved. That is, I suggest, the only possible answer if we define the
meter along Kripke’s lines. This would mean, however, that rod S had been deposed
from the role of standard meter. This role would no longer be served by
rod S but by a set of other measuring instruments (provided they all matched) or
their average or median length. Who could decide whether rod S had been
halved or whether the other measuring instruments had doubled in length? All
the evidence equally supports both hypotheses. Every situation could be interpreted
in such a way that the standard meter had remained unchanged, and to
take the role of the standard meter seriously means to accept this interpretation
in every case. So this is also the gist of my argument.
Kripke admits the possibility that the standard meter could have changed its
length; this possibility, however, cannot be detected in an incontestable way.
This is due to the fact that the paradigm of one meter is not a concrete object
(rod S) but an abstraction, i.e., the length which it accidentally had at t0.
We can try to put the point in a more formal way. Consider these formal transcriptions
of our three definitions:
(I) One-meter-longt(x) =df lengtht(x)=lengthtₒ(S)
(II) One-meter-longt(x) =df lengtht(x)=lengtht(S)
(III) One-meter-longt(x) =df (ιx) x=S
can, however, ask—as physicists actually do—whether the speed of light is constant or
changing in time, cf. the so-called variable speed of light hypothesis. The philosophical lesson
would be the same, although there is no natural intuition with respect to how such a variability
could be detected. The general question should be, thus, about whether a physical
unit can be defined at all.
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One-meter-long̕t(x) =df (y) (One-meter-longt(y) 
lengtht(x)=lengtht(y))645
One-meter-long is a predicate which applies to spatial objects and is parameterized
by time. Length is a primitive function which gives the length of a spatial
object (and is parameterized by time too). One may, however, wonder why we
are trying to define one meter when we already have what we wanted. If we
employ a function which gives the length of an object as a numerical value,
then the unit of length must already be defined.646
Instead of the function length(x), we can employ a primitive binary predicate of
same-length(x,y) which intuitively matches the possible coincidence of x and y
in their end-points. Now we can try to restate the formal definitions this way:
(I’) One-meter-longt(x) =df same-lengtht(x,Stₒ)
(II’) One-meter-longt(x) =df same-lengtht(x,S)
(III’) One-meter-longt(x) =df (ιx) x=S
One-meter-longt(x) =df (y) (One-meter-longt(y)  same-
lengtht(x,y))
We can see that definition (I’) is not correct, or we must give up the idea that S
is a rigid designator, because S is not a name or a definite description. S is a
function here. For the definition (II’) a problem arises when we apply the predicate
one-meter-long to the rod S itself. Then we would have to deal with the expression
same-lengtht(S,S). We can simply stipulate that the predicate or, better,
the relation same-length is reflexive. But then the predicate will no longer cor-
645
Definition (III) states that one-meter-long is the unique thing (ιx) that is identical with the
standard meter. One-meter-long’ (that is, one-meter-long in a derived sense) refers to everything
whose end-points would coincide with those of the standard meter when placed alongside
it.
646
Definitions (ii) and (iii)—and their formal counterparts (II) and (III)—require a possibility
of coincidence, not that the coincidence is actually taking place. These definitions demand
that the two rods must be comparable not compared. This means, however, that the
unit of length must be defined in advance. There is a parallel issue about defining the notion
of number in Waismann’s notes. It is argued here that one “cannot explain number by means
of correlation […] you can explain it by means of possible correlation, and this precisely
presupposes number” (WWK, pp. 164–5). See also PPO, p. 373.
205
respond to the possibility of the process of placing two objects side-by-side.
The third definition avoids such counterintuitive reflexive uses of the predicate
same-length, leaving it undefined.
If one cannot assert that rod S is one meter long, then a fortiori one cannot assert
that rod S has changed its length either, because—as demonstrated by the
thought-experiment above—every apparent change in (the length of) the standard
meter has to be interpreted as a change in all other objects. Both eventualities
(that the standard meter is one meter long and that it has changed its length)
are, however, perfectly conceivable and this makes Wittgenstein’s account
counterintuitive and puzzling. All change, to be sure, has to be conceived of
with respect to a stationary background, a hypokeimenon. When we recall the
dilemma discussed exposed in the previous section, by choosing S as the standard
meter we stated that all change is to be conceived of relative to S. The
standard meter is thus an arbitrarily chosen fixed point (or rather: a fixed vector)
in our (metric) system of measurement. The background (hypokeimenon)
here is—and this is Wittgenstein’s point—nothing metaphysical, but only a
chosen role in this system.647
That is, I think, a sufficient argument for the claim
that the standard meter cannot change its length.
The conjecture that the paradigm of a unit of length cannot change its length
does not free us from the effort of keeping the rod, ideally, in a constant environment.
The environment where the rod S is stored is actually a part of the definition
of the standard meter. Once it was The International Bureau of Weights
and Measures in Paris; now it is ‘a vacuum’.648
A definition of the standard meter
which does not take into account the influence of the environment is, of
course, possible; it is, however, as misguided as that of a word which changes
its meaning according to the days of the week—an example offered by Wittgenstein.
The consequences of this move must be clear: definitions of the standard
meter which differ in their references to the environment are distinct from one
another. The standard meter defined by the rod which is stored in a vacuum is
647
To recall footnote 644, what is the background against which we could measure the variability
of the speed of light? The variability of the speed of light makes sense only if the
remaining fundamental physical constants (these are the electron charge and Plank’s constant)
are fixed. For a detailed discussion see Uzan, 2003.
648
Cf. footnote 644.
206
different from the standard meter defined by the same rod stored in water. Having
two such definitions, we can compute the relative contributions of the environment
to thermal and volumetric expansion, etc.
16.3. The third definition
Heather Gert has tried, however, to demonstrate that it was not Wittgenstein’s
intention to assert that one cannot say of the standard meter that it is or is not
one meter long. Wittgenstein’s later writings actually have a peculiar dialogical
structure which implies that not all sentences express Wittgenstein’s own view.
Focusing on this view, Gert argues that
the claim that the standard meter cannot be said to be a meter long is introduced as
analogous to the claim that elements cannot be said to exist, a Tractarian claim the
later Wittgenstein clearly rejects. […] Wittgenstein does not merely reject his own
earlier claim that there are elements to which existence cannot be attributed, he rejects
the claim that existence cannot be attributed to an object within the languagegame
within which it plays the role of element.649
In any case, we can take this interpretation as an indication that a straightforward
attribution of definition (ii) to Wittgenstein might be problematic. Wittgenstein
employs the standard meter above all as an analogy to the more general
problem of whether primary elements can be described or named only. But
such a question is the wrong one. In order to describe an object, we must be
able to name it. So we have to assign a name to it in advance. In Wittgenstein’s
words: “For naming and describing do not stand on the same level: naming is a
preparation for description. Naming is so far not a move in the language-game–
any more than putting a piece in its place on the board is a move in chess”650
.
So a game of chess presupposes a preparatory game of the naming of the pieces.
In a similar way: a language-game of measuring presupposes a preparatory
game of the fixing of a unit of measure, its standard, and the whole method of
measurement.
649
Gert, 2002, pp. 65f.
650
PI §49.
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Gert argues, however, that we can use the word as a name and as a description
“even within a given language-game”651
. Then, by analogy, we could use the
expression ‘one meter long’ as a name (of the standard meter bar) and as a description
(of all other objects that are one meter long) within the same lan-
guage-game.
I do not find Gert’s line of reasoning convincing on this point. The naming of a
piece is not a move in chess, although it is a move in the language-game of ostensively
teaching the names of the chess pieces. Of course, nothing prevents us
from taking both activities as one language-game, for—as Jolley stated recently—“there
is […] no natural unit for the counting of language-games.”652
The
upshot is that we can take the whole of language as one language-game. But
Wittgenstein introduces the notion of a language-game inter alia in order to
separate out language activities that might have the same verbal manifestations.
Thus, the word ‘R’ can stand for the name of an element or for a description of
a simple complex consisting solely of precisely this element. This can lead to
philosophical confusion. Introducing different language-games—an introductory
game of naming and a game of describing—helps us to become aware of
what we are doing and therefore to get over this confusion. Here we have a special
case of the strategy outlined in §10.3. There I distinguish between the language-game
of teaching and the language-game of applying a rule. Naming is a
kind of teaching training and describing is the special case of applying a rule.
Thus, we can distinguish between two language-games in which the words ‘R’
or ‘one meter long’ appear in their different meanings, or where they appear in
two uses both within a single language-game.653
The latter option, however,
poses the danger that these two uses could be confused. Thus, the sentence ‘R
(exists)’ might stand for ‘There is an element called (or named) R’, but also
‘There is a complex consisting solely of R’. By analogy, ‘X is one meter long’
651
Gert, 2002, p. 63.
652
Jolley, 2010, p. 116. Language-games are like geographic regions. There are no facts of
the matter as to when we have two different regions or a single one. Thank to Jim Klagge
for this analogy.
653
Cf. Jolley, 2010, p. 116. A close reading of §50 of the Philosophical Investigations vindicating
the conjecture that the expression “one meter long” can appear in different meanings
is to be found in Gierlinger, 2010.
208
might mean ‘X is the rod that has been stipulated (or named) as being one meter
long’, but also that ‘X is as long as the standard meter’.
Definition (iii) is in accord with this analysis. In a preparatory language-game,
we have to choose a standard of measurement—specifically, we have to name
rod S as the standard meter.654
Such a naming must have a declarative character;
and as is noted in §10.3 above, it can be taken as an imperative. This is the
first step of the definition (iii): saying that rod S alone is one meter long.655
After
this is done, we can then play the language-game of measurement, which
takes the name of the standard meter as assigned and the length of its extension
(that is, rod S) as invariable. This means that in this language-game, the (literal)
meaning of the term ‘standard meter’ is borrowed from the preparatory game
and applied in a derived sense to all the other objects. There is, thus, a vertical
relation between these two language-games.
16.4. Internal and external relations
In this section I want to connect the problem discussed above with the distinction
between internal and external relations. This will lead us to a general characterization
of language-games in which the sentence ‘The standard meter is
one meter long’ is then found to lack any sense. Lacking sense means here that
this sentence is not a move in a language-game, as it cannot be asserted or ne-
gated.
In his conversations with Waismann,656
Wittgenstein says that the relation of
being longer than between two sticks is external, whereas the relation of being
greater than between their lengths is internal. Let us slightly modify Wittgen-
654
This preparatory language-game does not consist solely in the act of pure ostensive naming.
It is more likely that the reference of the expression “the standard meter” has to be fixed
by appropriate training. Moreover, as already noted, the environment in which the standard
meter is stored should also be fixed. The preparatory language-game is, thus, rather a complex
issue. See §10.3 for details.
655
Formulations of such a definition can be found in Wittgenstein’s manuscripts from the
early 1930s. Viz. Wittgenstein’s definitions “1m = die Länge des Pariser Urmeters” or “1m
ist die Länge des Pariser Urmeters” [1 m is the length of the standard meter in Paris] from
Ms 113, p. 23r; Ms 115, p. 58; Ts 211 p. 569f.; Ts 212, p. 704; Ts 213 (BT), p. 241.
656
WWK, p. 54; see §10.2 above for more examples.
209
stein’s example. Let rod A be as long as the standard meter. This, then, is an external
relation. The length of rod A is then the same as the length of the standard
meter; this is an internal relation which can be expressed as 1 = 1. Hence all objects
that are one meter long are instances of a property that is internally related
to the length of the standard meter by the relation of identity. By the same token,
all objects that are longer than one meter have a property that is internally
related to the length of the standard meter by the relation of being longer than.
What sort of relation is expressed by the predicate same-length? In definition
(I’), the second argument is parameterized by time, which gives us the idea that
the whole predicate same-length is temporal and according to the criterion of
temporality657
it thus expresses an external relation. The temporal parameter,
however, does not appear anywhere on the right-hand side of the second (II’) or
third definition (III’). Thus, the temporal parameter can be canceled out of these
definitions and the predicate same-length, according to these definitions, can be
regarded as timeless, i.e., as expressing an internal relation.
Before turning our attention to the question of whether propositions that express
internal relations can be asserted or negated, I am going to discuss a recent account
by Doron Avital, who employs the notion of an internal relation in order
to defend Wittgenstein’s claim about the standard meter against Kripke’s criticism.
Avital argues—in favor of definition (ii)—that the two end-points (taken
as numerical coordinates in Euclidean space) of rod S, which is just the standard
meter, exemplify an internal relation of measure. Another object R is one meter
long if its end-points exemplify the same internal relation as the end-points of S.
We can put this correspondence in terms of internal properties rather than internal
relations. Rod S (the standard meter) has the internal property of being one
meter long, while rod R is one meter long if it has the same property. This is,
essentially, the Tractarian model of picturing as examined in §9.2. A picture or
“a proposition describes reality by its internal properties.”658
One can also assert that ‘R is one meter long’, because R may have other endpoints
that exemplify another internal relation, e.g., being two meters distant.
But we cannot assert sentences like ‘S is one meter in length’ or ‘S is one meter
657
See §10.4.
658
TLP 4.023.
210
in length at t0’ because “they are true by necessity”659
. Such sentences cannot be
negated and hence they cannot be asserted either. Hence, in order to decide
whether a sentence can be asserted or not, Avital employs the bipolarity criterion
of meaning. If nothing else, this criterion or principle is problematic in Wittgenstein’s
later work.660
I have argued that there must be an internal relation between
the length of the standard meter and the lengths of all other objects that
are one meter long. Avital considers a Euclidean space whose unit vector is
marked by the standard meter. The internal relations that Avital has in mind are
different ones; they hold true between those points of Euclidean space which
are one meter apart. This distance is easy to compute by Pythagoras’ theorem.
Let’s take a two-dimensional Euclidean space and two arbitrary points A[a1,a2]
and B[b1,b2]. Then their distance is d(A,B)=((a1 – b1)2
+ (a2 – b2)2
). Avital advances
the suggestion of an internal relation between the abstract points (whose
distance is 1). I propose, however, to consider an internal relation (of identity)
between the lengths (whatever they are) of the objects that are being compared.
Clearly, if two pairs of points are related by the former relation, their distances
are related by the latter relation and vice versa.
659
Avital, 2008, p. 323.
