A Logical Calculus of the Ideas Immanent in Nervous Activity 395 



{E Zi)Zi . Priiz^j respectively; and a simple circuit, a, C2, ... , c,., 

 of ?2 links, each sufficient to excite the next, gives an expression 



A^„(zi) . = .M(0) . C.„ 



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 inotor-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 otliers, and not all of them. This is of interest as 

 affording a psychological justification of the Turing definition of 

 computability and its equivalents, Clhurch's X — definability and 

 Kleene's primitive recursiveness: If any number can be computed 

 by an organism, it is computable by these definitions, and con- 

 versely. 



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 regen- 

 erative 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 



