Bulletin of Mothemnticnl Biology Vol. 52, No. l/2. pp. 99-115. 1990.
Printed in Great Britain.
oo92-824OjW$3.OO+O.MI
Pergamon Press plc
Society for Mathematical Biology
A LOGICAL CALCULUS OF THE IDEAS IMMANENT IN
NERVOUS ACTIVITY*
n WARREN S. MCCULLOCH AND WALTER PITTS
University of Illinois, College of Medicine,
Department of Psychiatry at the Illinois Neuropsychiatric Institute,
University of Chicago, Chicago, U.S.A.
Because of the “all-or-none” character of nervous activity, neural events and the relations among
them can be treated by means of propositional logic. It is found that the behavior of every net can
be described in these terms, with the addition of more complicated logical means for nets
containing circles; and that for any logical expression satisfying certain conditions, one can find a
net behaving in the fashion it describes. It is shown that many particular choices among possible
neurophysiological assumptions are equivalent, in the sense that for every net behaving under
one assumption, there exists another net which behaves under the other and gives the same
results, although perhaps not in the same time. Various applications of the calculus are
discussed.
1. Introduction. Theoretical neurophysiology rests on certain cardinal
assumptions. The nervous system is a net of neurons, each having a soma and
an axon. Their adjunctions, or synapses, are always between the axon of one
neuron and the soma of another. At any instant a neuron has some threshold,
which excitation must exceed to initiate an impulse. This, except for the fact
and the time of its occurence, is determined by the neuron, not by the
excitation. From the point of excitation the impulse is propagated to all parts of
the neuron. The velocity along the axon varies directly with its diameter, from
< 1 ms-’ in thin axons, which are usually short, to > 150 ms- ’ in thick axons,
which are usually long. The time for axonal conduction is consequently of little
importance in determining the time of arrival of impulses at points unequally
remote from the same source. Excitation across synapses occurs predominantly
from axonal terminations to somata. It is still a moot point whether this
depends upon irreciprocity of individual synapses or merely upon prevalent
anatomical configurations. To suppose the latter requires no hypothesis ad hoc
and explains known exceptions, but any assumption as to cause is compatible
with the calculus to come. No case is known in which excitation through a
single synapse has elicited a nervous impulse in any neuron, whereas any
neuron may be excited by impulses arriving at a sufficient number of
neighboring synapses within the period of latent addition, which lasts
~0.25 ms. Observed temporal summation of impulses at greater intervals
* Reprinted from the Bulletin ofMathematicalBiophysics, Vol. 5, pp. 115-133 (1943).
99
loo W. S. McCULLOCH AND W. PITTS
is impossible for single neurons and empirically depends upon structural
properties of the net. Between the arrival of impulses upon a neuron and its
own propagated impulse there is a synaptic delay of > 0.5 ms. During the first
part of the nervous impulse the neuron is absolutely refractory to any
stimulation. Thereafter its excitability returns rapidly, in some cases reaching a
value above normal from which it sinks again to a subnormal value, whence it
returns slowly to normal. Frequent activity augments this subnormality. Such
specificity as is possessed by nervous impulses depends solely upon their time
and place and not on any other specificity of nervous energies. Of late only
inhibition has been seriously adduced to contravene this thesis. Inhibition is
the termination or prevention of the activity of one group of neurons by
concurrent or antecedent activity of a second group. Until recently this could
be explained on the supposition that previous activity of neurons of the second
group might so raise the thresholds of internuncial neurons that they could no
longer be excited by neurons of the first group, whereas the impulses of the first
group must sum with the impulses of these internuncials to excite the now
inhibited neurons. Today, some inhibitions have been shown to consume
< 1 ms. This excludes internuncials and requires synapses through which
impulses inhibit that neuron which is being stimulated by impulses through
other synapses. As yet experiment has not shown whether the refractoriness is
relative or absolute. We will assume the latter and demonstrate that the
difference is immaterial to our argument. Either variety of refractoriness can be
accounted for in either of two ways. The “inhibitory synapse” may be of such a
kind as to produce a substance which raises the threshold of the neuron, or it
may be so placed that the local disturbance produced by its excitation opposes
the alteration induced by the otherwise excitatory synapses. Inasmuch as
position is already known to have such effects in the cases of electrical
stimulation, the first hypothesis is to be excluded unless and until it be
subtantiated, for the second involves no new hypothesis. We have, then, two
explanations of inhibition based on the same general premises, differing only in
the assumed nervous nets and, consequently, in the time required for
inhibition. Hereafter we shall refer to such nervous nets as equivalent in the
extended sense. Since we are concerned with properties of nets which are
invarient under equivalence, we may make the physical assumptions which are
most convenient for the calculus.