660
The principle of bipolarity states that “every proposition must be capable of being true,
and capable of being false”. (Glock, 1996, p. 63) Wittgenstein rejected this criterion explicitly
as a “mythology of symbolism” (Z §211). Sentences that would have been recognized
and withdrawn as nonsensical according to the principle of bipolarity are conceived of in
Wittgenstein’s later work as rules of a game, as part of a kind of mythology. Some sentences
are of the form of empirical sentences, but they have been hardened and become rules. See
(OC §§95–99.) This is the case of the sentence ‘Rod S, which is the standard meter, is one
meter long’.
211
Figure 6
If rod S, which is just the standard meter, changes its length as a result of a
change in circumstances, e.g., in temperature, one would expect that the unit of
length would change. Avital argues, however, that such a change affects all other
objects too, for they are in the same space. If, for example, the temperature
goes up, such a change in the environment influences other objects besides the
standard meter. That is Avital’s main point. “Meanings in this respect are mate-
rialized”661
. The measuring rod must be in the same space as the measured object
(this is the maxim of “same space”662
) and they have to be subject to the
same physical law. This ensures that whatever the changes of circumstances are,
the relative ordering of spatial objects according to their lengths remains the
same. Hence, Avital can conclude: “The rod being sampled extends and shrinks
in tandem with the objects belonging to the sample as so does the Metre, as this
661
Avital, 2008, p. 323.
662
Cf. §9.2 where the maxim of same space is considered in the Tractarian framework.
B[1,0]Standard meter
An internal relation between the lengths of
the rods (my proposal)
d(A,B)
d(C,D)
C[0,1]
A[0,0] Standard meter
D[1,1]
B[1,0]
Measured object
y
x
C[0,1]
A[0,0]
D[1,1]
Measured object
y
x
Internal relations of measure between the endpoints
of the rods (Tractatus and Avital, 2008)
212
is materialized in the form of the chosen rod; hence, the measures it gives are
independent of the changing circumstances”663
.
I think that it is misguided to invoke Euclidean space at this point. Such a consideration
grants too much to Kripke’s analysis. Only if we ‘sublime’ our space
into an ideal Euclidean space, can we then consider extending and shrinking the
standard meter. Avital is right in his critique of Kripke that all changes in every
spatial object must be regarded in relation to the standard meter, and not to an
abstract space. Avital, however, needs this abstract space in order to enable the
possibility of the changing of the standard meter, although this possibility is not
detectable.
My proposal is to discard the idea of abstract space and consider the standard
meter to be unchanging all of the time. Moreover, the standard meter must be
taken to be unchanging even if its extending and contracting were governed by
different physical laws than other objects are. It might be changed by human
action; e.g., someone might cut a part of it off. Even in this case, we have to regard
the standard meter as unchanged if we want to meet all the requirements of
the role (of the standard meter).664
We cannot argue at this point that objects in
space and time are subject to the same law as regards their longitudinal variability.
It is possible, after all, that the same change of circumstances causes the
shrinking of one object and the extension of another. Raising the temperature
from 0°C to 3.99°C would simultaneously cause the contraction of a water column
and the extension of a mercury column.
To sum up, for Avital there is an internal relation of measure between the endpoints
of all objects that are one meter long. Then, by derivation, there is an internal
relation of having the same length between the lengths of all objects that
are one meter long. I propose to begin with this latter kind of internal relation,
and not to consider the idea of Euclidean space beyond our visible space.
663
Avital, 2008, p. 331.
664
In practice, if something like this happened, we would abolish this unit of measure and
define a new one. Cf. Dolev quoting Reichenbach: “if an earthquake should ever throw it
[the standard meter] out of its vault and deform its diameter, nobody would want to retain it
as the prototype of the meter” (Dolev, 2007, p. 133).
213
16.5. What can be said about the standard meter
It should now be clear from the previous discussion what it means for an object
to be one meter long. The situation of an object being one meter long amounts
to the object’s length being internally related to the length of the standard meter.
If we consider rod S, which is currently the standard meter, then to say that it is
one meter long would entail expressing an internal relation between its length
and its length. This means expressing that its length is identical with itself. Such
a reflexive use of an internal relation is suspect and possibly meaningless. Following
Cora Diamond’s paper,665
I can see precisely at this point what is problematic
about saying that the standard meter is one meter long.
Diamond points out Wittgenstein’s distinction between the transitive and intransitive
use of sentences containing the word ‘particular’ in his Brown Book: “On
the one hand, we may say, it is used preliminary to a specification, description,
comparison; on the other hand, as what one might describe as an emphasis”666
.
Let us illustrate this distinction by Diamond’s examples:
“This rod has a particular length, namely 39.37 inches.”
“This rod has a particular length,” said, for example, when one is concentrating on
the thing’s length, and not going on to specify the length or to compare it in length
with anything else.667
The first sentence relates the length of the rod to something else, namely to the
standard inch. A comparison is also being expressed between the lengths of the
rod and that of the standard inch. No such comparison is being expressed in the
second sentence. So the former sentence is an example of a transitive use, the
latter of an intransitive use (of the word ‘particular’). I want to add that an internal
relation in the transitive use of sentences is also being expressed. The
word ‘particular’ is, of course, an example here; this distinction also applies to
other words or terms including the word ‘meter’ or ‘one meter long’.
Philosophical confusion may arise when we confuse these two sorts of word
uses: “There are many troubles which arise in this way, that a word has a transi-
665
Diamond, 2001.
666
BBB, p. 158.
667
Diamond, 2001, p. 110.
214
tive and an intransitive use, and that we regard the latter as a particular case of
the former, explaining the word when it is used intransitively by a reflexive
construction”668
. Following Wittgenstein’s diagnosis, we have to look carefully
at the reflexive use of words and examine whether this is rather a case of their
intransitive use. The idea behind this is that “We often use the reflexive form of
speech as a means of emphasizing something. And in all such cases our reflexive
expressions can therefore be ‘straightened out’.” Wittgenstein’s examples
are: “If I can’t, I can’t” or “I am as I am”. These sentences can be ‘straightened
out’ into intransitive uses meaning ‘I can’t do something’ or an emphasis on the
way that I am. Wittgenstein’s next example is closer to our subject: “Suppose to
the question, ‘What’s a kilogram?’ I answered, ‘It is what a litre of water
weighs’, and someone asked, ‘Well, what does a litre of water weigh?’”669
In
order to straighten out these sentences, we have to realize that the expression
‘weight of a litre of water’ is used intransitively in this dialogue, which means
that its meaning cannot be given within this language-game. Following this definition
of kilogram, the situation of an object weighing one kilogram amounts
to the object’s weight being internally related to the weight of a liter of water
(by the relation of having the same weight). Thus, this case is perfectly analogous
to the case of asking how long the standard meter is. We can now formulate
the following tentative maxim of logical analysis: a reflexive use (a special
case of a transitive use) of a word should be straightened out into an intransitive
use. Or so it seems.
Diamond, however, does not see “anything wrong with representing a noncomparison
as a special case of a comparison; if it does lead to some philosophical
problem, it will be particularly important not to try to get rid of the problem
by simply ruling out such representations”670
. On the basis of this advice, Diamond
has to explain why we should retain transitive uses of the expression ‘one
meter long’ even in reflexive cases and not try to convert them to intransitive
uses.671
The core of her argument is based on her conviction that “we can imag-
668
BBB, p. 160.
669
BBB, p. 161.
670
Diamond, 2001, p. 120.
671
Recall the previous discussion where we could stipulate that the relation same-length is
reflexive in definition (II’). Yet this move would be drawing a veil over the problem of the
possible confusion of transitive and intransitive uses of the expression ‘one meter long’.
215
ine comparisons of length which we do not actually carry out”672
. Hence, she
admits that we can compare the meter rod with an imaginary or counterfactual
situation in which the rod has been heated and become longer.673
Even worse,
Diamond admits that: “We can easily imagine a situation in which we wake up
one morning and find that the rod that we have been using as a standard appears
to have become longer or shorter”674
. There is, of course, nothing wrong with
the mere appearance of such a change. Confusion may arise, though, if one
takes (or interprets) this appearance as a real change to the standard rod. But, as
I have argued above, there are two possibilities: The first is that such a change
is possible only after the rod has been deposed from the role of the standard meter.
From that moment onwards, the standard would be some object or objects
surrounding the former standard rod against which the change in the standard
rod was detected.675
The second possibility is that we would have to reconstruct
the situation along Kripke’s lines, because “we have separated the length of the
rod from the length ‘one meter’”676
. Then, however, we would not be comparing
the standard meter with the same rod in a counterfactual situation in which
it has been heated, but its actual length with its length after it had been heated.
After all, this means that the standard is not the rod, but its length at the moment
of establishing the standard, i.e., at the moment of baptism. This is pretty
much Kripke’s point. Diamond, then, insists on allowing a reflexive use of the
expression ‘one meter long’, for in her view, it is possible that it becomes a
genuine transitive use involving a real comparison.
By now, it should be clear that the sentence ‘A is one meter long’ involves a
comparison of rod A and the meter rod, and it expresses the internal relation between
the lengths of these two rods. The reflexive case ‘The meter rod is one
meter long’ (which would involve a comparison of the standard meter with itself)
should be straightened out into an intransitive use. This, however, means
672
Diamond, 2001, p. 116.
673
As noted in footnote 646 above, the comparison need not actually take place; it must be,
however, possible to carry it out. Diamond’s counterfactual situation goes over such a pos-
sibility.
674
Diamond, 2001, pp. 120–1.
675
If the standard meter appeared to have become longer or shorter, then there would have
to be at least one object that appeared to be unchanged.
676
Diamond, 2001, p. 121.
216
that the meaning of the phrase ‘one meter long’ cannot be defined in this very
same language-game. If it were possible, the phrase would be used transitively.
Since the sentence is not meaningless, the meaning of the phrase ‘one meter
long’ must come from elsewhere. It must be a rule (or a hinge proposition) of
this language-game which is inherited from another language-game (which I
called preparatory) where the meaning of the phrase ‘one meter long’ is to be
defined. It can be defined in various ways, e.g., by an ostensive definition
(‘This rod is one meter long’) or transitively using another unit of length (‘One
meter is 39.37 inches’) or—again transitively—as the distance travelled by light
in a vacuum during the time interval of 1⁄299 792 458 of a second.677
677
Wittgenstein mentioned this sort of ambiguity at the beginning of The Blue Book: “I
point this out to remove, once and for all, the idea that the words of the ostensive definition
predicate something of the defined; the confusion between the sentence ‘this is red’, attributing
the colour red to something, and the ostensive definition ‘this is called ‘red’’ (BBB, p.
2) See also PI §37 where Wittgenstein varies this argument with the sentence ‘That is blue’,
which is at one time a statement about an object (a case of predication) and at another time
an ostensive explanation of meaning: ‘That is called ‘blue’’.
217
17. Aspect-seeing and philosophy of psychology
Wittgenstein used the duck-rabbit figure678
to show an example of a rare phenomenon
which makes the expression ‘something is seen as something else’
meaningful in everyday language. That led him to distinguish between the
“continuous seeing” of an aspect and the “dawning” of an aspect:
Only through the phenomenon of change of aspect does the aspect seem to be detached
from the rest of the seeing. It is as if, after the experience of change of aspect,
one could say ‘So there was an aspect there!’679
However, aspects may change without achieving this specific experience, e.g.,
someone can fail to experience the ambiguity of the figure. Aspect-blind can
recognize that something is an ambiguous figure without being able to experience
it. They cannot speak of aspects, but something has changed which Wittgenstein
calls the “conception” or “way of taking” [“Auffassung”]: “If there
were no change of aspect then there would only be a way of taking”680
. Aspectblind
people cannot see aspects but only various conceptions. If they want to
report an aspect, they have to employ a conception. Thus: “An aspect is admittedly
called a conception, but a conception can persist without the persisting of
an aspect.”681
The aspect and the conception have the same expression in lan-
678
Figure 7, see PI II, p. 194.
679
Ts 229, p. 228; RPP I, §415.
680
Ms 137, p. 9b; RPP II, §436.
681
Ms 132, p. 182, my trans.
Figure 7. The duck-rabbit
218
guage.. The statement ‘It’s a duck’ can stand either for (an exclamation of the
dawning of) an aspect or only for (a report of) a conception.
Equipped with this distinction, we can now analyze the concept of aspect more
precisely. Wittgenstein says: “what I perceive in the dawning of an aspect is not
a property of the object, but an internal relation between it and other objects.”682
This is the one and only occurrence of the expression ‘internal relation’ in the
final version of the Philosophical Investigations and so we have to investigate
this remark very carefully. First of all, it is difficult to infer what the objects involved
are. Ter Hark considers three possibilities:
(i) One object is the geometrical constellation, the other is either the duck or the rabbit.
(ii) One object is the duck, the other is the rabbit. (iii) One object is the change of
aspect, the other is either the duck or the rabbit.683
Ter Hark argues against (i) as follows: (a) the duck-rabbit can be identified independently
of the duck or the rabbit and (b) the duck-rabbit is not necessary to
describe the aspects. Therefore, there has to be an external relation between the
duck-rabbit and the duck or the rabbit. These objections are valid only if the
constellation is seen neither as a duck, nor as a rabbit.684
But then the duck and
the rabbit would stand for conceptions, and not for aspects, and thus there
would be no relation at all, neither internal nor external. Ter Hark concludes in
favor of (iii). Surely there must be an internal relation between the experience
of the change of aspect and the conceptions involved. But this is not the relation
Wittgenstein means. The quotation above implies that one term in the relation is
the perceived object, i.e., the duck-rabbit. This consideration rules out the second
possibility as well. The formulation (i) should, however, be refined so that
in the dawning of the aspect an internal relation is perceived between the perceived
object (i.e., the duck-rabbit) and the duck-aspect or the rabbit-aspect re-
spectively.
682
PI II, p. 212; Ms 137, p. 128a.
683
Ter Hark, 1990, pp. 182f.
684
Cf. Jantschek, 1996, fn. 75.
