A^, 



A Logical Calculus of the Ideas Immanent in Nervous Activity 385 



1. A^p^ [zi] is a TPE, where Pi is a predicate-variable. 



2. If Si and So are TPE containing" the same free individual 

 variable, so are SS\, SivSo, Si.S-> and S,. ^-^ S2. 



3. Nothing else is a TPE. 



Theorem I 



Every net of order can be solved in terms of temporal propositional 

 expressions. 



Let Ci be any neuron of V^l with a threshold 6, > 0, and let Cn, 

 Ci2, . •• , (',p have respectively //,i, '?,2, •• • , n,p excitatory synapses 

 upon it. Let Cj], r,2, • • • , ^'jy have inhibitory synapses upon it. 

 Let Ki be the set of the subclasses of \n,i, n,2, •• • , fi,p\ such that 

 the sum of their members exceeds 6,. We shall then be able to 

 write, in accordance with the assumptions mentioned above, 



where the 'E' ^'^^ 'H' are syntactical symbols for disjunctions 

 and conjunctions which are finite in each case. Since an expression 

 of this form can be written for each C; which is not a peripheral 

 afferent, we can, by substituting the corresponding expression in 

 (1) for each A''^,,, or A'',,- whose neuron is not a peripheral afferent, 

 and repeating the process on the result, ultimately come to an 

 expression for A^, in terms solely of peripherally afferent A^, since 

 ^^l 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 II 



Every TPE is realizable by a net of order zero. 



The functor .9 obviously coi"nmutes with disjunction, conjunction, 

 and negation. It is obvious that the result of substituting any S,, 

 realizable in the narrow sense (i.n.s.), for the p{z) in a realizable 

 expression Si is itself realizable i.n.s.; one constructs the realizing 

 net by replacing the peripheral afferents in the net for Si by the 

 realizing" neurons in the nets for the Si. The one neuron net 