Many years ago one of us, by considerations impertinent to this argument,
was led to conceive of the response of any neuron as factually equivalnt to a
proposition which proposed its adequate stimulus. He therefore attempted to
record the behavior of complicated nets in the notation of the symbolic logic of
propositions. The “all-or-none” law of nervous activity is sufficient to insure
that the activity of any neuron may be represented as a proposition.
Physiological relations existing among nervous activities correspond, of
LOGICAL CALCULUS FOR NERVOUS ACTIVITY 101
course, to relations among the propositions; and the utility of the representation
depends upon the identity of these relations with those of the logic of
propositions. To each reaction of any neuron there is a corresponding assertion
of a simple proposition. This, in turn, implies either some other simple
proposition or the disjunction of the conjunction, with or without negation, of
similar propositions, according to the configuration of the synapses upon and
the threshold of the neuron in question. Two difficulties appeared. The first
concerns facilitation and extinction, in which antecedent activity temporarily
alters responsiveness to subsequent stimulation of one and the same part of the
net. The second concerns learning, in which activities concurrent at some
previous time have altered the net pe~anently, so that a stimulus which would
previously have been inadequate is now adequate. But for nets undergoing
both alterations, we can substitute equivalent fictitious nets composed of
neurons whose connections and thresholds are unaltered. But one point must
be made clear: neither of us conceives the formal equivalence to be a factual
explanation. Per contra!-we regard facilitation and extinction as dependent
upon continuous changes in threshold related to electrical and chemical
variables, such as after-potentials and ionic concentrations; and learning as an
enduring change which can survive sleep, anaesthesia, convulsions and coma.
The impo~ance of the formal equivalence lies in this: that the alterations
actually underlying facilitation, extinction and learning in no way affect the
conclusions which follow from the formal treatment of the activity of nervous
nets, and the relations of the corresponding propositions remain those of the
logic of propositions.
The nervous system contains many circular paths, whose activity so
regenerates the excitation of any participant neuron that reference to time past
becomes indefinite, although it still implies that afferent activity has realized
one of a certain class of configurations over time. Precise specification of these
implications by means of recursive functions, and determination of those that
can be embodied in the activity of nervous nets, completes the theory.
2. The Theory: Nets Without Circles. We shall make the following physical
assumptions for our calculus.
(1) The activity of the neuron is an “all-or-none” process.
(2) A certain fixed number of synapses must be excited within the period of
latent addition in order to excite a neuron at any time, and this number is
independent of previous activity and position on the neuron.
(3) The only significant delay within the nervous sytem is synaptic delay.
(4) The activity of any inhibitory synapse absolutely prevents excitation of
the neuron at that time.
(5) The structure of the net does not change with time.
102 W. S. McCULLOCH AND W. PI-ITS
To present the theory, the most appropriate symbolism is that of Language
II of Carnap (1938), augmented with various notations drawn from Russell and
Whitehead (1927), including the Principia conventions for dots. Typographical
necessity, however, will compel us to use the upright ‘E’ for the existential
operator instead of the inverted, and an arrow (“-+“) for implication instead of
the horseshoe. We shall also use the Carnap syntactical notations, but print
them in boldface rather than German type; and we shall introduce a functor S,
whose value for a property P is the property which holds of a number when P
holds of its predecessor; it is defined by “S(P) (t) .s . P(Kx) . t =x’)“; the
brackets around its argument will often be omitted, in which case this is
understood to be the nearest predicate-expression [Pr] on the right. Moreover,
we shall write S2Pr for S(S(Pr)), etc.
The neurons of a given net JV may be assigned designations “ci”, “c2”, . . . ,
“c,“. This done, we shall denote the property of a number, that a neuron ci fires
at a time which is that number of synaptic delays from the origin of time, by
“N” with the numeral i as subscript, so that N,(t) asserts that ci fires at the time
t. Ni is called the action of ci. We shall sometimes regard the subscripted
numeral of “N” as ifit belonged to the object-language, and were in a place for a
functoral argument, so that it might be replaced by a number-variable [z] and
quantified; this enables us to abbreviate long but finite disjunctions and
conjunctions by the use of an operator. We shall employ this locution quite
generally for sequences of Pr; it may be secured formally by an obvious
disjunctive definition. The predicates “N,“, “N2”, . . . , comprise the syntactical
class “N”.