219
Here is another version of this remark: “By noticing an aspect one perceives an
internal relation (between objects).”685
The internal relation does not really hold
between objects (perhaps that is why Wittgenstein struck “between objects”
out), but between concepts that are involved in their descriptions. The reason is
expressed at the beginning of Chapter 10. Any talk of internal relations between
objects has to be understood as talk of internal relations between concepts describing
these objects.686
The experience of the change of aspect is not involved
in the internal relation, although one perceives the internal relation owing to
(durch) the experience. Rather, it consists in noticing the new and previously
unnoticed aspect. Let me slightly modify the example. Instead of the duckrabbit
we now have a chaotic tangle of lines which we have no actual concept
of (i.e., we possess no description for precisely this tangle of lines, for this
shape). After a while, however, we notice that there is a rabbit in the tangle of
lines:
The expression of the dawning of an aspect is: “Now it’s this—now it’s that.” The
expression of noticing the rabbit in the tangle of lines is: “There is a rabbit here.” We
have not noticed something and now we do; there’s nothing paradoxical about this.687
Where is the internal relation in this scenario? If we possess no description of
the tangle of lines, there is nothing to relate it to. We can, however, say that after
noticing the rabbit-aspect in the tangle, we can perceive an internal property
of the tangle. We could call this the property of rabbitness.
Let me be completely clear about the internal relation in aspect-seeing: the expression
of the dawning of an aspect is as follows: ‘Now I am seeing A as B.’
An internal relation is perceived between concepts A and B.
I would now like to emphasize a connection between the concepts of internal
relation and organization. There are many kinds of internal relations and many
kinds of aspects. In seeing-as, we are dealing with aspects of organization:
685
Ms 138, p. 5a, my trans. “Man [erkennt im Aufleuchten eine | nimmt durch das Bemerken]
des Aspekts eine interne Relation (von Objekten) wahr.” The strikeouts are Wittgen-
stein’s.
686
LFM, p. 73.
687
LWPP I, §520.
220
“One kind of aspect might be called ‘aspects of organization’.”688
In one of his
manuscripts, Wittgenstein notes in a cryptic remark: “The internal relation of
structures is the organization which generates the one from the other.”689
We
can infer that in an internal relation, one term is organizing the other.
Next we have to look carefully at reflexive cases of such an internal relation.
What could it possibly mean when A is seen as A? Wittgenstein is very clear
about this question: “It would have made as little sense for me to say ‘Now I am
seeing it as…’ as to say at the sight of a knife and fork ‘Now I am seeing this as
a knife and fork’”690
. Or: “I cannot try to see a conventional picture of a lion as
a lion, any more than an F as that letter”691
. Therefore a reflexive case of the internal
relation of seeing A as A makes no sense. I cannot see a knife as a knife, a
lion as a lion or an F as an F. 692
These cases can be straightened out into the
sentences ‘I see a knife’, ‘I see a lion’ or ‘I see the letter F’.693
17.1. Fitting
The concept of fitting [passen] is briefly discussed in §11.3. Two objects fit together
if a single description holds for both. If a piston and a cylinder fit together,
then a single description must hold for their shapes. The same point can be
made with pieces of a jigsaw puzzle: they fit together if they have (at least in
part) the same shape. This means that there is an internal relation between objects
that fit together or, more precisely, between their shapes.
In this section, I want to focus on a different kind of fitting which is predominant
in Wittgenstein’s latest texts. Let us call it, tentatively, fitting underlined—
or rather conditioned—by a certain experience, by a feeling. As stated above in
the case of aspect-seeing, one perceives an internal relation owing to the experience
of a change of aspect. But in which sense do we mean ‘owing to’?
688
PI II, p. 208, italics original.
689
Ms 127, p. 215, my trans.
690
PI II, p. 195.
691
PI II, p. 206.
692
We can imagine an extraordinary language-game in which these sentences were nevertheless
meaningful. Then, of course, no straightening out would be necessary.
693
This section draws on my book (Mácha, 2010, pp. 138–142) and my paper (Mácha,
2009).
221
In Wittgenstein’s later remarks on the philosophy of psychology, the expression
‘to fit’ is intended as a substitute for the concept of psychological association.694
Psychological association should be understood causally and thus as external,
but fitting, by contrast, should be understood formally and thus as internal.
Wittgenstein shows that two phenomena can fit together in numerous examples.
The name Schubert fits together with Schubert’s works,695
Beethoven’s face fits
together with his Ninth Symphony,696
the word ‘Goethe’ fits together with its
atmosphere and with the color yellow,697
my long-familiar furniture fits together
with my room698
or two motives necessarily fit together in a musical composition,
two figures fit naturally together in a poem.699
Now, the gist of these examples
is that these connections are not psychological associations, despite the
fact that psychological associations and other causal connections might occur
here as well.700
These objects do not need to fit together as long as they are conceived
of as isolated objects. They are, rather, phenomena that fit into the whole
of our experience: “That is how this piece fits into the world of our thoughts &
feelings.”701
If two things fit together and, hence, are internally connected, then
they make up a whole.702
Let me illustrate this kind of feeling by analyzing a
longer, dense remark:
Look at a long familiar piece of furniture in its old place in your room. You would
like to say: “It is part of an organism.” Or “Take it outside, and it’s no longer at all
the same as it was”, and similar things. And naturally one isn’t thinking of any causal
dependence of one part on the rest. Rather it’s like this: I could give this thing a name
and say that it is shifted from its place, has a stain, is dusty; but if I tried taking it
quite out of its present context, I should say that it had ceased to exist and another
had got into its place.
One might even feel like this: “Everything is part and parcel of everything else” (internal
and external relations). Displace a piece and it is no longer what it was. Only in
694
Cf., for example, RPP I §337.
695
PI, p. 215.
696
RPP I, §338.
697
Ms 131, p. 149.
698
RPP I, §339.
699
CV, p. 65; Ms 134, p. 78.
700
LWPP I, §76.
701
CV, p. 65; Ms 134, p. 78.
702
RPP I, §341.
222
this surrounding is this table this table. Everything is part of everything. [I believe
Hegel meant something like this.] Here we have the inseparable atmosphere. And
what is anyone saying, who says this? What sort of method of representation is he
proposing? Isn’t it that of the painted picture? If, for example, the table has moved, you
paint a new picture of the table with its surrounding.703
Wittgenstein describes a common experience here: one is used to a certain arrangement
of everyday objects, e.g., to an arrangement of furniture in one’s
own room. The furniture may be arranged completely randomly without any
aesthetic consideration, or it may be done by someone else without taking into
account the feelings of the occupant of the room. It does not matter whether the
pieces of furniture fit together like pieces of a jigsaw puzzle.
In the second part of the remark, Wittgenstein presents two interconnected ideas,
at least on my reading: (i) We might have a feeling of the unity of our experience
which can be expressed as:
Everything is part and parcel of everything else.(104)
(ii) Objects as phenomena are what they are only within the world of our feelings,
i.e., in (felt) relations to other objects. Focusing on the first idea, we may
notice a certain resemblance to the notion of ‘feeling’ or the ‘feeling base’ in
Bradley. For Bradley, a feeling that is given in immediate experience is—or at
least can be—so rich that it can give us a sense of its identity with the whole or
Absolute.704
A feeling can transcend immediate experience towards the Absolute.
In Bradley’s words:
[Immediate experience] is a positive non-relational non-objective whole of feeling.
Within my immediate experience falls everything of which in any sense I am aware,
so far at least as I am aware of it.705
This claim has to be understood in the context of Bradley’s theory of judgment.
If one makes a judgment about one’s immediate experience (e.g., ‘This is my
table’), one has to focus on a certain part of reality. Relations to other parts of
703
RPP I, §339. The parenthesis in square brackets occurs only in an earlier manuscript Ms
131, p. 154.
704
Cf. Ferreira, 1999, pp. 10, 86 & passim.
705
Bradley, 1914, p. 189.
223
reality enter into this judgment as well (The table is what it is only in its familiar
place, in its surroundings). Although we can try to abstract all the relations
out of the judgment, the “feeling […] remains after relations have been abstracted
out of it.”706
Feeling is in this sense a feeling base which cannot be abstracted,
because it is non-relational or even non-conceptual. The feeling base
thus transcends all experience.
I do not claim that Wittgenstein was influenced by Bradley here. My point is
rather that both thinkers pertain to the same kind of philosophical intuition. Can
we say, accordingly, that Wittgenstein was an adherent of monism like Bradley?
We have to go back to Wittgenstein’s method of analysis as elaborated in §1.
Wittgenstein’s aim was to differentiate, to show what our experience is, what it
is like and what form it takes. In short, his aim was to analyze phenomena. This
analysis may take into account our—rather indeterminate—feeling that everything
is part and parcel of everything else. Wittgenstein, however, introduces
this kind of feeling with the preamble “One might even feel like this”. This
simply means that people might have this feeling. One of the tasks of the philosophical
analysis is to qualify (104). The actual question is whether it is really
the case that for any two single phenomena we might experience a feeling that
they fit together or that one is an aspect of the other. Wittgenstein’s reflections
show that this is not the case: we cannot imagine certain combinations of col-
ors;707
we feel aesthetic discomfort with certain combinations of phenomena.708
We have a feeling of unity that ultimately turns out to be differentiated. Bradley
would say that we experience differentiated reality which turns out to be ultimately
a unity.

Wittgenstein ascribes to Hegel the idea that objects are what they are only in
their familiar surroundings. I do not want to overemphasize this casual remark.
Here is my brief suggestion regarding what Wittgenstein might have been referring
to in Hegel. In the “Sense-Certainty” chapter of Phenomenology of Spirit,
Hegel provides a complicated argument that every demonstrative act indicates
706
Ferreira, 1999, p. 120.
707
Cf. §15.3.
708
Cf. §18.
224
our immediate experience, every ‘This’ is always mediated by a universal.
When we try to point out a single thing, we realize that:
The Here pointed out, to which I hold fast, is similarly a this Here which, in fact is
not this Here, but a Before and Behind, an Above and Below, a Right and Left. The
Above is itself similarly this manifold otherness of above, below, etc.709
Consider an object that we want to point out as ‘Here’ or ‘This’. We can refer to
this object as the object in front of something else, above something else, to the
left of something else, etc. Every ostensive ‘This’ means ‘this thing I am pointing
at’ or ‘this thing before me’, etc. Every demonstrative act indicating an object
is, hence, dependent on the object’s relations to other objects one may point
at. In Wittgenstein’s terms: every demonstrative act occurs against a background
of demonstrative practices. 710
Without this background, knowledge
would be impossible. What appears at the outset to be the most immediate and
certain knowledge (“the richest kind of knowledge” in Hegel’s wording) proves
to be, in fact, a very low (“the most abstract and poorest”) kind of knowledge.
Knowledge is, hence, mediated by universality, by our shared (and thus universal)
practices.
What, then, has the status of the most certain and immediate knowledge in this
scenario if it is not a single intuition? Hegel, in adopting his characteristic synthetic
stance, says that if anything about the sense-certainty is maximally immediate
knowledge, it is the sense-certainty as a whole:
Thus it is only sense-certainty as a whole which stands firm within itself as immediacy
and by so doing excludes from itself all the opposition which has hitherto obtained.
711
709
Hegel, 1968, p. 64.
710
Here is a ‘Wittgensteinian’ summary of the “Sense-Certainty” chapter by Willem deVries
(2008, p. 74): “Hegel’s argument brings to the fore the fact that there are no lone isolated
demonstrative acts, and therefore no lone isolated intuitions. Every demonstration, and
therefore every intuition, is the determinate act it is because it occurs within and against a
background of demonstrative practices that license and indeed ultimately demand the normative
assessment of the individual demonstrative acts.”
711
Hegel, 1968, p. 62.
225
If one strives (as Hegel does, but Wittgenstein does not) for the most adequate
knowledge (of an object), one has to take into account all other objects and relations
to it. Wittgenstein says that if one takes an object in isolation, one may
have a feeling that the object “is part and parcel of everything else,” that it fits
into our experience. For Hegel, such an object (when taken in isolation) will be
incomplete and will stimulate a feeling of desire.712
There is a neat summary of the “Sense-Certainty” chapter by Philip Kain:
“Sense-certainty is as opposed to a doctrine of internal relations as anything can
be.”713
Sense-certainty is an account of knowledge that is founded on single intuitions,
single demonstrative acts, and single objects as its most certain and
immediate elements. It is the most utterly pluralistic and atomistic account of
knowledge. Sense-certainty is equivalent to the Doctrine of External Relations.
The main argument of the “Sense-Certainty” chapter is merely that the Doctrine
of External Relations is an inadequate account of knowledge. Wittgenstein
gives virtually the same argument against the Doctrine of External Relations as
Hegel does. “Sense-Certainty” is, however, only the first chapter of the Phenomenology,
and Hegel still has to produce many more arguments in order to
establish the Doctrine of Internal Relations. Wittgenstein does not follow Hegel
in this respect.714, 715
17.2. Metaphor as seeing-as
In this chapter, I shall develop the claim made earlier that in seeing A as B, an
internal relation is perceived between the concepts A and B. Many authors have
noticed a link between metaphor and perception. Aristotle says in his Poetics
“to make metaphors well is to observe what is like [something else]”716
. The
712
Cf. Kain, 2005, p. 45: “If the doctrine of internal relations is correct and the reality of
things involves the totality of their relations—the absolute—then to cut things off from the
absolute will create in them an absence or lack, which in those things with consciousness
will stimulate desire.”
713
Kain, 2005, p. 27.
714
Cf. Taylor, 1975, p. 143 for a statement of an explicit similarity between Hegel’s and
Wittgenstein’s arguments.
715
Section 17.1 draws on my paper (Mácha, 2014).
716
Aristotle, 1987, 1459a.
226
most significant recent studies on this topic, by Max Black and Donald Davidson,
conclude that metaphor is to be likened to seeing-as. Davidson furthermore
mentions Wittgenstein’s ‘duck-rabbit’ and maintains that “seeing as is not
seeing that”717
. In the metaphor ‘A is B’ subject A is thus seen as predicate B.
Certainly, such a comparison may be conceived of as a metaphor as well. The
seeing-as in a metaphor should be similar or somehow analogous to the seeingas
in visual perception.
My intention here is to elaborate an account of how such an analogy is to be
conceived. How far does the analogy between these two similar structures go?
Or are we misled by that analogy? These are the general questions about philosophical
inquiry which Wittgenstein asks himself in his Blue Book.718
Seeing-as has, in my view, several characteristics. Firstly, aspect-seeing is holistic.
