1
Consequences
Everybody, sooner or later,
sits down to a banquet of consequences.
– Robert Louis Stevenson
‘Watch what you say,’ my mother often advised, ‘because what
you say has consequences.’ She was right, and doubly so. There
are two senses in which what one says has consequences. One
sense, not terribly relevant for present purposes, is captured in the
familiar dictum that actions have consequences. To say something
is to do something, and doing something is an action. Actions,
in turn, are events, and events, as experience tells, have consequences,
namely, their causal effects. (Example: a consequence—
a causal effect—of your drinking petrol is your being ill, at least
other things being equal.) So, in the causal effects sense of ‘consequences’,
my mother was perfectly right, but that sense of ‘consequence’
has little to do with logic.
For present purposes, there is a more relevant sense in which
what one says has consequences. What one says, at least in the
declarative mode,1
has logical consequences, namely, whatever
logically follows from what one said, or whatever is logically implied
by what one said. Suppose, for example, that you’re given
the following information.
1. Agnes is a cat.
2. All cats are smart.
1For purposes of this book, a declarative sentence (or a sentence used
in the ‘declarative mode’) is one that is used (successfully or otherwise) to
declare or state something about the world. This is hardly a precise definition,
but it’ll do. (Example. Each of ‘You are reading a book’, ‘Obama is the first
black US president’, and ‘1+1=5’ are declarative sentences, but sentences
such as ‘Shut that door!’ and ‘Do you like Vegemite?’ are not declarative,
since they fail to declare or state anything about the world.)
Consequences4
A consequence of (1) and (2), taken together, is that Agnes is
smart. In other words, that Agnes is smart logically follows from
(1) and (2); it is implied by (1) and (2), taken together.
1.1 Relations of support
Logical consequence is a relation on sentences of a language,
where ‘sentence’, unless otherwise indicated, is short for ‘meaningful,
declarative sentence’.2
Logical consequence is one among many relations over the
sentences of a language. Some of those relations might be called
relations of support. For example, let A1, . . . , An and B be arbitrary
sentences of some given language—say, English. For some
such sentences, the various Ai jointly ‘support’ B in the following
sense.
S1. If all of A1, . . . , An are true, then B is probably true.
Consider, for example, the following sentences.
3. Max took a nap on Day 1.
4. Max took a nap on Day 2.
5. Max took a nap on Day 3.
...
n. Max took a nap on Day n (viz., today).
m. Max will take a nap on Day n + 1 (viz., tomorrow).
On the surface, sentences (3)–(n) support sentence (m) in the
sense of (S1): the former, taken together, make the latter more
likely. Similarly, (6) supports (7) in the same way.
6. The sun came up every day in the past.
7. The sun will come up tomorrow.
If (6) is true, then (7) is probably true too.
The relation of support given in (S1) is important for empirical
science and, in general, for rationally navigating about our world.
Clarifying the (S1) notion of ‘support’ is the job of probability
theory (and, relatedly, decision theory), an area beyond the range
of this book.
2Taking consequence to be relation on sentences simplifies matters a great
deal, and sidesteps the issue of so-called ‘truth bearers’, an ongoing issue in
philosophy of logic. For present purposes, simplicity is worth the sidestep.
Logical consequence: the basic recipe 5
1.2 Logical consequence: the basic recipe
Logical consequence, the chief topic of logic, is a stricter relation
of support than that in (S1). Notice, for example, that while (7)
may be very likely true if (6) is true, it is still possible, in some
sense, for (6) to be true without (7) being true. After all, the sun
might well explode later today.
While (S1) might indicate a strong relation of support between
some sentences and another, it doesn’t capture the tightest relation
of support. Logical consequence, on many standard views, is
often thought to be the tightest relation of support over sentences
of a language. In order for some sentence B to be a logical consequence
of sentences A1, . . . , An, the truth of the latter needs to
‘guarantee’ the truth of the former, in some suitably strong sense
of ‘guarantee’.
Throughout this book, we will rely on the following (so-called
semantic) account of logical consequence, where A1, . . . , An and
B are arbitrary sentences of some given language (or fragment of
a language).
Definition 1. (Logical Consequence) B is a logical consequence of
A1, . . . , An if and only if there is no case in which A1, . . . , An are
all true but B is not true.
Notice that the given ‘definition’ has two parts corresponding to
the ‘if and only if’ construction, namely,
• If B is a logical consequence of A1, . . . , An, then there is no
case in which A1, . . . , An are all true but B is not true.
• If there is no case in which A1, . . . , An are all true but B is
not true, then B is a logical consequence of A1, . . . , An.
Also notable is that the given ‘definition’ is really just a recipe.
In order to get a proper definition, one needs to specify two key
ingredients:
• what ‘cases’ are;
• what it is to be true in a case.
Once these ingredients are specified, one gets an account of logical
consequence. For example, let A1, . . . , An and B be declarative
sentences of English. If we have a sufficiently precise notion of
possibility and, in turn, think of ‘cases’ as such possibilities, we
Consequences6
can treat ‘true in a case’ as ‘possibly true’ and get the following
account of logical consequence—call it ‘necessary consequence’.