Let us define the peripheral uferents ofJf as the neurons of _,Vwith no axons
synapsing upon them. Let N, , . . . , N, denote the actions of such neurons and
N NP+z,...,p+1, N,, those of the rest. Then a solution of-W will be a class of
sentences of the form Si: N,, ,(z,) .=. Pri(N,, N,, . . . , N,, z,), where Pri
contains no free variable save z1 and no descriptive symbols save the N in the
argument [Arg], and possibly some constant sentences [sa]; and such that
each Si is true of JV. Conversely, given a Pr,(‘p:, ‘pi, . . . , ‘pj, zl, s),
containing no free variable save those in its Arg, we shall say that it is realizable
in the narrow sense if there exists a net ,Y and a series of Ni in it such that
N,(z,).=.PR,(N,,N,, . . . , zl, ml) is true of it, where sa, has the form N(0).
We shall call it realizable in the extended sense, or simply realizable, if for some
n S”(Pr,)(P,, . . . , pp, zl, s) is realizable in the above sense. cpi is here the
realizing neuron. We shall say of two laws of nervous excitation which are such
that every S which is realizable in either sense upon one supposition is also
realizable, perhaps by a different net, upon the other, that they are equivalent
assumptions, in that sense.
The following theorems about realizability all refer to the extended sense. In
some cases, sharper theorems about narrow realizability can be obtained; but
LOGICAL CALCULUS FOR NERVOUS ACTIVITY 103
in addition to greater complication in statement this were of little practical
value, since our present neurophysiological knowledge determines the law of
excitation only to extended equivalence, and the more precise theorems differ
according to which possible assumption we make. Our less precise theorems,
however, are invariant under equivalence, and are still sufficient for all
purposes in which the exact time for impulses to pass through the whole net is
not crucial.
Our central problems may now be stated exactly: first, to find an effective
method of obtaining a set of computable Sconstituting a solution of any given
net; and second, to characterize the class of realizable Sin an effective fashion.
Materially stated, the problems are to calculate the behavior of any net, and to
find a net which will behave in a specified way, when such a net exists.
A net will be called cyclic if it contains a circle, i.e. if there exists a chain ci,
ci+19. . . of neurons on it, each member of the chain synapsing upon the next,
with the same beginning and end. If a set of its neurons ci , c2, . . . , cp is such
that its removal from JV leaves it without circles, and no smaller class of
neurons has this property, the set is called a cyclic set, and its cardinality is the
order of JV”.In an important sense, as we shall see, the order of a net is an index
of the complexity of its behaviour. In particular, nets of zero order have
especially simple properties; we shall discuss them first.
Let us define a temporal propositional expression (a TPE), designating a
temporal propositional function (TPF), by the following recursion.
(1)
(2)
(3)
A ‘p’[zJ is a TPE, where p1 is a predicate-variable.
If S, and S, are TPE containing the same free individual variable, so are
SS,, S,vS,, S, .S, and Si. - .S,.
Nothing else is a TPE.
THEOREM 1. Every net of order 0 can be solved in terms of temporal propositional
expressions.
Let ci be any neuron of J1’ with a threshold /Ii> 0, and let cil , ciz, . . . , cig
have respectively ni, , n,, , . . . , nipexcitatory synapses upon it. Let cjl, cj2, . . . ,
cjq have inhibitory synapses upon it. Let rcibe the set of the subclasses of {nil,
ni2,. . . , ni,} such that the sum of their members exceeds ei. We shall then be
able to write, in accordance with the assumptions mentioned above:
Ni(Zl)*s.S ?I
f - Njm(Zl) * C n Ni,(Zl)
I9 (1)
m=l ClEKi sea
where the “r and “II” are syntactical symbols for disjunctions and
conjunctions which are finite in each case. Since an expression of this form can
be written for each ci which is not a peripheral afferent, we can, by substituting
104 W.S.McCULLOCHANDW.PITTS
the corresponding expression in (1) for each Njn,or Ni, whose neuron is not a
peripheral afferent, and repeating the process on the result, ultimately come to
an expression for Ni in terms solely of peripherally afferent N, since _,V is
without circles. Moreover, this expression will be a TPE, since obviously (1) is;
and it follows immediately from the definition that the result of substituting a
TPE for a constituent p(z) in a TPE is also one.
THEOREM 2.Every TPE is realizable by a net of order zero.