Aspects of the ambiguous figure have to be mutually exclusive: one can
successfully see it either way, but one can never see it both ways at once just as
one cannot see one part of the picture under one aspect and another part under
the other aspect. Secondly, a concrete instance of seeing-as can be triggered by
a literal statement. The statement ‘It’s a duck’ can cause one to see the figure as
a duck. Aspect-seeing has, thus, a certain causal feature. These causal and holistic
characteristics of aspect-perception are to be transposed to the metaphor.
However, there are some problems which hold this immediate transposition
back. There are metaphors concerning abstract terms which cannot be literally
seen. How can justice be seen as a blind woman with a set of pendulum scales?
Another difficulty is the author’s intentional use of the ambiguous duck-rabbit
figure. Would it mean that all metaphors are ambiguous in our analogy as well?
There are three items: the duck, the rabbit, and the duck-rabbit figure. What
therefore corresponds to them in our analogy? Let me first discuss an account
by Marcus Hester.719
He claims that in Wittgenstein’s example we are given the
duck-rabbit and the problem is to see the duck and the rabbit in it. In the metaphor,
on the other hand, we are given the duck and the rabbit and the problem is
to see the duck-rabbit. In the metaphor ‘A is B’, the concepts (or images of) A
and B should blend in order for us to discover the common Gestalt between
717
Davidson, 2001, p. 263.
718
Cf. BBB, p. 28.
719
Hester, 1967, p. 179.
227
them. For example, in Keats’ metaphor of his imagination as a monastery720
both elements should merge into a single image which can be seen as imagination
or a monastery. This resembles Francis Galton’s process of composite photography
in which several portraits are merged into one in order to reveal common
qualities of the group.
Hester’s account cannot deal with abstract terms: how can we merge an image
of imagination with an image of a monastery? It cannot be an image which will
have the common properties of both terms, viz. properties common to both imagination
and a monastery. There are no such properties for most metaphors.
This is the question from the very beginning and Hester’s account gives us no
answer to it. Furthermore, both aspects are mutually exclusive and so the
merged image cannot be seen both ways simultaneously, for then the holistic
trait of the aspect would not be preserved.
Another account of the analogy has been offered by Roger White:
We may […] regard the metaphorical sentence as a ‘Duck-Rabbit’; it is a sentence
that may simultaneously be regarded as presenting two different situations; looked at
one way, it describes the actual situation, and looked at the other way, an hypothetical
situation with which that situation is being compared.721
We are supposed to take the abovementioned metaphor of Keats, in analogy to
the duck-rabbit ambiguity, as presenting in one reading the imagination (that is,
the actual situation) and in another reading a monastery (a hypothetical situation).
The holistic trait of the aspect remains preserved here. But the recipient
won’t be dubious about the two aspects. Both of them are given together with
the duck-rabbit. And now we are told that both situations, i.e., both aspects,
should be compared. What the analogy also shows is that in the metaphor ‘A is
B’ both the terms should be compared. If all three elements are already given,
why should the reader compare the situations? I do not want to deny that White
gives a plausible explanation of such a comparison, but this explanation is not a
consequence of this analogy.
720
“My imagination is a monastery, and I am its monk.” Letter to Percy Bysshe Shelley, August
1820.
721
White, 1996, p. 115.
228
Nevertheless, both accounts share, in my view, the same defect: two situations/aspects
are given which should be compared or merged. But we do not
know how. Furthermore, both authors do not use Wittgenstein’s subsequent reflections
about the dawning of an aspect and about the role played by concepts
in perception. A dawning of an aspect is for Wittgenstein “half visual experience,
half thought,” it is “an amalgam of the two” 722
. These considerations have
to be employed in our analogy.
We have to get rid of the intentional ambiguity of the duck-rabbit figure. A
spectator does not need to know about the ambiguity of the figure. They might
consider it at first as a duck and only later on experience the change of aspect.
In such cases they might say: ‘Now I see this duck as a rabbit’ or, more metaphorically,
‘This duck is now a rabbit’. Anyway, we do not need to suppose that
a spectator would identify the whole figure as a duck-rabbit at all. They would
initially conceive the figure as a tangle of lines.723
I propose analyzing this analogy as follows: subject A of the metaphor ‘A is B’
corresponds to the duck-rabbit and predicate B is one of the aspects, e.g., the
duck. We already know that in seeing A as B, an internal relation is perceived
between A and B. Then what is perceived in the metaphor ‘A is B’ is an internal
relation between subject A and predicate B insofar as they are both perceived
and thought of. Moreover, a conceptual relation is perceived between the terms
involved which has an irreducible subjective side as well. This means that in the
metaphor, the predicate B organizes the subject A. In our example above, the
concept of a monastery organizes the concept of Keats’ or even someone else’s
imagination.
Due to the notion of an aspect, the causal as well as the holistic feature of the
seeing-as is preserved in the analogy. The first consequence for my theory of
metaphor is that metaphors cannot be fully paraphrased in literal language because
of the subjective experience of the change of aspect. Furthermore, any
internal relations cannot be expressed. The consequence is that a secondary
metaphorical meaning cannot be expressed in the metaphor. The main objection
against theories of metaphorical meaning is that they are reducing the aspect to
722
PI II, p. 197.
723
Cf., for example, Ms 137, p. 14b; Ms 144, p. 47.
229
a conception and leaving aside the subjective experience of the change of aspect.
On the other hand, there are theories that see a function of the metaphor in
the evocation of an emotive or perlocutionary effect. They are reducing the aspect
just to its subjective component, leaving the linguistic component aside.
Furthermore, if the point of a metaphor were an experience of the change of aspect,
then only an external relation would be perceived in the metaphor here,
because the experience is a concrete event which is causally linked to the meta-
phor.
The aim of my analysis here has been to demonstrate that elaborating Wittgenstein’s
notion of the seeing of an aspect allows a useful analogy between seeing-as
and metaphors. Let me conclude with a paraphrase of Aristotle: to make
metaphors well is to observe internal relations.724
724
This section draws on my own paper (Mácha, 2009).
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18. Aesthetics and art
There is a certain duality in Wittgenstein’s reflections on both aesthetics and the
theory of art. On the one hand, aesthetics and art exist in a cultural context.
They belong to the language-game which Wittgenstein calls ‘culture’. His account
is then, in this sense, normative and institutional. To recognize an object
(or a phenomenon) as a work of art at all, it needs to be internally related to the
culture of its time. This only means that any work of art has to follow some
rules. I will call this aspect of art its transitive understanding. On the other
hand, beauty and art are also about expression. A work of art displays an extraordinary
talent, in which even a genius may transgress traditional rules. In
this sense, Wittgenstein’s account of art is expressive. If a work of art is thus
conceived of as an expression, it does not convey or refer to something else.
Following Wittgenstein, I will call this aspect of art intransitive understanding.
18.1. Transitive understanding: culture
The understanding of a work of art is manifested in a response or resonance to
it. Let me call this aesthetic reaction. The fact that people understand a work of
art becomes apparent when they respond to it in a particular manner. In this
sense, one could say that they respond correctly. The concept of aesthetic correctness
is indeed central to Wittgenstein’s aesthetics. A work of art is correct or
right as long as it follows common aesthetic rules (concerning, for instance,
harmony, composition, or ideal proportions).
The aesthetic reaction is not just limited to predicates like ‘nice’ or ‘pretty’
which Wittgenstein regards rather as interjections.725
Such simple aesthetic
judgments, which are used by children or by less educated people, are replaced
by more complex judgments which need a deeper context (which may be provided
by education). Wittgenstein calls the context for aesthetic judgments ‘culture’.
I want to take culture as a language-game, although the only explicit evidence
for this claim is less reliable: “What belongs to a language game is a
whole culture.”726
This suggestion is justified by the fact that Wittgenstein uses
725
LA, p. 3.
726
LA, p. 8.
231
the notion of a language-game as a context. For additional evidence, we can focus
on Wittgenstein’s remarks from the Zettel727
that deal with music. In §164
Wittgenstein explores how the expression ‘expressive playing’ can be explained.
For such an explanation, one needs to be educated in a particular culture.
I take §175 as Wittgenstein’s denial that musical themes (or works of art
more generally) have any intrinsic aesthetic features. A theme is always embedded
in the context; it “makes an impression on me which is connected with the
things in its surroundings—e.g., with our language and its intonations; and
hence with the whole field of our language-games.” Thus, based on these remarks,
it is sufficiently justified and indeed fruitful to speak of culture itself as a
language-game. After all, the label ‘culture’ as such is not that important. We
would lose nothing by simply talking of a language-game with aesthetic expres-
sions.
To produce a correct aesthetic reaction to a work of art presupposes acquaintance
with an (actual) culture. Wittgenstein uses the term ‘culture’ very loosely
here. I shall examine how an aesthetic judgment is involved in it. One can understand
culture as a network of connections. 728 A work of art fits (passt
zusammen) into this network due to its correctness. An aesthetic judgment is
supposed to express this fit.729
As already elaborated in §§11.3 & 17.1 above,
this fitting expresses an internal relation. Two things or phenomena fit together
if there is an internal relation between them, or more precisely, if there is an internal
relation between their forms. In the same vein, Wittgenstein employs the
distinction between motive or reason and cause in this context: “There is a
‘Why?’ to aesthetic discomfort not a ‘cause’ to it.”730
Or: “an aesthetic explanation
is not a causal explanation.”731
And again, this suggests that an aesthetic
727
Z §§157–175.
728
“Culture” might be on a list of expressions that Wittgenstein borrowed from Spengler.
Indeed, Wittgenstein sometimes uses the expression “culture” in Spengler’s sense, i.e., as
opposed to “civilization”. (See Ms 109, pp. 204f.; Ms 110, pp. 12f.; Ms 136, p. 18b; Ms
183, p. 46; Ts 211, p. 157.) I want to place emphasis on the fact that in Wittgenstein’s conversations
on aesthetics, the expression “culture” is used in a slightly different (although
related) sense as a context for aesthetic judgments that can also take place within Spengler’s
civilization.
729
RPP II, §501.
730
LA, p. 14; see LS §908.
731
LA, p. 18.
232
explanation seeks to express an internal relation, and not a causal relation which
is, of course, external.
Now back to the investigation of aesthetic reactions to a work of art. They, the
work of art and any reaction to it, must fit together; there must be an internal
relation between them. Such a reaction need not be a verbal one; it can simply
be a gesture or even another work of art. One could find a fitting musical accompaniment
for a poem, or a fitting dance figure for a melody.732
One may now ask how a correct aesthetic reaction to a work of art can be identified
(or become known). This would be the same as asking how to find a missing
part from a whole phenomenon. The answer lies in, or can be deduced from,
Wittgenstein’s account of aspect-seeing. The phrase ‘to see (or hear) something
as something else’ is used in art very often, e.g., “You have to hear these bars as
an introduction.”733
Such aspect-perceiving, or the possibility of an eventual
change of aspect is, says Wittgenstein, essential in aesthetics.734
The phrases
‘see as’ and ‘fit together’ are closely related. If something is seen as another
thing, then both things fit together.735 A phenomenon is, however, not seen as
another thing all the time. The aspect has to dawn and in such dawning of an
aspect one perceives an internal relation. The correctness of an aesthetic reaction,
therefore, can be affirmed in the dawning of the aspect and in the related
astonishment or surprise [Staunen].736
These reflections lead us to the next tentative
conclusion: an aesthetic reaction expresses an aspect of a work of art.
As we know from §17, in aspect-seeing, the perceived object must be organized.
To organize a phenomenon means seeing that its parts fit together in a
particular way, namely correctly. Now we have to distinguish the following.
Parts of a work of art fit together (I shall call this fitting immanent). This fitting
is the basis of the fitting together (what I call the transcendent fitting) of the
work of art and the reactions to it. An aesthetic reaction is correct insofar as it
732
Cf. Ms 137, p. 20b where a gesture and a dancing figure are taken as simple explanations
of a musical phrase.
733
Z §209.
734
LS §634.
735
Viz. LS §654: “If I see it this way, it fits this, but not that.” The seeing ‘this way’ is a variation
of aspect-seeing. See §17 above or the quotation at the beginning of §18.2.
736
See PI II, p. 199 where the German expression ‘Staunen’ is translated as ‘surprise’.
233
expresses an aspect of the work of art. An aesthetic judgment reveals to us an
internal relation between the work of art and other objects, by which we understand,
as noted, that they share a common form. But what are these other objects?
I want to suggest that they are other works of art. This enables us to regard
complex aesthetic reactions as works of art.
The fact that there is an internal relation between two objects—two works of
art—implies that they are, or could be, parts of a whole. They are a total work
of art (Gesamtkunstwerk), one might say. There are many kinds of aesthetic reaction
to a work of art and so they can be internally related to a variety of phenomena.
These reactions can be very simple (‘It’s nice!’) or significantly more
complex. A reaction can reveal, for example, a deep affinity between two artists,
e.g., between Brahms and Keller.737
More precisely, a network of such
connections is what Wittgenstein calls a culture. It is the total or ultimate work
of art. The sole culture in toto prescribes the rules for itself. An object becomes
a work of art insofar as it expresses (or is used as an expression of) an aspect of
the culture. This does not mean that every work of art has to meet all the rules
of the convention, but that these rules have an impact on the aesthetic reactions
to a work of art, especially to an object’s being taken as a work of art at all.738
If we take culture as the praxis with art and the sum of all works of art, then the
internal relations within a culture are all that matters. In this sense, Wittgenstein’s
aesthetics is institutional.
737
It is not obvious (at least for me, not being familiar with Keller’s works) what Wittgenstein
could have in mind when he says this. In Ms 183, p. 59, he writes that the same principles
of good and right (from the period) are embodied in the works of these artists. Hence,
due to these principles, there is an internal relation between these works. There is an extensive
commentary on their affinity: “I often found that certain themes of Brahms were extremely
Kellerian.” (LA, p. 32) This could simply mean that they lived at the same time or
in the same culture of the time. Such a relation is external. Nevertheless, there might be
something hidden in this utterance. It might express a similarity between the works of the
artists that is difficult to describe. Wittgenstein likens this to a similarity between two faces
that is not obvious at first but can quickly be found. Such a relation would be internal.
738
Once again, no artist can break all rules at the same time: “You can say that every composer
changed the rules, but the variation was very slight; not all the rules were changed.”