• B is a (necessary) consequence of A1, . . . , An if and only if there
is no possibility in which A1, . . . , An are all true but B is not
true. (In other words, B is a consequence of A1, . . . , An if and
only if it is impossible for each given Ai to be true without B
being true.)
Presumably, this account has it that, as above, ‘Agnes is smart’
is a consequence of (1) and (2). After all, presumably, it’s not
possible for (1) and (2) to be true without ‘Agnes is smart’ also
being true. On the other hand, (7) is not a necessary consequence
of (6), since, presumably, it is possible for (6) to be true without
(7) being true.
Of course, taking ‘cases’ to be ‘possibilities’ requires some
specification of what is possible, or at least some class of ‘relevant
possibilities’. The answer is not always straightforward. Is
it possible to travel faster than the speed of light? Well, it’s not
physically possible (i.e., the physical laws prohibit it), but one
might acknowledge a broader sense of ‘possibility’ in which such
travel is possible—for example, coherent or imaginable or the like.
If one restricts one’s ‘cases’ to only physical possibilities, one gets
a different account of logical consequence from an account that
admits of possibilities that go beyond the physical laws.
In subsequent chapters, we will be exploring different logical
theories of our language (or fragments of our language). A logical
theory of our language (or a fragment thereof) is a theory
that specifies the logical consequence relation over that language
(or fragment). Some fragments of our language seem to call for
some types of ‘cases’, while other fragments call for other (or
additional) types. Subsequent chapters will clarify this point.
1.3 Valid arguments and truth
In general, theses require arguments. Consider the thesis that
there are feline gods. Is the thesis true? An argument is required.
Why think that there are feline gods? We need to examine the
argument—the reasons that purport to ‘support’ the given thesis.
Arguments, for our purposes, comprise premises and a conclusion.
The latter item is the thesis in question; the former pur-
Valid arguments and truth 7
port to ‘support’ the conclusion. Arguments may be evaluated
according to any relation of support (over sentences). An argument
might be ‘good’ relative to some relation of support, but not
good by another. For example, the argument from (6) to (7) is
a good argument when assessed along the lines of (S1); however,
it is not good when assessed in terms of (say) necessary consequence,
since, as noted above, (7) is not a necessary consequence
of (6).
In some areas of rational inquiry, empirical observation is often
sufficient to figure out the truth. Suppose that you want to
know whether there’s a cat on the table. One reliable method
is handy: look at the table and see whether there’s a cat on it!
Of course, ‘real empirical science’ is much more complicated than
checking out cats, but empirical observation—empirical testing—
is nonetheless a critical ingredient.
What about other pursuits for which there is little, if any,
opportunity for settling matters by observation? Consider, for
example, pure mathematics or philosophy. In such areas, theses
cannot be empirically tested, at least in general. How, then, do we
figure out the truth in such areas? Argument is the only recourse.
When argument is the only recourse, as in mathematics or (at
least much of) philosophy, it makes sense to invoke the strictest
relation of support—namely, logical consequence. Traditionally,
an argument is said to be valid—strictly speaking, logically valid—
if its conclusion is a logical consequence of its premises. We will
follow suit.
Of course, a valid argument needn’t be a proof of anything.
After all, the ‘definition’ (or, for now, ‘recipe’) of logical consequence
doesn’t require that any of the premises be true. Rather,
the given account requires only the absence of any ‘counterexample’,
where these are defined as follows.
Definition 2. (Counterexample) A counterexample to an argument
is a case in which the premises are true but the conclusion is not
true.
We can say that B is a logical consequence of A1, . . . , An if
and only if there is no counterexample to the argument from
(premises) A1, . . . , An to (conclusion) B. In turn, an argument is
valid just if there is no counterexample to it.
Consequences8
Accordingly, an argument may be valid—that is, its conclusion
be a logical consequence of its premises—even though none
of its premises are true. In mathematics and philosophy, validity
is a necessary condition on suitable arguments; it is not sufficient.
What is sufficient, for such pursuits, is a so-called sound
argument.
Definition 3. (Sound Argument) A sound argument is valid and
all its premises are true.
Suppose that, among the ‘cases’ in our definition of validity (or
logical consequence), there is an ‘actual case’ @ such that A is
true-in-@ just if A is true (i.e., actually true). On such an account,
every sound argument has a true conclusion. After all, a sound
argument, by definition, has all true premises. By supposition, a
sentence is true just if true-in-@, and so all premises of a sound
argument are true-in-@. But a sound argument, by definition, is
also valid, and so, by definition, if its premises are true in a case,
then so too is its conclusion. Since, as noted, the premises of any
sound argument are true-in-@, so too is its conclusion.
Logic, in the end, serves the pursuit of truth; however, it does
not principally concern itself with truth. Instead, logic, as above,
has its chief concern with consequence—logical consequence. Logic
aims to precisely specify valid arguments. Once the valid arguments
are in order, rational inquiry may proceed to discern the
sound arguments. For our purposes in this book, we will focus on
different accounts of logical consequence, and some of the phenomena
that motivate the various accounts.