The functor S obviously commutes with disjunction, conjunction, and
negation. It is obvious that the result of substituting any Si, realizable in the
narrow sense (i.n.s.), for thep(z) in a realizable expression S, is itself realizable
i.n.s.; one constructs the realizing net by replacing the peripheral afferents in
the net for S, by the realizing neurons in the nets for the Si. The one neuron net
realizes pi(zi) i.n.s., and Fig. la shows a net that realizes S’,(z,) and hence
SS,, i.n.s., if S, can be realized i.n.s. Now if S, and S, are realizable then SmSz
and S”S, are realizable i.n.s., for suitable m and IZ.Hence so are Smf2S2 and
Sm+“S3. Now the nets of Figs lb-d respectively realize S(p,(z,) v p2(z,)),
S(p,(z,) .p2(z,)), and zs(pl(zl . -p,(z,))i.n.s. Hence Sm+n+l (S, v S,), Smfn+l
(S, .S,), and Sm+n+l (S, .-S,) are realizable i.n.s. Therefore S, v
s,s, .s,s, .- S, are realizable if S, and S, are. By complete induction, all
TPE are realizable. In this way all nets may be regarded as built out of the
fundamental elements of Figs la-d, precisely as the temporal propositional
expressions are generated out of the operations of precession, disjunction,
conjunction, and conjoined negation. In particular, corresponding to any
description of state, or distribution of the values true and false for the actions of
all the neurons of a net save that which makes them all false, a single neuron is
constructible whose firing is a necessary and sufficient condition for the validity
of that description. Moreover, there is always an indefinite number of
topologically different nets realizing any TPE.
THEOREM 3.Let there be a complex sentence S, built up in any manner out of
elementary sentences of theform p(z, - zz) where zz is any numeral, by any of the
propositional connections: negation, disjunction, conjunction, implication, and
equivalence. Then S, isa TPE and only ifit isfalse when its constituent p(zl - zz)
are all assumed false-i.e. replaced by false sentences-or that the last line in its
truth-table contains an ‘F-or there is no term in its Hilbert disjunctive normal
form composed exclusively of negated terms.
These latter three conditions are of course equivalent (Hilbert and
Ackermann, 1938). We see by induction that the first of them is necessary, since
p(zl - zz) becomes false when it is replaced by a false sentence, and S, v S, ,
S’.S, aridS,.- S, are all false if both their constituents are. We see that the
last condition is sufficient by remarking that a disjunction is a TPE when its
constituents are, and that any term:
LOGICAL CALCULUS FOR NERVOUS ACTIVITY 105
(e
(i 1
Figure 1. The neuron ci is always marked with the numeral i upon the body of the
cell, and the corresponding action is denoted by “N” with is subscript, as in the text:
(a) N*(t) .=.N,(t- 1);
(b) N,(t).s.N,(t-l)vN,(t-1);
(c) N3(t).s.N1(t-1).N2(t-1);
(d) N3(t).= N,(t-l).-N,(t-1);
(e) N,(t):=:N,(t-l).v.N,(t-3).-N,(t-2);
N&).=.N2(t-2).N2(t-1);
(f) N4(t):3: --N,(t-l).N,(t-l)vN,(t-l).v.N,(t-1).
N,(t-l).N,(t-1)
NJt):=: -N,(t-2).N,(t-2)vN,(t-2).v.N,(t-2).
N,(t-2).N,(t-2);
(g) N,(t).=.NN,(t-2).-N,(t-3);
(h) N,(t).=.N,(t-l).N,(t-2);
(i) N,(t):=:Nz(t-l).v.N,(t-l).(Ex)t-1 .N,(x).N,(x).
106 W. S. McCULLOCH AND W. PITTS
s, .s, ...Sm.4m+l .-. . .- s,,
can be written as:
(S, .s, ...Sm).N(Sm+lVS,+ZV...VS,),
which is clearly a TPE.
The method of the last theorems does in fact provide a very convenient and
workable procedure for constructing nervous nets to order, for those cases
where there is no reference to events indefinitely far in the past in the
specification of the conditions. By way of example, we may consider the case of
heat produced by a transient cooling.
If a cold object is held to the skin for a moment and removed, a sensation of
heat will be felt; if it is applied for a longer time, the sensation will be only of
cold, with no preliminary warmth, however transient. It is known that one
cutaneous receptor is affected by heat, and another by cold. If we let N, and N,
be the actions of the respective receptors and N3 and N4 of neurons whose
activity implies a sensation of heat and cold, our requirements may be written
as:
N,(t):=:N,(t-l).v.N,(t-3).-N&-2),
N&) .E. N&-2). N,(t- l),
where we suppose for simplicity that the required persistence in the sensation of
cold is say two synaptic delays, compared with one for that of heat. These
conditions clearly fall under Theorem 3. A net may consequently be
constructed to realize them, by the method of Theorem 2. We begin by writing
them in a fashion which exhibits them as built out of their constituents by the
operations realized in Figs la-d, i.e. in the form:
N&) -= .SW,WVXW,(t)). - N,(t)l)
N&) .=. S{[SN#)] .N,(t)}.
First we construct a net for the function enclosed in the greatest number of
brackets and proceed outward; in this case we run a net of the form shown in
Fig. la from c2 to some neuron c,, say, so that:
N,(t). = .SN,(t).
Next introduce two nets of the forms lc and Id, both running from c, and c2,
and ending respectively at c1 and say cb. Then:
N&J. =. WV,(t). N,(t)] .=. S[SN&)). N&)1.,
LOGICALCALCULUSFORNERVOUSACTIVITY 107
iv&t) .3 .S[N,(t).-N,(t)] .=.S[(Shqt)) .-N,(t)].