(LA, p. 6)
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18.2. Intransitive understanding: expression and genius
The preceding section can be taken as describing the first step in our analysis of
the language of aesthetics and art. In this first step, we had to distinguish between
internal and external relations. What makes an object a work of art is the
object’s internal relation to its culture. However, in this section, our concern
will be the reflexive cases of internal relations. This takes up the second step of
the analysis. The point of departure is Wittgenstein’s critique of Tolstoy’s theory
of art:
There is much that could be learned from Tolstoy’s false theorizing that the work of
art conveys ‘a feeling’.—And you really might call it, if not the expression of a feeling,
an expression of feeling, or a felt expression. And you might say too that people
who understand it to that extent ‘resonate’ with it, respond to it. You might say: The
work of art does not seek to convey something else, just itself.739
Suppose one is forced to answer questions like ‘What is this picture supposed to
mean?’ or ‘What does this melody seek to convey to us?’ If there is really nothing
that one may point at, then a reflexive construction may help. A work of art
seeks to convey or express itself—or it just means itself, it refers to itself. These
formulations are reflexive cases of internal relations. The key issue is: either the
work of art is a medium for grasping something else or it just exists for itself.740
The something that is supposed to be grasped may take various forms: for instance,
it may be a feeling as Tolstoy (and the Expressive theory of art) suggests;
it may be a hidden message (in structuralist accounts) or a political action
(in Marxism). Wittgenstein is opposed to all these views of art. A work of art
expresses itself. But the maxim of no reflexive uses of internal relations urges us
to get rid of this vague reflexive construction.741
However, the question is how
to straighten it out and give a positive account of art.
739
CV, p. 67.
740
Saying that a work of art just exists for itself is another reflexive construction. It seems
that language itself is trying to impose this on us.
741
Insisting on this reflexive construction can lead to absurdities. Schroeder reports Tolstoy’s
response to the question of what he meant by his novel Anna Karenina. He responded
that “if he were to put into words what he meant to express by his novel, he would have to
write down the whole novel again.” (Schroeder, 2001, p. 225) It would be absurd for Tolstoy
235
One model of this straightening out may be what Wittgenstein calls intransitive
understanding. He applies this label to pictures, but we can broaden it to works
of art in general.742
Wittgenstein writes in the Philosophical Grammar:
If I say: “I understand this picture” the question arises: do I mean “I understand it like
that”? With the “like that” standing for a translation of what I understand into a different
expression? Or is it a sort of intransitive understanding? When I’m understanding
one thing do I as it were think of another thing? Does understanding, that is, consist
of thinking of something else? And if that isn’t what I mean, then what’s understood
is as it were autonomous, and the understanding of it is comparable to the understanding
of a melody.743
I argued in §9.3 that a picture has to depict something else. If it does not, it is
not a picture at all (like a map that is identical with the territory it depicts). We
usually speak of understanding a picture when we recognize what it represents.
Pictures in art or works of art in general may represent something (a still life
represents a particular scene, for instance), but their representing role is not essential
to this kind of understanding. A theme in music does not represent anything.
If we look aesthetically at real (and not depicted) things like a landscape
or a flower, we do not ponder what they represent. In art we are interested in the
things themselves. Wittgenstein quotes Goethe’s exclamation in this context:
“Don’t look for anything behind the phenomena; they themselves are the theo-
ry.”744
This is why this kind of understanding is intransitive.
What, then, is art all about? Why are we interested in works of art at all? The
work of art is “an expression of [a] feeling, or a felt expression”. The very act
of expressing or grasping it is a matter of experiencing it. Or as Johannessen
puts it: “the understanding sought here is a way of experiencing.”745
Works of
art are expressions of feeling in the way that a smile is an expression of joy.
to have to write out the novel again to satisfy his questioner. If we resist the temptation to
hunt for hidden meanings in art, we can then direct our attention to the work’s real aesthetic
values, its appropriateness and its aspects.
742
See Johannessen, 1994 for a detailed analysis of Wittgenstein’s account of intransitive
understanding in philosophy and art.
743
PG, p. 79.
744
RPP I, §889.
745
Johannessen, 1994, p. 237.
236
Works of art are usually, however, much more complex. Nobody is tempted to
say that by smiling one is conveying a feeling of joy. Analogously, Sophocles’
Oedipus Rex or Beethoven’s Fifth Symphony cannot be said to be conveying the
feeling of the inexorability of fate. Wittgenstein can thus say that works of art
“give us pleasure, occupy our minds”746
but without conveying these feelings.
The next model of intransitive understanding concerns the expression ‘meaning’.
Even this expression demonstrates the ambiguity between transitive and
intransitive uses. Wittgenstein imagines the situation of being impressed by a
bed of pansies:
I could have used the expression “Each of these colour patterns has meaning”; but I
didn’t say “has meaning”, for this would provoke the question, “What meaning?”,
which in the case we are considering is senseless. We are distinguishing between
meaningless patterns and patterns which have meaning; but there is no such expression
in our game as “This pattern has the meaning so and so”.747
Hence, we can use the word ‘meaning’ transitively as ‘X has the meaning so and
so’. The expression ‘so and so’ stands for an explanation of meaning. Such an
explanation is actually a rule which expresses an internal relation. Intransitive
uses of the expression ‘meaning’ are close to aesthetics. What, then, is the intransitive
meaning of ‘meaning’? The following classification given by Severin
Schroeder may be helpful here. He identifies three ways to use ‘meaning’ intransitively
(although they may overlap): “denoting (1) value, (2) a specific Gestalt,
or (3) an (apparent) appropriateness.”748
‘Meaning’ may be used to highlight
the (typically personal) value of something: ‘This tune has meaning for
me’ or even ‘Life has meaning’. In these contexts ‘meaning’ is used intransitively
without implying that the meaning is so and so. Expressing a specific Gestalt
seems to indicate a transitive use of an internal relation. ‘I see this picture
as a duck’ expresses a duck-aspect of the picture, which amounts to expressing
an internal relation between the shape of this picture and the shape of a duck.749
But we can also find a specific Gestalt or aspect, for example, in a tune while
knowing it is not really there:
746
PI §524.
747
BBB, p. 179.
748
Schroeder, 2001, p. 224.
749
See Chapter 17.
237
we say “This tune says something”, and it is as though I had to find what it says. And
yet I know that it doesn’t say anything such that I might express in words or pictures
what it says. And if, recognizing this, I resign myself to saying “It just expresses a
musical thought”, this would mean no more than saying “It expresses itself”.750
But how can we see an aspect of an object and suspect that this aspect is not really
contained in that selfsame object? The aspect is not part of its meaning, although
the object means something. A plausible explanation is that various aspects
are switching (dawning and receding), which involves a feeling of astonishment
as mentioned a few pages previously.751
The object means this at one
moment and that at another moment. Now a few words about Schroeder’s third
intransitive way of using ‘meaning’. Something is appropriate if it fits into its
surroundings (i.e., its transitive understanding) or if its parts fit together. Noticing
such appropriateness is, in fact, akin to the noticing or dawning of an aspect.
This is the case of immanent fitting as mentioned in the previous section
18.1.

In this chapter, we have so far focused on the receptive side of art, that is to say
on the explanation of how and why art is perceived and apprehended. Now we
are going to focus on the productive side of art. How do works of art emerge? If
the work of art forms an aspect of an actual culture, how, then, can it be explained
that this culture can undergo a change? What is the source of these cultural
dynamics? Wittgenstein admits that there are works of art that cannot be
judged only according to their correctness. He calls such works of art tremen-
dous.752
Impressionist paintings were first judged disparagingly because they
did not follow the academic rules of the period. However, such negative attitudes
endured for less than a generation, and impressionist works then became
the canon of subsequent art movements. A work of art is called tremendous because
it occupies and forms a tremendous part of a culture. Although such
works of art express aspects of the culture, they contribute to the culture in a
different way.
750
BBB, p. 166.
751
Wenzel (2010) connects this Wittgensteinian aspect-switching with the Kantian ‘free
play’ of imagination and understanding which is central to Kant’s aesthetics.
752
LA, p. 8.
238
My suggestion is that tremendous works of art enrich a culture with new rules.
An artist who has the talent to enforce new rules is—in the Kantian sense—a
genius.753
“Genius is the talent in which the character expresses itself,”754
says
Wittgenstein. This reference to character is a crucial point that goes beyond the
institutional conception of art.755
The enrichment of a culture with new rules
does not happen all at once. New rules are, at first, only implicitly incorporated
in the works of artistic geniuses, and it takes some time before they are recognized
and thus become a part of culture. After this happens, another languagegame,
‘culture’, is played, for every language-game is defined by its rules.
Culture is embedded in our very form of life.756
Each child eventually manages
to integrate its primitive expressions of pain into complex language-games.
Nevertheless, one needs to have talent to master the complex language-game of
culture and to transpose one’s feelings into this extraordinary language-game.
To express common feelings in our language, we do not need to introduce new
rules. But by contrast, it may happen that the prevailing culture cannot satisfy
the artist’s demands, which is precisely why new rules are then introduced.
However, every artist needs to address the usual rules of their culture and they
may indeed violate some of these. If they violated all the rules of their culture,
we would have no reason to regard their products as works of art. They must
express, through their art, the aspects of their culture. In this sense, a work of
art should be understood as a model of culture.
A work of art stands in two main relationships—in a vertical relationship to our
feelings and life, and in a horizontal relationship to other works of art within the
language-game ‘culture’.757
If the horizontal relationship is accentuated, Witt-
753
Ms 162b, p. 22.
754
Ms 136, p. 59a.
755
This idea is embodied in Özlem, 2010.
756
In the context of The Brown Book, the expressions ‘culture’ and ‘form of life’ are very
close. See Glock, 1996, p. 125.
757
Instead of two relationships, we can speak of two tendencies or movements in the history
of art. Art movements that preferred the horizontal relation to culture are called ‘academism’,
‘classicism’, and the like. Romantic, revolutionary, or avant-garde art movements
strived rather for an expression of feelings. One can see this point more clearly if we consider
extreme examples of these tendencies. It is sometimes said that works of academic art
are without feeling and that expressive works of art are destructive or rude.
239
genstein’s conception of art is institutional; if we emphasize, on the other hand,
the vertical relationship, one can instead understand art as being expressive.
V. Conclusion
My interpretation of Wittgenstein’s writings is now complete. In this concluding
Part V, I want to finish by recapitulating the main principles and insights
that have driven my interpretation. I want to resist formulating any general theory
extracted from Wittgenstein’s philosophy. However, if my interpretation is
admissible, its main principles do coincide with the main principles of Wittgenstein’s
logical and grammatical analysis of language. These principles remain
mostly tacit in Wittgenstein’s texts. Nevertheless, some quite explicit formulations
of these principles can also be found.758
These principles, or rather their
guidelines, are:
(I) To insist on the distinction between internal and external relations in the
depth grammar.
(II) That reflexive cases of internal relations are in fact those cases of direct expression
where no relation at all is expressed.
The first principle goes back to Frege’s distinction between concepts and objects.
It is to be found throughout Wittgenstein’s writings, while the second
principle manifests itself mainly in Wittgenstein’s later philosophy. In the following
chapters, I am going to look back at the topics discussed above in the
light of these principles. The following chapter addresses why we express internal
relations at all. Expressions of internal relations are far from nonsensical.
They express current or proposed logical or grammatical rules. The final chapter
will discuss the maxim of no reflexive uses of internal relations. It will be
indicated there that this maxim is analogous to Bradley’s insistence that all relations
must be both internal and external.
758
I have in mind Wittgenstein’s talk of “the confusion between internal relations and relations
proper” in the Tractatus (4.122, see §6.2 above) and the very first paragraph of the
Remarks on Colour.
241
19. Internal relations as imperatives
Having distinguished between internal and external relations, we can ask for
what actual reasons do we express internal relations. Expressions of internal
relations do not depict any states of affairs and they are not moves in languagegames.
I am strongly opposed to the view (maintained by some resolute readers)
that expressing an internal relation is plain nonsense. Internal relations say
something about the logic or grammar of our language. This statement is, however,
also ambiguous, as internal relations express both what the logic or grammar
of our language is and what it ought to be. Expressing an internal relation
can function as a kind of reminder to someone who is not aware of the logic of
our language or it can function as a stimulus to improve our logic or grammar.
In short: expressing an internal relation has normative force and can also be
taken as an imperative.
Here is an outline of how such imperatives may work. Let us turn first to the
Tractatus. In the logically adequate language which is outlined in the Tractatus,
it is impossible to express any internal relation by means of a proposition. Formulating
an internal relation in a proposition is an indication that we have not
yet reached a logically adequate language. In §7.1 I stated that the signs for relations
ought to be eliminated from our logical notation in favor of an internal
relation within the signifying fact. This idea is further elaborated in §7.2 where
I give an example of how a logical relation between two states of affairs can be
incorporated into our logical notation. The very possibility of expressing an internal
relation (a logical implication, for example) can be taken as a direction to
improve our manner of expression so that it comes closer to a logically adequate
language.
Another example of this method is given in §9.1. Expressing an internal relation
(or a property, in this case) invites us to coordinate the combinatorial powers of
names and objects. Some internal relations can thus be converted into propositional
variables. A similar idea is elaborated in §9.3. It is demonstrated there
how the supposed expression of the internal relation of depicting between a
proposition and a fact can be transformed into a propositional variable.
It is obvious that once an order is fulfilled, it does not need to be repeated. By
the same token, once the task of internal relations is achieved (i.e., once they
242
are incorporated into logical grammar), they can be left behind.759
But there is
something else on top of that. Since an internal relation modifies the very language
in which it is expressed, it cannot be expressed in this modified language.
Hence, once the task of internal relations has been achieved, these relations are
left behind, for there are no propositions that express them.

What is different, then, in Wittgenstein’s later work? A striking difference is that
nothing prevents us from expressing internal relations, even if there is a surveyable
representation of the grammatical rules. We can express internal relations
in order to remind someone of actual grammar, to change an existing rule, or to
introduce a new rule. Expressing an internal relation is, however, not a move in
a language-game. Let me go through some such reminders and imperatives case
by case.
My first case concerns the situation of teaching someone an existing grammatical
rule, e.g., teaching the name of a thing. In §10.3 I distinguished between the
language-game of teaching and the language-game of applying a rule. The sen-
tence
This is called X.(105)
expresses an external relation in the former game and an internal relation in the
latter. If we take these two language-games as a single language-game, (105)
would be seen as ambiguous because it expresses both an internal and an external
relation. There is nothing wrong here if we are aware of such an ambiguity.
