A Logical Calculus of the Ideas Immanent in Nervous Activity 393 



The class !-iv,,(/c) is formed from k in analogy with H\(>'), but by 

 repeated apphcation not only of the logical operations but also 

 of that which replaces a class of properties P e a by S{P) e S ^^ a. 

 We shall then have the 



Lemma 



Priipu Pi. • . . , p.,. Zi) is a TPE if and only if 



(Zl) (pu ... , pra) {Ep„, + i) : />,„+! e ir^^eilpl, p2, • • • , P,n] ) 



A„+i(zi) = PuiPuPi, ... ,A»,Zl) (13) 



is true; and it is a TPE not involving \S" if and only if this holds 

 when '<-R,.' is replaced by 'f-R', and we then obtain 



Theorem IX 



A series of classes ai, a-^, ... a, is a series of f)rehensible classes if and 

 only if 



(Em) (En) (p)n(i) ('V) : . i.r)ni';^ix) = Ov •b{x = 1 :^ : (E^) 

 {Ey)m . 'M^) = . fSeiillyiiEi) . y = a,)) . v . {x)m . 

 ^(.r) = . l3efk[yaE,) . y = «,)] : (0 (0) : ^ea. . (14) 



'i4>, nt + p) . ^ . (Ef) . fef3 . {w)m{.v.)t - 1 . 



(t){n{t + 1) + p, nx + p, iv) = f(nt + p, nx + p, iv). 



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 t];'s being the charac- 

 teristic functions of the S„ and the (3 for each -^ being the class 

 whose designation has the form JJ ^r, J J PTj, where Pr,, denotes 



I'-i J-4i,-, 



a,, for all k. Conversely, we may write an expression of the form 

 (4) for a net VX fulfilling prehensible classes satisfying (14) by putting 

 for the Pra Pr denoting the ']j's and a Pr, written in the analogue 

 for classes of the disjunctive normal form, and denoting the a 

 corresponding to that '4^, 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 set of whose neurons requires the peripheral afferents to 