Finally, run a net of the form lb from c1 and cb to c3, and derive:
N3(t). = . S[N,(t) v N*(t)]
.5qv,(t) v S[SN,(t)] .-N,(t)).
These expressions for N3(t) and N4(t) are the ones desired; and the realizing net
in toto is shown in Fig. le.
This illusion makes very clear the dependence of the correspondence
between perception and the “external world” upon the specific structural
properties of the intervening nervous net. The same illusion, of course, could
also have been produced under various other assumptions about the behavior
of the cutaneous receptors, with corresponding different nets.
We shall now consider some theorems of equivalence, i.e. theorems which
demonstrate the essential identity, save for time, of various alternative laws of
nervous excitation. Let us first discuss the case of relative inhibition. By this we
mean the supposition that the firing of an inhibitory synapse does not
absolutely prevent the firing of the neuron, but merely raises its threshold, so
that a greater number of excitatory synapses must fire concurrently to fire it
than would otherwise be needed. We may suppose, losing no generality, that
the increase in threshold is unity for the firing of each such synapse; we then
have Theorem 4.
THEOREM 4.Relative and absolute inhibition are equivalent in the extended sense.
We may write out a law of nervous excitation after the fashion of (I), but
employing the assumption of relative inhibition instead; inspection then shows
that this expression is a TPE. An example of the replacement of relative
inhibition by absolute is given by Fig. If. The reverse replacement is even
easier; we give the inhibitory axons afferent to ci any sufficiently large number
of inhibitory synapses apiece.
Second, we consider the case of extinction. We may write this in the form of a
variation in the threshold Oi; after the neuron ci has fired; to the nearest
integer-and only to this approximation is the variation in threshold
significant in natural forms of excitation-this may be written as a sequence
Bi+ b, forj synaptic delays after firing, where bj= 0 forj large enough, say j = M
or greater. We may then state Theorem 5.
THEOREM 5.Extinction is equivalent to absolute inhibition.
For, assuming relative inhibition to hold for the moment, we need merely
run M circuits Y1, Yz, . . . FM containing respectively 1,2, . . . , A4 neurons,
such that the firing of each link in any is sufficient to fire the next, from the
neuron ci back to it, where the end of the circuit Yj has just bj inhibitory
synapses upon ci. It is evident that this will produce the desired results. The
108 W.S.McCULLOCHANDW.PITTS
reverse substitution may be accomplished by the diagram of Fig. lg. From the
transitivity of replacement, we infer the theorem. To this group of theorems
also belongs the following well-known theorem.
THEOREM 6. Facilitation and temporal summation may be replaced by spatial
summation.
This is obvious: one need merely introduce a suitable sequence of delaying
chains, of increasing numbers of synapses, between the exciting cell and the
neuron whereon temporal summation is desired to hold. The assumption of
spatial summation will then give the required results (see e.g. Fig. lh). This
procedure had application in showing that the observed temporal summation
in gross nets does not imply such a mechanism in the interaction of individual
neurons.
The phenomena of learning, which are of a character persisting over most
physiological changes in nervous activity, seem to require the possibility of
permanent alterations in the structure of nets. The simplest such alteration is
the formation of new synapses or equivalent local depressions of threshold. We
suppose that some axonal terminations cannot at first excite the succeeding
neuron; dut if at any time the neuron fires, and the axonal terminations are
simultaneously excited, they become synapses of the ordinary kind, henceforth
capable of exciting the neuron. The loss of an inhibitory synapse gives an
entirely equivalent result. We shall then have
THEOREM 7.Alterable synapses can be replaced by circles.
This is accomplished by the method of Fig. li. It is also to be remarked that a
neuron which becomes and remains spontaneously active can likewise be
replaced by a circle, which is set into activity by a peripheral afferent when the
activity commences, and inhibited by one when it ceases.