1.4 Summary, looking ahead, and reading
Summary. Logical consequence is the chief concern of logic. An
argument is valid just if its conclusion is a logical consequence
of its premises. Logical consequence, in this book, will be understood
as absence of counterexample, where a counterexample
is a ‘case’ in which all the premises are true but the conclusion
not true. One of the chief concerns of logic, broadly construed,
is to figure out which ‘cases’ are involved in specifying the consequence
relation on a given language (or fragment thereof). In
subsequent chapters, we will look at different accounts of logical
Summary, looking ahead, and reading 9
consequence—different logical theories of our language (or fragments
thereof)—and some of the phenomena that have motivated
them.
Looking Ahead. The next two chapters are devoted to stagesetting.
Chapter 2 discusses features of language that are relevant
to logic, and also discusses the general ‘model-building’ enterprise
of formal logic. Chapter 3 briefly—and, for the most part,
informally—introduces some useful set-theoretic notions. These
two chapters will make subsequent discussion easier.
Further Reading. For related, accessible discussion of logic, see
Read 1995, Haack 1978; Haack 1996. (And see the bibliographies
therein for a host of other sources!) For a more advanced discussion
of the ‘recipe’ of logical consequence, see Beall and Restall
2005.
Exercises
1. What is an argument?
2. What is a valid argument?
3. What is a sound argument?
4. What is the general ‘recipe’ for defining logical consequence
(or validity)? What are the two key ingredients that one
must specify in defining a consequence relation?
5. Consider the ‘necessary consequence’ relation, which takes
cases to be possibilities. Assume, as is reasonable (!), that
our actual world is possible—that is, that whatever is true
(actually true) is possibly true. Question: on this account
of logical consequence, are there any sound arguments that
have false conclusions? If so, why? If not, why not?
6. As noted in the chapter, ‘if and only if’ (which is often abbreviated
as ‘iff’) expresses two conditionals: ‘A iff B’ expresses
both of the following conditionals.3
• If A, then B.
• If B, then A.
3Strictly speaking, what is expressed is the ‘conjunction’ of the two conditionals,
but we leave the notion of conjunctions for the next chapter.
Consequences10
For our purposes, a biconditional ‘A iff B’ is true so long as
A and B are either both true or both false (and such biconditionals
are false otherwise). With this in mind, consider
the necessary consequence relation. Is the following argument
valid (where, here, validity is necessary consequence)?
If it is valid—if its conclusion is a necessary consequence of
the premises—explain why it is valid. If not, explain why
not.
(a) Max is happy if and only if Agnes is sleeping.
(b) Agnes is sleeping.
(c) Therefore, Max is happy.
What about the following argument?
(d) Max is happy if and only if Agnes is sleeping.
(e) Agnes is not sleeping.
(f) Therefore, Max is not happy.
7. Using the ‘necessary consequence’ account of validity, specify
which of the following arguments are valid or invalid.
Justify your answer.
(a) Argument 1.
i. If Agnes arrived at work on time, then her car
worked properly.
ii. If Agnes’s car worked properly, then the car’s ignition
was not broken.
iii. The car’s ignition was not broken.
iv. Therefore, Agnes arrived at work on time.
(b) Argument 2.
i. Either the sun will rise tomorrow or it will explode
tomorrow.
ii. The sun will not explode tomorrow.
iii. Therefore, the sun will rise tomorrow.
(c) Argument 3.
i. If Max wins the lottery, then Max will be a mil-
lionaire.
ii. Max will not win the lottery.
iii. Therefore, Max will not be a millionaire.
Summary, looking ahead, and reading 11
(d) Argument 4.
i. If Beetle is an extraterrestrial, then Beetle is not
from earth.
ii. Beetle is an extraterrestrial.
iii. Therefore, Beetle is not from earth.
Sample answers
Answer 5. On the necessary-consequence sense of ‘validity’ (the
sense in question), an argument is valid iff every possibility (e.g.,
possible circumstance) in which the premises are all true is one
in which the conclusion is true. Hence, if the actual world—the
‘real’ world, the way things really are—counts as a possibility,
then it itself cannot be a case in which the premises of a valid
argument are true but the conclusion false. But, then, any sound
argument—that is, a valid argument whose premises are all (actually)
true—is one in which the conclusion is true, and so not
false.4
Answer 6. The argument from (6a) and (6b) to (6c) is valid in
the necessary-consequence approach to validity: it is not possible
for both of (6a) and (6b) to be true without (6c) being true. After
all, recall that (6a) expresses not only that if Max is happy
then Agnes is sleeping; it also expresses that if Agnes is sleeping
then Max is happy. Now, consider any possibility (and possible
circumstance) in which both (6a) and (6b) are true, that is, a
possible circumstance in which not only Agnes is sleeping, but
if Agnes is sleeping (in that circumstance), then Max is happy
(in that circumstance). Well, then, no matter what possible circumstance
we choose, it’ll be one in which Max is happy if it’s
one in which both (6a) and (6b) are true. (Of course, there are,
presumably, many possibilities in which neither (6a) nor (6b) are
true, but this does not affect the necessary-consequence sense in
which the given argument is valid. Why?)
4This last step—from true to not false—is something that some logical
theories reject, but these theories are left for later chapters.