8.1 The Structure of Deduction

   Let S be any set, and let I={I1, I2, …, In} be a subset of S, called the set of assumptions. Let
SN denote the Cartesian product SxSxSx…xS, taken N times. And let T={T1,T2,…,Tn} be a set of
transformations; that is, a set of functions each of which maps some subset of SN into some
subset 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 assumptions
I 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 form
x=Ti(A1,…,Am), for some i, where each Ak=Ij for some j? Obviously, these elements are simple
transformations of the assumptions; they should be elements of D(I,T) as well. Let us call these
the depth-one elements of D(I,T). Similarly, we may define an element x of S to be a depth-n
element of D(I,T) if x=Ti(A1,…,Am), for some i, where each of the Ak is a depth-p element of
D(I,T), for some p<n. Finally, D(I,T) may then be defined as the set of all x which are depth-n
elements of D(I,T) for some n.
   Deductive reasoning is nothing more or less than the construction of elements of D(I,T), given
I and T. If the T are the rules of logic and the I are some set of propositions about the world, then
D(I,T) is the set of all propositions which are logically equivalent to some subset of I. In this case
deduction is a matter of finding the logical consequences of I, which are presumably a small
subset of the total set S of all propositions.
Kaynak: A New Mathematical Model of Mind
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