3. The Theory: Nets with Circles. The treatment of nets which do not satisfy
our previous assumption of freedom from circles is very much more difficult
than that case. This is largely a consequence of the possibility that activity may
be set up in a circuit and continue reverberating around it for an indefinite
period of time, so that the realizable Pr may involve reference to past events of
an indefinite degree of remoteness. Consider such a net Jlr, say of order p, and
letc,,c,, . . . , cp be a cyclic set of neurons of Jlr. It is first of all clear from the
definition that every N3 of JV can be expressed as a TPE, of N1, N2, . . . , N,
and the absolute afferents; the solution of X involves then only the
determination of expressions for the cyclic set. This done, we shall derive a set
of expressions [A]:
N&z,). =. Pri[SnilN1(zl), SnizNz(zl), . . . , LWNp(q)],
where Pri also involves peripheral afferents. Now if n is the least common
LOGICAL CALCULUS FOR NERVOUS ACTIVITY 109
multiple of the ylij,we shall, by substituting their equivalents according to (2) in
(3) for the Nj, and repeating this process often enough on the result, obtain Sof
the form:
Ni(Zl) *= *Prl[S”Nl(Zl)y SnNz(Zl)y * + * 9 S"Np(Zl)]* (3)
These expressions may be written in the Hilbert disjunctive normal form as:
Ni(zl)*z. C S, fl S”Nj(Zl) JJ N S”Nj(z1), for suitable K,
BEK jw .k&
B&K
(4)
where S, is a TPE of the absolute afferents of N. There exist some 2p different
sentences formed out of the @Viby conjoining to the conjunction of some set of
them the conjunction of the negations of the rest. Denumerating these by
Xr(z,), X&,)9 **f 7X&z,), we may, by use of the expressions (4), arrive at an
equipollent set of equations of the form:
Xi(z,) .-. $J I+&,). S”Xj(zl ).
j=t
(5)
Now we import the subscripted numerals i,j into the object-language, i.e.
define Pr, and Pr, such that PrI(zzl, z,) .= .&(z,) and Pr2(zz1, zz2,
Z,) .E. PrijfZl) are provable whenever zzr and zzZdenote i and j respectively.
Then we may rewrite (5) as:
(zl)zzp:Prltzl~z,)
.- .tEz,)zz,-Pr,(zl , z2, z3 - zz,).Pr, (z, , z3 - zz,), (6)
where zz, denotes y1and zz, denotes 2p. By repeated substitution we arrive at an
expression:
(zl)zzp:Pr,(zl, zz,zz2) .=. (Ez2)zzp(Ez,)~~, . . . (Ez,)zz,.
Pr2(z1 , z2, zz,(zz, - 1)) *Pr,(z,, 23) ZZ”(ZZ3 - 1)) *. . (7)
Pr,(z, _ 1, z,, O), for any numeral zz2 which denotes s. This is easily shown by
induction to be equipollent to:
(z, )zz,: *Pr,(z1, zz,zz,): = : (Ef) (z2)zzz- lJ(Z,ZZ”)
S zz, .f(zz,zz,) = z1 . Pr2(~(zz~(z~+ l)),
f(zz,zd).Pr, (f(o), Oh (8)
110 W. S.McCULLOCH AND W. PI-I-E
and since this is the case for all zz2, it is also true that:
w tz,&J: WZl, z,) -= *(Ef) (z,) (z.$--1).f(z,)
~zz,.f(24)=Zlf(Z4)=Zl JWY(z~+ l),f(Zz), z,].
4 Cf(res(z,, zz,)), res(z,, zz,)], (9)
where tz, denotes n, resfr, s) is the residue of r mod s and zz, denotes 2p. This
may be written in a less exact way as:
N,(t). 3. (Ed) (x)t- 1 . am 2’ I ~(t)=i.
me+ 117444 *F#&vl~
where x and t are also assumed divisible by n, and Pr, denotes P. From the
preceding remarks we shall have Theorem 8
THEOREM 8. The expression (9)for neurons ofthe cyclic set ofa net J1’ together
with certain TPE expressing the actions ofother neurons in terms of them,
constitute a solution of Jlr.
Consider now the question of the realizability of a set of Si. A first necessary
condition, demonstrable by an easy induction, is that:
should be true, with similar statements for the other freep in Si, i.e. no nervous
net can take account of future peripheral afferents. Any Si satisfying this
requirement can be replaced by an equipollent S of the form:
(Ef) (z~)z~(z~)zz~:~~r~~
:f(z,, z2, z3)= 1=.pza(z2), (11)
where zz, denotes p, by defining:
Pr,i=.&) (z,)z1(z3)zz~: .ftz,, z2, z~)=O.~YZ~~ z2,z3)
=I:.&, zz, z3)=1 .~.p,,(z$-KS,].
Consider now these series of classes aj, for which:
N,(t):=: (II@) (X)t(m)q:#&ai:N,(X) .G. #(t, X, m)= 1.
fi=q+ 1, , . . , M”j (121
holds for some net. These will be called prehensible classes. Let us define the
Boolean ring generated by a class of classes K as the aggregate of the classes
LOGICAL CALCULUS FOR NERVOUS ACTIVITY 111
which can be formed from members of ICby repeated app~cation of the logical
operations, i.e. we put:
9?(K)= &(a, p): a&K
+a.ka, /M .-+. -a, a .j?, av/M].
