A Logical Calculus of the Ideas Immanent in jYervous Activity 383 



To present the theory, the most appropriate symbolism is that 

 of Language II of R. Carnap (1938), augmented with various 

 notations drawn from B. Russell and A. N. Whitehead (1927) 

 including the Pnncipia conventions for dots. Typographical neces- 

 sity, however, will compel us to use the upright 'E' for the existen- 

 tial 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) (/) . = . P(A'.v) . / = .v')'; 

 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 S-Pr for S{S{Pr)), etc. 



The neurons of a given net '^ may be assigned designations 

 '^I'j '^2', . . . , 'c„'. This done, we shall denote the property of a 

 number, that a neuron c, fires at a time which is that number of 

 synaptic delays from the origin of time, by ^A^' with the numeral 

 i as subscript, so that N ,{t) asserts that c, fires at the time t. N, is 

 called the action of c,. We shall sometimes regard the subscripted 

 numeral of ' N' as if it 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 abbre- 

 viate 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 dis- 

 junctive definition. The predicates '.Vi', '.V^', . . . , comprise the 

 syntactical class ' N\ 



Let us define the peripheral afferents of V)! as the neurons of ^^I 

 with no axons synapsing upon them. Let N,, . . . ^ N^ denote the 

 actions of such neurons and A^,,+i, N,^,, . . . , N„ those of the rest. 

 Then a solution of VX will be a class of sentences of the form S- 

 A^p+i (21) . ^ . Pr, {N,, N,, ... , N„ 2i), where Pr, contains no 

 free variable save Zi and no descriptive symbols save the A'' in the 

 argument [Arg], and possibly some constant sentences [sa]; and 

 such that each S, is true of VX. Conversely, given a Pvi {^i, ^2 

 ^p\, Zi, 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 9l 