This sentence can be employed in the process of teaching as a reminder of the
rule: “You don’t know what this is called, do you? It’s called X”.
Next, we can slightly modify our scenario. Now we want to introduce a new
rule, e.g., a name for a newly designed object (that is termed baptizing). The
verbal expression may be the same as in the previous case or more explicit:
From now on, this shall be called X.(106)
Both (105) and (106) express an internal relation. The difference between them
is that the former sentence is used as a reminder of an existing rule and the lat-
759
Cf. the remark by McGinn quoted on p. 12.
243
ter one as an imperative that establishes a new rule. This schema is totally universal;
it works for every rule. What comes next are some special cases of this
schema, viz. the rules concerning the standards of measure and color concepts.
In §15.2, I pointed out that the sentence ‘This is red’ can mean ‘There is a red
square’ or ‘This color is called red’. There is an ambiguity here between describing
(expressing an external property) and naming (expressing an internal
property). Expressing an internal property means, in fact, expressing a rule.
And again, ‘This color is called red’ can be used to remind someone of an existing
rule or to introduce a new rule. Something analogous can also be said about
standards of measure.760
‘This rod is one meter long’ can mean ‘This rod is as
long as the standard meter’ or ‘This rod is the standard meter’. The latter sentence
expresses an internal relation. It can inform someone about the actual
standard or it can be used for proclaiming a new standard of measure, meaning
roughly ‘This rod shall be the standard meter’.
Color concepts and standards of measure are, thus, introduced by means of paradigmatic
samples. The same is true of numerals, i.e., concepts for numbers. We
can give meaning to the numeral 3 by the following definition:
The list | | | means 3.(107)
As stated in §14.2, the list | | | serves in this sense as a yardstick. The numeral 3
is a substitution or, rather, an abbreviation for the list | | |. The same schema applies
here. (107) can be used in a teaching process to point out the existing rule
for the numeral 3,761
or it can be used as an imperative to introduce the meaning
of ‘3’.
Mathematical propositions are statements of internal relations. Expressing such
a proposition (e.g., that 68 + 57 = 125) says that this statement is a rule within
our system of arithmetic. Although mathematical conjectures may have the
same surface form as mathematical propositions, they express external relations.
They can be confirmed or refuted empirically or heuristically. Mathematical
propositions have, therefore, a normative force over mathematical conjectures.
We can express a mathematical proposition in order to say that some con-
760
See §16.3.
761
I remember having learned the meaning of ‘3’ by reference to a paradigmatic sample
consisting of three pears.
244
jecture is true or false—or that its empirical or heuristic confirmation is correct
or incorrect.
Proving a new mathematical proposition is tantamount to introducing a new
rule. What was only a conjecture (expressing an external relation) then becomes
a mathematical proposition (expressing an internal relation) through its proof.
Mathematical proofs aim to integrate mathematical conjectures into the system
of already-proven mathematical propositions. The mathematical proof is a calculation
that functions as a paradigmatic sample here. The proof says that this
proposition shall be a mathematical proposition from now on, which means it
shall be connected with our system of mathematics. A mathematical proposition
is, thus, like an order, and its proof is like the fulfillment of the order.
This is the first cluster of cases illustrating how expressions of internal relations
are used. All these cases are variations on the teaching situation (reminding of
an existing rule or introducing a new rule). The second cluster of cases is related
to aspect-seeing. Such expressions of internal relations are underlined by a
certain experience, typically by a feeling. Suppose someone reported an experience
of the change of an aspect:
Now I see this picture as a duck.(108)
This is an exclamatory sentence. However, we can use it in the imperative
mood as:
See this picture as a duck!(109)
Expressed in either of these ways, both sentences imply that the experience of
the change of aspect is possible, including an internal relation between the
shape in this picture and the shape of a duck. This means that there is a smooth
transition between these shapes, which can also be expressed as a case of one
shape organizing the other. The experiencing of such a transition is, however,
not a private feeling of its being possible. It is shown in our praxis. The imperative
sentence (109) and the indicative sentence (108) both claim that our praxis
is such that it allows a smooth transition between these shapes, amounting to
experiencing an internal relation between them.
This universal scheme has several instances which I am going to sum up in
turn. The first case (discussed in §15.3) is about experiencing mixtures of col-
245
ors. Experiencing orange as a mixture of red and yellow amounts to seeing orange
as reddish yellow or as yellowish red. This means that a smooth transition
between these internally related colors can be experienced.
The second instance of this scheme concerns the making of a metaphorical
statement. Consider Keats’ metaphor (already mentioned above):
My imagination is a monastery.(110)
This metaphor can be rephrased into the form of an aspect expression:
My imagination can be seen as a monastery.(111)
Here, the poet is expressing his feelings about a certain state of his imagination.
He expresses an internal relation between the concepts of his imagination and a
monastery that is backed up by his experience. Again, this amounts to saying
that the concept of a monastery organizes the concept of Keats’ or the reader’s
imagination and that there is a smooth transition between them.
Paul Ricoeur said that a metaphor could be taken as a poem in miniature. A
metaphor is, in fact, a small work of art in its own right. What is true of metaphors
is also true of works of art in general. A work of art is an aesthetic reaction
to other works of art whose totality is called culture. A work of art expresses
an aspect of (current) culture. Works of art are thus internally related to culture
and a kind of aspect-switching between various works of art can be experi-
enced.
246
20. The maxim of no reflexive uses of internal relations
Having identified an internal relation expressed in a sentence, the second step in
logical analysis is to look at its reflexive use. Let me say some words about the
notion of a reflexive relation. A relation R is reflexive if and only if R(x,x) holds
true for every element in its domain. What I have in mind when speaking of a
reflexive use of an internal relation is a single case R(a,a). Therefore, it is not
enough to say that an internal relation must not be reflexive. To avoid the reflexive
uses, one has to say that an internal relation must be irreflexive, or rather
that it must exclude its reflexive cases from its domain.
The methodological principle that I am calling the maxim of no reflexive uses of
internal relations says that a reflexive use of an internal relation might be a
failed case of emphasis. One should consider whether straightening it out into
an intransitive use (where no relation is expressed at all) would make the language-game
more plausible. This method is to be found throughout Wittgenstein’s
writings. Before turning to his own instances of this method, which have
been discussed above, I want to provide a simple illustration of why a reflexive
case of an internal relation may be nonsensical.
There are signposts near roads that indicate directions and distances to nearby
places. Consider now the following situation. Someone who wants to reach
place A has lost their way. Remember that “A philosophical problem always has
the form: ‘I simply don’t know my way about.’”762
Now, the lost person is, in
fact, already in A without realizing it. The lost person is looking for direction
signs, but cannot find any sign indicating how to reach A. Only a reflexive direction
sign could be helpful in this situation. There are, however, no such signposts
and it makes little sense to set them up. A reflexive direction sign would
be a weird way of indicating the name of an actual place; it would simply be a
place-name sign. In fact, nothing relational could help them to know their way
around. The right thing to do is not to look for a direction sign, but for the lost
person to get to know where they actually are. This means they have to look for
a place-name sign. They have to “see what is right in front of [their] eyes!”763
A
762
BT, p. 310e.
763
CV, p. 44.
247
reflexive case of an internal relation is analogous to a signpost indicating the
way to the very place where the signpost actually is.
Now let us look at Wittgenstein’s own instances of the maxim of no reflexive
uses of internal relations.
(1) The first example is Wittgenstein’s treatment of identity, which is an internal
relation par excellence. The sentence ‘a = a’ or ‘a is a’ is, as an expression of
pure identity, uninformative and useless. In the Tractatus 5.473, Wittgenstein
offers the sentence “Socrates is identical” as an example of nonsense. His point
here764
is that the unary predicate ‘identical’ has a different meaning from a sign
for identity. This line of thinking is taken up again in the Philosophical Investigations
where Wittgenstein offers the sentence “A thing is identical with itself”
as an example of a useless sentence.765
This example is then subsequently presented
in several variations: “Every thing fits into itself” or “Every thing fits
into its own shape”. We can now restate these sentences in a reflexive way: the
form of A is (identical with) the form of A. These sentences are seemingly not
meaningless; they are, however, useless. We can straighten such sentences out
into something like ‘Every thing has a particular form’ and consider whether
this straightened-out sentence would be better suited for the speaker’s original
intention.
(2) The second example concerns the internal relation of depicting, which was
discussed in Chapter 9. A reflexive use of this relation amounts to saying that a
picture represents itself. In §9.3, I illustrated the point that a picture cannot represent
itself with the map-territory relationship. A map that is identical with the
territory it represents is useless. It is, in fact, not a map at all, as aptly recounted
by Lewis Carroll and Jorge Luis Borges.
(3) The next application of the method: expectation and its fulfillment (namely,
the intentional object) are internally related as elaborated in §11.2. There are
cases of expectation without any intentional object, e.g., when someone says
that they have a feeling of expectation without being able to specify what exactly
they are expecting. We cannot say that the object (that is to say, the fulfillment)
is in some mysterious way contained within the expectation. That would
764
TLP 5.4733.
765
PI §216.
248
be as if an expectation referred to itself. Such a reflexive case should be
straightened out into a direct expression of the feeling of expectation as argued
in §11.4.
(4) A similar scheme is employed in Wittgenstein’s critique of the Principle of
Sufficient Reason:766
“For every fact F, there must be a reason why F is the
case.” If R is the reason for F, R and F are internally related. There are, however,
some facts whose reasons are missing or at least are not obvious. One can
save the Principle of Sufficient Reason by postulating a reflexive construction
saying that the fact is its own reason, e.g., ‘This is simply what I do.’ This is,
again, a reflexive case of an internal relation. Instead of using such reflexive
constructions, Wittgenstein is willing to reject the Principle of Sufficient Reason.
Facts without reason express grammatical rules. They are termini ad quem
of a justification.
(5) The next instance of the maxim of no reflexive uses of internal relations is
that we cannot say that a paradigmatic sample has the property of which it is a
sample. We cannot in the same language-game say that a paradigmatic sample
of X has property X. It is nonsense to say that the paradigmatic sample | | | of the
number 3 has three strokes, because 3 is an internal property of this list. We
cannot say that the paradigmatic sample of blue is blue. Blue is an internal
property of this sample. Finally, we cannot say that the standard meter is one
meter long. Being one meter long is an internal property of the standard meter.
We have to assign the internal property to the sample in a preparatory languagegame
which precedes the game of applying the sample. In this way, we can define
a rule in one game and apply it in another game.
(6) If seeing A as B involves expressing an internal relation between A and B,
we have to consider reflexive cases here as well. As indicated in Chapter 17, it
makes little sense to say that I see this knife as a knife, or a lion as a lion, or a
letter F as an F. If these reflexive cases mean anything, they have to be straightened
out into intransitive cases that express an emphasis: This really is a knife.
This really is a lion. This really is an F.
(7) Another reflexive construction is saying that a work of art expresses itself.
This construction represents, in fact, an intransitive understanding of art. A
766
Cf. §12.1.
249
work of art is a direct expression of a feeling but without conveying or even
communicating any particular feeling. By saying that a work of art has meaning,
we do not imply that it has the meaning so and so. ‘Meaning’ is used here
intransitively without the implication that the meaning is so and so. The intransitive
meaning of a work of art may be its value, its specific Gestalt or its ap-
propriateness.
(8) I would like now to add one further example, not mentioned by Wittgenstein.
In his Italian Journey, Goethe makes the following observation: “The Venetian
was forced to become a new creature; and thus Venice can only be compared
with itself. The large canal, winding like a serpent, yields to no street in
the world […].”767
It would make no sense to compare Venice with itself. The
point of uttering such a strange sentence is that—as Goethe realized—every attempt
to compare Venice with another town must fail. To say that Venice can be
compared only with itself means, in fact, that Venice could not be compared
with anything at all. A reflexive use of a comparison is thus straightened out into
an intransitive case of a rejection of any possible comparison.

My final reflection is devoted to making some deeper sense of the maxim of no
reflexive uses of internal relations. The requirement that an instance of an internal
relation not be reflexive demands, in fact, that there must be a difference
between its terms. An internal relation is not one of pure identity. The terms of
an internal relation are in a certain sense the same (qua internal) and they are
also different (qua relation). The demand for a difference between these terms
suggests that there must be an external relation between these terms which
would account for their difference. Hence, if two terms a and b are internally
related by the relation Rint, there must be an external relation Rext ensuring that a
and b are different:
(Rint)(a,b) ( (Rint(a,b)  (Rext) Rext(a,b) )(112)
This is also a formal expression of the maxim of no reflexive uses of internal
relations. However, even this expression might be misleading due to the nature
of the terms a and b. The expressions ‘a’ and ‘b’ are de dicto in Rint(a,b); they
767
Goethe, 2008, p. 123.
250
are however de re in Rext(a,b). This means that internal relations are held between
concepts, whereas external relations are held between objects as indicated
in §10.2.
The idea behind the maxim is not Wittgenstein’s sole invention. Several other
philosophers have developed various strategies for how to cope with the paradoxes
of identity. Let me mention Fichte’s ground of distinction [Unterscheidungs-Grund].
If two entities are asserted—or ‘posited’ [gesetzt]—to be the
same, they must be different in some respect, i.e., there must be some ground
for their difference. This ground lies in the very presupposition that there must
be two entities posited as identical. If there were only one entity, any positing
would make little sense.
Frege, in his seminal paper On Sense and Reference, proposes a different solution
to the paradox of identity. Why, he asks, is the proposition ‘a = b’ of any
cognitive value if a and b refer to the same object? Why does the proposition
that ‘The Morning Star is the Evening Star’ say something substantial if ‘the
Morning Star’ as well as ‘the Evening Star’ refer to Venus? Frege’s solution
rests in finding out that the sides of the equation are nevertheless different. They
have the same reference, but different senses. ‘The Morning Star’ refers to Venus
as she appears in the morning sky; ‘the Evening Star’ refers to the same
planet as she appears in the evening sky. The idea lingering behind this is similar
to that in Fichte. Any statement of identity presupposes some difference in
its terms. Any identity, if it should be informative, is only a partial identity.