We shall also define:
and:
B(K). = .92(K) - z‘p‘- <‘K,
%‘,(rcb,= p‘X[(a, fi): a.m-+a& .--) . -a, a. p, m-p, S‘ael]
L!&,(K)= Be(K) - t&f- “K,
The class W,(K) is formed from k: in analogy with L%(X),but by repeated
application not only of the logical operations but also of that which replaces a
class of properties P&aby S(P)~S”a. We shall then have the following lemma.
LEMMA. Pr,(p,, p2, . . . , p,, zl) is a TPE if and only $
(zl)(pl,. . .T ~m)(E~m+1):~rn+1~~)e((~Ir ~23.. .y pm>)
(13)
P~+~(z~)~~~(P~, P21+f *7Pm, z,),
is true; and it is a TPE not involving “S” if and only if this holds when “3?‘, and we
then obtain Theorem 9.
THEOREM 9. A series ofclasses a,, a2, . . . aYisa series of prehensible classes ifand
only if:
(Em) (En) (p)n(i~ (t,&):. (x)m~(~)=Ov~~x)= l:+: (Efl)
(Ey)m.$(y)=O.&S?[~((Ei).y=ai)).v. (x)m.
1(/(x)=0 .fisB[f((Ei) . y = ai)]: (t) (4): @ai.
~(4, nt + p) .--+.(Ef) .fs/?h (w)m(x)t- 1.
Ql(n(t+l)+p, nx+p, w)=f(nt+p, nx+p, w).
(14)
Proof: The proof here follows directly from the lemma. The condition is
necessary, since every net for which an expression of the form (4) can be written
obviously verifies it, the I/S being the characteristic functions of the S, and the b
for each $ being the class whose designation has the form niea Pri nj&flaPrj,
where Pv, denotes akfor all k. Conversely, we may write an expression of the
form (4) for a net N fulfilling prehensible classes satisfying (14) by putting for
112 W. S. McCULLOCH AND W. PI-I-I-S
the Pr, Pr denoting the Il/‘s,and a Pr, written in the analogue for classes of the
disjunctive normal form, and denoting the c( corresponding to that $,
conjoined to it. Since every S of the form (4) is clearly realizable, we have the
theorem.
It is of some interest to consider the extent to which we can by knowledge of
the present determine the whole past of various special nets, i.e. when we may
construct a net the firing of the cyclic of whose neurons requires the peripheral
afferents to have had a set of past values specified by given functions +i. In this
case the classes ai of the last theorem reduced to unit classes; and the condition
may be transformed into:
(Em, n) (p)n(i, $) (I$): . (x)m: $(x) = O.v.tj(x) = 1:
~i&a(ll/,nt+p):~: (W)m(X)t-l .~i(n(t+ 1)
+p, nX+p, W)=cbj(nt+P, nx+p, w)”
(u9 v, (w)ma$i(n(u+ ‘)+p, n”+PY w,
=~i(n(v+l)+p, n”+p, w)’
On account of limitations of space, we have presented the above argument
very sketchily; we propose to expand it and certain of its implications in a
further publication.
The condition of the last theorem is fairly simply in principle, though not in
detail; its application to practical cases would, however, require the
exploration of some 22” classes of functions, namely the members of
=Q%&,.**, a,>). Since each of these is a possible p of Theorem 9, this result
cannot be sharpened. But we may obtain a sufficient condition for the
realizability of an S which is very easily applicable and probably covers most
practical purposes. This is given by Theorem 10.
THEOREM 10. Let us dejne a set K off3 by thefollowing recursion: (1) any TPE
and any TPE whose arguments have been replaced by members of K belong to K;
(2) if Pr,(z,) is a member of K, then (zq)zl .Pr,(z,), (Ez,)z,. Pr,(z,), and
C,,(z,) . s belong to it, where C,, denotes the property of being congruent to m
modulo n, m < n; (3) The set K has nofurther members.
Then every member of K is realizable. For, if Pr, (z,) is realizable, nervous
nets for which:
Ni(Zl).~.Pr,(Z,).SN,(Z,),
Ni(Z,) - =. Pr,(z,) V SNi(Z,)y
are the expressions of equation (4), realize (z,)z, .Prl(z2) and (Ez,)z, . Pr,(z,)
LOGICAL CALCULUS FOR NERVOUS ACTIVITY 113
respectively; and a simple circuit, cr , c2, . . . , c,, of y1links, each sufficient to
excite the next, gives an expression:
for the last form. By induction we derive the theorem.
