Let S be any set, and let I={I1, I2, ..., In} be a subset of S, called the set of assumptions. LetSN denote the Cartesian product SxSxSx...xS, taken N times. And let T={T1,T2,...,Tn} be a set oftransformations; that is, a set of functions each of which maps some subset of SN into somesubset of S. For instance, if S were a set of propositions, one might have T1(x,y)= x and y. Let us now define the set D(I,T) of all elements of S which are derivablefrom the assumptionsI via the transformations T. First of all, it is clear that I should be a subset of D(I,T). Let us call the elements of I the depth-zero elements of D(I,T). Next, what about elements of the formx=Ti(A1,...,Am), for some i, where each Ak=Ij for some j? Obviously, these elements are simpletransformations of the assumptions; they should be elements of D(I,T) as well. Let us call thesethe depth-one elements of D(I,T). Similarly, we may define an element x of S to be a depth-nelement of D(I,T) if x=Ti(A1,...,Am), for some i, where each of the Ak is a depth-p element ofD(I,T), for some pelements of D(I,T) for some n. Deductive reasoning is nothing more or less than the construction of elements of D(I,T), givenI and T. If the T are the rules of logic and the I are some set of propositions about the world, thenD(I,T) is the set of all propositions which are logically equivalent to some subset of I. In this casededuction is a matter of finding the logical consequences of I, which are presumably a smallsubset of the total set S of all propositions. Kaynak: A New Mathematical Model of Mind belgesi-954