In my view, the clearest expression of this idea is to be found in Bradley. As already
discussed in detail in §4.2, for Bradley every relation must be partly internal
and partly external. In particular, there are no wholly internal relations. If
there were such a relation, we could not distinguish between its terms. They
would lose their individuality. Then, however, if there were no distinct terms,
there would be nothing to relate.
Let me highlight this affinity between Bradley and Wittgenstein. Bradley argues
that every relation must be both internal and external. Endorsing an analytic attitude,
Wittgenstein does not say that a single relation must be both internal and
external. He insists on distinguishing between internal and external relations.
The maxim of no reflexive uses of internal relations says that every internal re-
251
lation presupposes an external relation that accounts for any difference between
its terms.

Wittgenstein’s first choice for the motto of the Philosophical Investigations was
Joseph Butler’s “Everything is what it is, and nothing else.” He did not use this
sentence because G. E. Moore had already used it as the motto for his Principia
Ethica.768
“Everything is what it is” is another reflexive construction saying that
everything is related to itself. This construction can therefore be straightened
out and understood intransitively as placing an emphasis on things themselves
as opposed to what they stand for.769
Then this construction would have a similar
sense as Goethe’s “Don’t look for anything behind the phenomena; they
themselves are the theory.”770
Now, everything is nothing else. If a thing were (or stood for, referred to,
meant) some other thing, these two things would be internally related.771
But if
everything were nothing else, there would be no internal relations. Thus, internal
relations do not—in the end, in the final analysis—belong to things; they are
not constitutive of things. They are the means of representation of things. Internal
relations can be—in an unattainable ideal—simply left behind.
768
Reported in Kreisel, 1978, p. 13.
769
Most commentators have interpreted Wittgenstein’s fondness for this epitaph as an expression
of his anti-reductionism.
770
RPP I, §889; see §18.2.
771
Hegel would say that everything is indeed something else; everything “is the unity of
itself and its opposite” (Hegel, 1977, p. 220).
252
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Index
a priori, 28, 49, 70, 72, 89, 90, 117,
152, 170, 202, 256
album, ix
Ambrose, A., xiii
analogy, xi, 2, 3, 17, 73, 74, 79, 82,
83, 84, 85, 87, 90, 112, 127, 128,
129, 130, 131, 132, 140, 142, 181,
199, 200, 206, 207, 226, 227, 228,
229
analysis, x, xi, 1, 2, 3, 4, 7, 14, 16,
31, 38, 53, 54, 55, 58, 59, 63, 65,
67, 69, 70, 72, 82, 97, 104, 105,
110, 119, 130, 132, 133, 135, 165,
167, 168, 172, 181, 190, 202, 208,
212, 214, 223, 229, 234, 235, 240,
246, 251
analytic philosophy, x, 1, 8, 16
Anscombe, E., xiii, xiv, xv, 127,
136, 252
Aristotle, 225, 229, 252
art, 14, 17, 26, 27, 164, 230, 231,
232, 233, 234, 235, 237, 238, 245,
248, 252
aspect, 17, 23, 24, 26, 82, 94, 166,
176, 179, 181, 185, 194, 195, 196,
217, 218, 219, 220, 223, 226, 227,
228, 229, 230, 232, 233, 235, 236,
237, 238, 244, 245
blindness, 195, 217
dawning of, 194, 195, 217, 218,
219, 228, 232, 237
seeing, 194, 219, 220, 226, 232,
244
Augustus, 99
Avital, D., 201, 209, 210, 211, 212,
252
Baker, G. P., xiv, 106, 115, 116, 156,
157, 158, 159, 201, 202, 252
Basile, P., 63, 253
Baumann, P., xii, 111, 119
Beethoven, L. van, 101, 221, 236
behavior cycle, 120, 121, 122, 130,
134
Beran, O., xii, 50, 192
Black, M., 99, 226, 252
Borges, J. L., 88, 247, 252
Bosanquet, B., 17, 31
Bradlex, F. H.
Regress Argument, 7, 23, 24, 25,
37, 260
Bradley, F. H., x, 7, 8, 12, 15, 17, 18,
19, 20, 21, 22, 23, 24, 25, 27, 28,
29, 31, 32, 33, 34, 35, 36, 37, 38,
39, 50, 53, 56, 61, 63, 142, 222,
223, 240, 250, 252, 253, 254, 257,
260
Appearance, 17, 18, 19, 20, 21,
24, 27, 29, 33, 34, 35, 252
feeling base, 20, 222, 223
Reality, 17, 18, 19, 20, 21, 26, 27,
28, 29, 33, 34, 35, 37, 75, 252
Bradley, R., 63, 64, 65, 66, 67, 69,
77, 252
Brahms, J., 233
Breitenbach, A., 152, 253
Butler, J., 251
262
Caesar, G. J., 99
Candlish, S., 16, 17, 20, 21, 24, 32,
35, 56, 63, 253
Cappelen, H., 108, 253
Carroll, L., 88, 247, 253
certainty, xi, 124, 224, 225
Sense-certainty, 223, 224, 225,
253
color, 3, 7, 18, 47, 62, 81, 87, 90, 94,
99, 103, 112, 113, 164, 178, 180,
181, 182, 183, 184, 185, 186, 187,
188, 189, 190, 191, 192, 193, 194,
195, 196, 197, 198, 199, 216, 221,
223, 236, 243, 245
mixture, 190, 192, 193, 194, 244
simple, 190, 192, 193, 194, 195,
196, 197, 198
space, 48, 99, 113, 183, 184, 191,
192, 194, 196, 198
community, 158, 159, 162, 192, 197
concept-script (Begriffsschrift), x, 2,
44, 55
Crane, T., 126, 127, 253
criterion of temporality, 93, 96, 106,
107, 108, 109, 110, 111, 118, 130,
131, 137, 139, 140, 150, 183, 192,
198, 202, 209
culture, 27, 230, 231, 233, 234, 237,
238, 245
Davidson, D., 141, 153, 154, 155,
226, 253
de dicto, 5, 44, 63, 65, 67, 68, 69,
76, 78, 201, 249
de re, 5, 44, 63, 64, 66, 67, 69, 76,
77, 110, 201, 250
definition, x, 20, 21, 22, 24, 28, 31,
37, 47, 48, 49, 50, 51, 86, 90, 94,
95, 98, 100, 111, 114, 116, 161,
169, 170, 171, 172, 180, 184, 201,
202, 203, 204, 205, 206, 208, 209,
214, 216, 243
Democritus, 63
Descartes, R., 141, 151
deVries, W. A., 224, 253
Diamond, C., xiii, 12, 13, 73, 74,
179, 213, 214, 215, 253
doctrine
of degrees of truth, 19
of external relations, 5, 6, 15, 37,
39, 40, 53, 57, 59, 225
of internal relations, 16, 17, 21,
24, 31, 33, 35, 38, 225
of showing, 37, 55
Dolev, Y., 201, 212, 254
Donellan, K., 254
Drury, M. O’C., xiii, 4
duck-rabbit figure, 79, 217, 218,
219, 226, 227, 228, 236, 244
Erbacher, Ch., xii, 254
essence, 16, 57, 73, 94, 151
ethics, xi, 44, 82
Ewing, A. C., 9, 254
fact, x, 4, 6, 7, 13, 19, 22, 23, 24, 26,
32, 33, 36, 38, 39, 40, 46, 50, 53,
54, 55, 56, 57, 61, 62, 64, 67, 70,
73, 75, 76, 78, 79, 80, 82, 83, 87,
93, 96, 99, 100, 103, 104, 113,
114, 116, 118, 121, 122, 123, 124,
125, 126, 131, 134, 136, 140, 141,
142, 144, 145, 146, 150, 154, 156,
158, 159, 160, 163, 165, 166, 168,
169, 173, 182, 185, 188, 189, 192,
194, 196, 198, 199, 201, 202, 203,
207, 221, 224, 230, 231, 233, 237,
240, 241, 243, 245, 246, 247, 248,
249
complex, 78
family resemblance, 169, 173
263
feeling, 20, 120, 121, 126, 130, 133,
134, 198, 220, 221, 222, 223, 225,
234, 235, 237, 238, 244, 247, 249
Ferreira, P., 19, 24, 25, 27, 34, 222,
223, 254
Fichte, J. G., 11, 250
fitting (passen), 1, 12, 27, 62, 127,
128, 129, 130, 131, 161, 186, 220,
221, 222, 223, 225, 231, 232, 237,
247
Floyd, J., 175, 253, 254
Fogelin, R. J., 201, 254
form
logical, 1, 2, 7, 50, 55, 56, 64, 74,
84, 87, 88, 89, 107, 108
of life, 91, 93, 118, 158, 160, 164,
165, 238
Frascolla, P., 169, 178, 254
Frege, G., 1, 2, 4, 8, 21, 44, 55, 57,
86, 87, 88, 89, 90, 130, 148, 169,
170, 240, 250, 253, 254, 258, 259
Galton, F., 227
Garver, N., 254
Geach, P., xiv, 192, 253
generality, 167, 168
Gert, H. J., 206, 207, 254
Gestalt, 226, 236, 249
Gierlinger, F., 207, 254
Glock, H.-J., 104, 116, 139, 210,
238, 254
Goethe, J. W. von, 221, 235, 249,
251, 254
grammar, 1, 2, 13, 14, 51, 70, 93, 94,
95, 104, 106, 108, 115, 117, 119,
129, 131, 133, 136, 145, 151, 158,
173, 176, 181, 190, 191, 192, 240,
241, 242
Hacker, P. M. S., xii, 63, 64, 77, 106,
115, 116, 149, 156, 157, 158, 159,
201, 202, 252, 254, 255
Hegel, G. W. F., x, 4, 16, 17, 32,
222, 223, 224, 225, 251, 253, 255,
257, 259
Hering, E., 193, 196, 255
Hester, M., 226, 227, 255
hieroglyphs, 79
Hintikka, J., 106, 255
Hochberg, H., 6, 255
Hylton, P., 38, 59, 253, 255
Hymers, M., 115, 122, 255
idealism, 15, 16, 17, 254, 255, 257
identity, 3, 7, 11, 12, 13, 20, 25, 27,
40, 45, 87, 94, 101, 103, 145, 146,
157, 172, 189, 209, 210, 222, 247,
249, 250, 253
identity theory of truth, 253
infinity, 7, 25, 113, 141, 142, 144,
167, 169
intentionality, 119, 120, 122, 129,
130, 132, 133, 134, 135, 136, 226,
228, 247
causal conception of, 112, 120,
122, 123, 130, 134, 136, 137,
140, 141, 142, 144, 146, 147,
151, 152, 153, 154, 155, 156,
221, 226, 228, 231
desire, x, 27, 119, 120, 121, 122,
134, 159, 160, 176, 225
fulfillment, 119, 120, 123, 124,
125, 126, 127, 128, 129, 130,
131, 132, 134, 135, 138, 244,
247
interjections, 230
Jantschek, T., 218, 255
Johannessen, K. S., 235, 255
Johnston, C., 70, 72, 255
264
Jolley, K. D., 207, 255
judgment, 7, 18, 19, 20, 21, 26, 27,
33, 34, 35, 50, 138, 152, 197, 222,
230, 231, 233
Kain, P. J., 16, 225, 255
Kant, I., 1, 8, 113, 114, 151, 152,
237, 238, 253, 255
Keats, J., 227, 228, 245
Keller, G., 233
Klagge, J., xi, xiv, 86, 117, 149, 150,
207, 255
Köhler, W., 149
Korzybski, A., 88
Kreisel, G., 251, 255
Kremer, M., 57, 58, 255
Kripke, S., 156, 157, 158, 159, 160,
161, 162, 201, 202, 203, 209, 212,
215, 256
Krüger, W., xii, 162, 163, 165, 256
Kuusela, O., 2, 3, 73, 256
Landini, G., 55, 56, 57, 67, 256
language-game, 11, 13, 14, 93, 95,
102, 104, 105, 106, 107, 108, 111,
113, 115, 117, 118, 136, 138, 142,
143, 144, 145, 146, 150, 151, 155,
165, 182, 183, 186, 187, 188, 189,
199, 200, 206, 207, 208, 214, 216,
220, 230, 238, 241, 242, 246, 248
of applying a rule, 60, 105, 112,
125, 176, 187, 188, 189, 207,
242, 248
of training, 105, 106, 165, 207,
208
Lee, A., 194, 197, 198, 256
Lee, D., xiii, 70
Leibniz, G. W., 63, 141, 169
Lepore, E., 108, 253
Lin, M., 141, 257
Livingston, P., 148, 149, 256
logically adequate language, 2, 14,
44, 51, 54, 55, 56, 60, 61, 63, 67,
70, 71, 75, 81, 241
Loomis, E., 201, 202, 256
Loprieno, A., 79, 256
Majetschak, S., 257
Manser, A. R., 21, 28, 29, 257
Marx, K., 4, 257
mathematics, 13, 14, 48, 79, 108,
112, 113, 131, 138, 161, 164, 166,
167, 168, 169, 170, 171, 172, 173,
174, 175, 176, 177, 178, 179, 180,
181, 183, 191, 192, 194, 197, 243,
244, 258
number, xv, 16, 90, 94, 98, 99,
100, 101, 103, 113, 114, 148,
158, 161, 162, 164, 166, 167,
168, 169, 170, 171, 172, 173,
174, 175, 178, 179, 180, 181,
