A Logical Calculus of the Ideas Immanent in Nervous Activity 391 



negations of the rest. Denumerating these by A'i(zi), ^'2(21), . .., 

 X-2p{zi), we may, by use of the expressions (4), arrive at an equi- 

 pollent set of equations of the form 



X,{z,) . ^ .ZPruiz,) . S^Xjiz,). (5) 



Now we import the subscripted numerals i,j into the object- 

 language: i.e., define Pri and Pr2 such that Pri(zzi,Zi) . = . X,{zi) 

 and Prj(zzi,zz2,Zi) . = . Pr,j{zi) are provable whenever zzi and 

 ZZ2 denote i and / respectively. 

 Then we may rewrite (5) as 



(zi)zzp : Pri(zi, Z3) 

 . = . {EZ'{)zZp . Pr-iizi. Zo, z-i - zzn) . Priizo, Zz - zZn) (6) 



where zz^ denotes n and zZp denotes 2'\ By repeated substitution 

 we arrive at an expression 



(zi)zZp : Pri(zi, zz„zzo) . = . {Ez-z)zZp {Ezz)zZp . . . {Ez„)zZp 



Pr-zizi, z-2, ZZn {zzo — 1)) . Pr-i{z2,Zz,zZn {zz2 - 1)) (7) 



Pr2(z„_i,z„,0) . Pri(Zn,0), for any numeral ZZ2 which denotes s. 



This is easily shown by induction to be equipollent to 



{zi)zzp : . Pri{Zi,zZnZZ2) : = : (Ef) (Z2) zzo — l/(zoZZ„) 



^ ZZp . fiZZnZZ2) = Zi . Pr2{f(,ZZn (Z2 + 1)), (8) 



f(zz,a-z)) . PrAf {0),0) 

 and since this is the case for all ZZ2, it is also true that 



(Z4) {z,)zzp : Pn{z,,z,) . = . (Ef) (Z2) (Z4 - 1) ./(Z2) 



^ ZZp . /(Z4) = Zi/(Z4) - zi . Pro[/(z2 + 1),/(Z2), Z2] . (9) 



Pri[/(res (Z4. zZn)), res (Z4, zz,,)], 



where zz„ denotes n, res {r,s) is the residue of /• mod s and zZp 

 denotes 2''. This may be written in a less exact way as 



N^t) . ^ . (Ecf>) ix)t - 1 . <^(.r) ^ 2' . 0(0 = i . 



P[0(.f+ l),0(.r) ..V,(o^ (0)], 



where a and t are also assumed divisible by n, and Pr2 denotes P. 

 From the preceding remarks we shall have 