One more thing is to be remarked in conclusion. It is easily shown: first, that
every net, if furnished with a tape, scanners connected to afferents, and suitable
efferents to perform the necessary motor-operations, can compute only such
numbers as can a Turing machine; second, that each of the latter numbers can
be computed by such a net; and that nets with circles can be computed by such a
net; and that nets with circles can compute, without scanners and a tape, some
of the numbers the machine can, but no others, and not all of them. This is of
interest as affording a psychological justification of the Turing definition of
computability and its equivalents, Church’s A-definability and Kleene’s
primitive recursiveness: if any number can be computed by an organism, it is
computable by these definitions, and conversely.
4. Consequences. Causality, which requires description of states and a law of
necessary connection relating them, has appeared in several forms in several
sciences, but never, except in statistics, has it been as irreciprocal as in this
theory. Specification for any one time of afferent stimulation and of the
activity of all constituent neurons, each an “all-or-none” affair, determines
the state. Specification of the nervous net provides the law of necessary
connection whereby one can compute from the description of any state that of
the succeeding state, but the inclusion of disjunctive relations prevents
complete determination of the one before. Moreover, the regenerative activity
of constituent circles renders reference indefinite as to time past. Thus our
knowledge of the world, including ourselves, is incomplete as to space and
indefinite as to time. This ignorance, implicit in all our brains, is the
counterpart of the abstraction which renders our knowledge useful. The role of
brains in determining the epistemic relations of our theories to our
observations and of these to the facts is all too clear, for it is apparent that every
idea and every sensation is realized by activity within that net, and by no such
activity are the actual afferents fully determined.
There is no theory we may hold and no observation we can make that will
retain so much as its old defective reference to the facts if the net be altered.
Tinitus, paraesthesias, hallucinations, delusions, confusions and disorientation
intervene. Thus empiry confirms that if our nets are undefined, our facts are
undefined, and to the “real” we can attribute not so much as one quality or
“form.” With determination of the net, the unkowable object of knowledge, the
“thing in itself,” ceases to be unknowable.
To psychology, however defined, specification of the net would contribute all
114 W. S.McCULLOCH AND W. PIT-K.3
that could be achieved in that field-even if the analysis were pushed to
ultimate psychic units or “psychons,” for a psychon can be no less than the
activity of a single neuron. Since that activity is inherently propositional, all
psychic events have an intentional, or “semiotic,” character. The “all-or-none”
law of these activities, and the conformity of their relations to those of the logic
of propositions, insure that the relations of psychons are those of the twovalued
logic of propositions. Thus in psychology, introspective, behavioristic
or physiological, the fundamental relations are those of two-valued logic.
Hence arise constructional solutions of holistic problems involving the
differentiated continuum of sense awareness and the normative, perfective and
resolvent properties of perception and execution. From the irreciprocity of
causality it follows that even if the net be known, though we may predict future
from present activities, we can deduce neither afferent from central, nor central
from efferent, nor past from present activities--conclusions which are
reinforced by the contradictory testimony of eye-witnesses, by the difficulty of
diagnosing differentially the organically diseased, the hysteric and the
malingerer, and by comparing one’s own memories or recollections with his
contemporaneous records. Moreover, systems which so respond to the
difl’erence between afferents to a regenerative net and certain activity within
that net, as to reduce the difference, exhibit purposive behavior; and
organisms are known to possess many such systems, subserving homeostasis,
appetition and attention. Thus both the formal and the final aspects of
that activity which we are wont to call mental are rigorously deduceable from
present neurophysiology. The psychiatrist may take comfort from the
obvious conclusion concerning causality-that, for prognosis, history is
never necessary. He can take little from the equally valid conclusion that his
observables are explicable only in terms of nervous activities which, until
recently, have been beyond his ken. The crux of this ignorance is that
inference from any sample of overt behavior to nervous nets is not unique,
whereas, of imaginable nets, only one in fact exists, and may, at any moment,
exhibit some unpredictable activity. Certainly for the psychiatrist it is more
to the point that in such systems “Mind” no longer “goes more ghostly than a
ghost.” Instead, diseased mentality can be understood without loss of scope
or rigor, in the scientific terms of neurophysiology. For neurology, the
theory sharpens the distinction between nets necessary or merely sufficient
for given activities, and so clarifies the relations of disturbed structure to
disturbed function. In its own domain the difference between equivalent nets
and nets equivalent in the narrow sense indicates the appropriate use and
importance of temporal studies of nervous activity: and to mathematical
biophysics the theory contributes a tool for rigorous symbolic treatment of
known nets and an easy method of constructing hypothetical nets of required
properties.
LOGICAL CALCULUS FOR NERVOUS ACTIVITY 115
LITERATURE
Carnap, R. 1938. The Logical Syntax ofLanguage. New York: Harcourt-Brace.
Hilbert, D. and W. Ackermann. 1927. Grundtige der Theoretischen Logik. Berlin: Springer.
Russell, B. and A. N. Whitehead. 1925. Principa Mathematics. Cambridge University Press.