182, 183, 184, 190, 192, 204,
243, 248
cardinal, 168, 169
Fibonacci, 113, 114
natural, 161, 162, 164, 167,
169, 174, 182
proof, 32, 161, 166, 167, 173,
174, 175, 176, 177, 178, 179,
180, 244
maxim of no reflexive uses of
internal relations, 13, 88, 234,
240, 246, 247, 248, 249, 250
McGinn, C., 112, 257
McGinn, M., 14, 38, 57, 58, 77, 79,
242, 256, 257
McGuinness, B. F., xiv, 51
McManus, D., 4, 11, 13, 61, 73, 74,
257
meaning, ix, 19, 21, 28, 55, 73, 74,
77, 78, 79, 80, 86, 87, 90, 95, 105,
265
108, 116, 123, 126, 127, 128, 129,
132, 140, 154, 156, 160, 162, 170,
172, 173, 174, 175, 176, 180, 184,
186, 192, 199, 200, 205, 208, 210,
213, 214, 216, 217, 220, 228, 236,
237, 243, 247,249
experiencing, 135
measuring rod, 84, 127, 138, 200,
211
Melamed, Y., 141, 257
metaphor, 1, 20, 23, 25, 62, 127,
130, 145, 225, 226, 227, 228, 229,
245
metaphysics, 3, 5, 7, 8, 17, 19, 21,
39, 62, 63, 65, 66, 67, 73, 77, 79,
119, 141, 144, 145, 151, 153, 205
method, x, xi, 1, 3, 4, 5, 13, 16, 59,
78, 79, 84, 87, 91, 96, 97, 105,
109, 111, 114, 119, 140, 152, 155,
165, 185, 199, 206, 222, 223, 241,
246, 247
monism, 7, 20, 28, 34, 37, 62, 223
Moore, G. E., x, 1, 5, 6, 8, 9, 12, 15,
16, 17, 21, 28, 31, 32, 36, 37, 38,
39, 40, 43, 45, 51, 53, 57, 59, 61,
62, 64, 94, 251, 257
Principia Ethica, 37, 251, 257
motive, reason, ground, 10, 11, 12,
23, 34, 45, 46, 48, 60, 79, 91, 93,
95, 104, 106, 108, 129, 136, 137,
138, 139, 140, 141, 142, 143, 144,
145, 149, 150, 153, 154, 155, 158,
159, 161, 162, 176, 187, 188, 189,
203, 219, 221, 231, 238, 241, 248,
250, 257
Moyal-Sharrock, D., 105, 257, 259
Mulligan, K., 6, 253, 255
name, 1, 3, 37, 53, 54, 55, 56, 58,
64, 66, 67, 68, 69, 70, 74, 75, 76,
77, 78, 79, 80, 82, 83, 84, 106,
119, 133, 186, 204, 206, 207, 208,
221, 241, 242, 243, 246
proper, 1, 2
necessity, x, 9, 44, 59, 64, 66, 110,
146, 210
object, 6, 8, 11, 12, 15, 18, 22, 23,
27, 28, 29, 30, 36, 37, 38, 40, 44,
45, 47, 50, 54, 55, 56, 58, 62, 63,
64, 65, 67, 68, 69, 70, 71, 72, 74,
75, 76, 77, 78, 80, 82, 83, 84, 87,
90, 93, 95, 96, 97, 98, 102, 103,
104, 106, 110, 111, 112, 116, 118,
120, 121, 126, 127, 128, 129, 131,
132, 133, 134, 135, 136, 148, 152,
160, 165, 169, 170, 178, 179, 180,
181, 183, 184, 185, 188, 189, 191,
194, 198, 199, 200, 201, 202, 203,
204, 205, 206, 207, 208, 209, 210,
211, 212, 213, 214, 215, 216, 218,
219, 220, 221, 222, 223, 224, 225,
230, 232, 233, 234, 237, 240, 241,
242, 247, 250
of comparison, 165, 179, 200
simple, 5, 41, 44, 47, 53, 54, 58,
62, 63, 64, 65, 66, 67, 69, 70,
71, 72, 76, 77, 78, 80
Ogden, C. K., xiv, 2, 12, 15, 49, 51,
120
ontology, 7, 8, 16, 17, 20, 28, 34, 35,
36, 62, 63, 120, 160
Özlem, F., 238, 258
pain, 25, 115, 116, 117, 238
paradigmatic sample, 164, 173, 174,
178, 182, 184, 185, 186, 187, 188,
189, 195, 197, 198, 211, 243, 244,
248
color, 185, 189
266
paradox, 11, 94, 148, 157, 158, 160,
162, 219, 250
of identity, 11, 250
rule-following, 156, 157
Pears, D., xiv, 28, 30, 51, 259
Philosophical Investigations, ix, xi,
xiv, 1, 3, 12, 90, 91, 94, 95, 104,
106, 116, 123, 129, 130, 131, 135,
138, 143, 147, 158, 160, 163, 164,
170, 175, 180, 186, 187, 188, 194,
199, 200, 206, 207, 216, 217, 218,
220, 221, 228, 232, 236, 247, 251,
252, 254, 255, 256, 258, 260
picture, 6, 49, 50, 53, 73, 74, 75, 83,
86, 87, 88, 89, 90, 120, 123, 128,
132, 142, 147, 156, 163, 166, 167,
170, 176, 177, 178, 179, 184, 200,
209, 220, 222, 226, 234, 235, 236,
237, 244, 247
conception of intentionality, 120,
123, 128, 132
picture theory, 53, 73, 74, 86, 87,
123, 132, 142, 156
playing chess, 206, 207
pluralism, 5, 7, 37, 39
Pollock, W. J., 201, 202, 258
Potter, M., 38, 56, 258
principle
of causal mediation, 150, 151,
153, 155
rebus, 79
Proops, I., 38, 40, 49, 59, 258
proposition, xi, 2, 3, 6, 7, 12, 28, 32,
35, 39, 40, 43, 44, 45, 48, 49, 50,
51, 53, 54, 56, 58, 59, 60, 61, 64,
70, 73, 74, 75, 76, 77, 78, 79, 80,
82, 83, 84, 87, 89, 90, 93, 94, 95,
96, 98, 99, 103, 105, 107, 108,
109, 110, 111, 113, 117, 118, 123,
124, 126, 127, 128, 129, 137, 142,
143, 144, 161, 165, 167, 168, 173,
174, 175, 176, 177, 183, 186, 188,
190, 191, 200, 202, 209, 210, 216,
241, 242, 243, 244, 250
complex, 40, 57, 58, 59, 60, 66,
75, 76, 78, 80
elementary, 40, 41, 53, 56, 57, 58,
59, 60, 61, 70, 74, 75, 76, 78,
80
hinge, xi, 216
Protagoras, 82
Quine, W. V. O., 19, 255
Ramsey, F. P., xiv, 2, 12, 15, 49, 51,
72
Read, R., 10, 254
Reichenbach, H., 212
relation
asymmetrical, 33, 34, 35, 36
binary, 26, 31, 35, 46, 112, 204
external, ix, x, xi, 3, 4, 6, 7, 8, 9,
10, 12, 13, 14, 15, 16, 21, 22,
23, 24, 25, 26, 27, 28, 29, 30,
31, 32, 33, 36, 37, 38, 39, 40,
41, 43, 44, 46, 48, 50, 51, 52,
53, 54, 55, 56, 57, 59, 61, 62,
63, 64, 77, 78, 82, 87, 93, 96,
98, 99, 100, 101, 102, 103, 104,
106, 107, 109, 110, 111, 112,
114, 115, 117, 118, 119, 120,
121, 123, 125, 127, 130, 131,
134, 135, 136, 137, 139, 140,
142, 144, 146, 147, 153, 156,
157, 159, 160, 163, 166, 167,
168, 169, 171, 174, 176, 177,
179, 182, 183, 186, 187, 189,
199, 202, 208, 209, 218, 221,
229, 232, 233, 234, 240, 241,
242, 243, 244, 249, 250
267
grammatical, 10, 13, 94, 124, 192
internal, ix, x, xi, 3, 4, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 21,
22, 24, 25, 26, 27, 28, 29, 30,
31, 32, 33, 34, 35, 36, 37, 38,
39, 40, 41, 43, 44, 45, 46, 47,
48, 49, 50, 51, 52, 53, 54, 55,
56, 57, 59, 60, 61, 62, 63, 64,
66, 67, 68, 69, 70, 71, 72, 73,
74, 75, 77, 78, 79, 80, 81, 82,
84, 85, 87, 88, 90, 93, 94, 95,
96, 98, 99, 100, 101, 102, 103,
104, 106, 109, 110, 111, 112,
113, 114, 115, 116, 117, 118,
119, 120, 123, 124, 125, 126,
127, 128, 129, 130, 131, 132,
135, 136, 137, 138, 139, 140,
142, 143, 144, 146, 147, 148,
150, 151, 153, 156, 157, 159,
160, 161, 162, 163, 164, 165,
166, 167, 168, 169, 170, 171,
172, 173, 174, 175, 177, 178,
179, 181, 182, 183, 184, 186,
187, 189, 190, 191, 192, 194,
195, 197, 198, 199, 202, 208,
209, 212, 213, 214, 215, 218,
219, 220, 221, 225, 228, 229,
230, 231, 232, 233, 234, 236,
240, 241, 242, 243, 244, 245,
246, 247, 248, 249, 250, 251
intransitive, 133, 134, 135, 213,
214, 215, 230, 235, 236, 237,
246, 248, 249
multiple, 35, 69, 105
reflexive, 13, 38, 88, 130, 135,
144, 162, 163, 172, 173, 189,
200, 204, 213, 214, 215, 220,
234, 240, 246, 247, 248, 249,
251
symmetrical, 33, 34, 69
transitive, 133, 134, 213, 214,
230, 236, 237
vertical, 106, 144, 208, 238
resolute reading, ix, 63, 74, 241
Ricoeur, P., 245
Rodríguez-Consuegra, F., 258
Rodych, V., 258
Rorty, R., 9, 258
rule, xi, 3, 12, 13, 14, 48, 79, 95, 96,
103, 104, 105, 106, 109, 110, 111,
113, 114, 116, 118, 120, 125, 126,
129, 138, 139, 142, 143, 144, 148,
150, 151, 156, 157, 158, 159, 160,
161, 162, 163, 164, 166, 169, 172,
173, 174, 177, 185, 186, 187, 188,
189, 191,194, 197, 198, 199, 200,
207, 210, 216, 218, 230, 233, 236,
237, 238, 240, 242, 243, 244, 248
following, 12, 120, 143, 148, 156,
157, 160, 194, 197
Runge, P. O., 198
Russell
Theory of Knowledge, 1, 40, 59,
259
Russell, B., x, 1, 2, 4, 5, 6, 8, 12, 15,
16, 17, 21, 22, 26, 31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 43, 51, 53,
57, 58, 59, 61, 62, 65, 69, 70, 120,
121, 122, 133, 136, 140, 141, 148,
169, 170, 171, 173, 253, 255, 256,
258, 259
Schaffer, J., 31, 62, 259
Schroeder, S., 234, 236, 237, 259
Schubert, F., 221
Schulte, J., 259
Searle, J., 105, 148, 259
sentence, 1, 3, 19, 46, 69, 72, 96, 98,
99, 100, 102, 103, 104, 105, 106,
268
107, 108, 109, 110, 111, 115, 116,
117, 127, 129, 131, 139, 140, 144,
145, 150, 172, 177, 186, 187, 188,
189, 192, 194, 199, 200, 206, 207,
208, 209, 210, 213, 214, 215, 216,
220, 227, 242, 243, 244, 246, 247,
249, 251, 254
Shanker, S. F., 176, 259
Sheffer stroke, 60
Shelley, P. B., 227
showing, 2, 4, 10, 24, 25, 37, 40, 43,
44, 45, 50, 51, 52, 55, 57, 61, 64,
72, 74, 79, 91, 93, 94, 104, 126,
145, 158, 160, 161, 167, 171, 173,
174, 181, 190, 195, 202, 217, 221,
223, 227, 244, 254
sign, 3, 48, 53, 54, 55, 59, 60, 70,
72, 74, 77, 78, 79, 80, 81, 82, 87,
158, 161, 162, 163, 166, 168, 169,
170, 241, 246, 247
Sluga, H., 86, 253, 259
Smith, D., xii, 106, 107, 259
Socrates, 65, 247
Spengler, O., 231
Spinoza, B., 141, 142, 144, 145, 151
standard meter, 82, 83, 84, 85, 87,
103, 127, 128, 129, 170, 178, 179,
180, 184, 187, 200, 201, 202, 203,
204, 205, 206, 207, 208, 209, 210,
211, 212, 213, 214, 215, 243, 248
Stanley, J., 107, 108, 259
state of affairs, 49, 50, 55, 56, 57,
58, 59, 62, 63, 64, 68, 70, 71, 72,
75, 76, 77, 78, 80, 97, 102, 119,
120, 121, 122, 125, 132, 133, 134,
166, 177, 179, 241
substance, 17, 18, 39, 56, 141, 143
super-mechanism, 145, 147
supervenience, 25, 112, 149
surveyable representation
(übersichtliche Darstellung), x,
183, 192, 242
symbol, symbolism, 43, 51, 53, 54,
74, 78, 79, 81, 146, 166, 170, 178,
210
tautology, 44, 45, 46, 49, 115
Taylor, Ch., 225, 259
Taylor, J., 137
Ter Hark, M., 106, 157, 193, 218,
259
The Big Typescript, xiii, 111, 114,
124, 126, 127, 129, 137, 138, 145,
172, 190, 191, 193, 208, 246
thinkability, x, 22, 23, 27, 48, 49, 57,
62, 78, 80, 93, 95, 96, 98, 109,
111
Tolstoy, L. N., 234
Tractatus Logico-Philosophicus, ix,
x, xiv, 1, 2, 3, 4, 8, 9, 10, 11, 12,
13, 14, 15, 22, 30, 32, 37, 40, 41,
43, 44, 45, 46, 47, 48, 49, 50, 51,
52, 53, 54, 55, 56, 58, 59, 60, 61,
62, 63, 64, 65, 66, 67, 68, 72, 73,
75, 76, 77, 78, 79, 80, 81, 82, 83,
84, 86, 87, 89, 90, 94, 95, 98, 99,
111, 112, 123, 132, 133, 156, 166,
167, 178, 181, 182, 200, 206, 209,
211, 240, 241, 247, 254, 255, 256,
257, 258
transcendent, 232
trope theory, 29
use, 1, 2, 4, 21, 23, 47, 49, 56, 65,
71, 74, 84, 88, 90, 91, 97, 103,
104, 105, 107, 109, 123, 130, 132,
133, 134, 135, 141, 144, 155, 160,
162, 163, 164, 165, 168, 170, 173,
180, 184, 185, 186, 187, 188, 189,
194, 195, 201, 207, 213, 214, 215,
269
226, 228, 236, 244, 246, 247, 249,
251
Uzan, J.-P., 205, 260
Vallicella, V., 24, 26, 260
variable
propositional, 81, 82, 89, 241
structured, 81
Waismann, F., xiv, 55, 171, 204,
208, 260
Weininger, O., 82
Wenzel, Ch. H., 237, 260
White, R., 227
Wollheim, R., 28, 30
Wright, G. H. von, xiii, xiv, xv