# 4 Belge Chapter 8: Deduction

## 8.3 Deduction Cannot Stand Alone

When deduction is formulated in the abstract, in terms of assumptions and transformation, it isimmediately apparent that deductive reasoning is incapable of standing on its own. In isolation, itis useless. For why would there be intrinsic value in determining which x lie in D(I,T)? Whocares? The usefulness of deduction presupposes several things, none of…

## 8.2 Paraconsistency

Contemporary mathematical logic is not the only conceivable deductive system. In fact, Isuggest that any deductive system which relies centrally upon Boolean algebra, withoutsignificant external constraints, is not even qualified for the purpose of general mentaldeduction. Boolean algebra is very useful for many purposes, such as mathematical deduction. Iagree that it probably plays an…

## 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. 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…

## 8.0 Deduction and Analogy in Mathematics

In mathematical logic, deduction is analyzed as a thing in itself, as an entity entirelyindependent from other mental processes. This point of view has led to dozens of beautiful ideas:Godel’s Incompleteness Theorem, the theory of logical types, model theory, and so on. But itslimitations are too often overlooked. Over the last century, mathematical logic…