Contemporary mathematical logic is not the only conceivable deductive system. In fact, I

suggest that any deductive system which relies centrally upon Boolean algebra, without

significant external constraints, is not even qualified for the purpose of general mental

deduction. Boolean algebra is very useful for many purposes, such as mathematical deduction. I

agree that it probably plays an important role in mental process. But it has at least one highly

undesirable property: if any two of the propositions in I contradict each other, then D(I,T) is the

entire set S of all propositions. From one contradiction, everything is derivable.

The proof of this is very simple. Assume both A and -A. Then, surely A implies A+B. But

from A+B and -A, one may conclude B. This works for any B. For instance, assume A="It is true

that my mother loves me". Then -A="It is not true that my mother loves me". Boolean logic

implies that anyone who holds A and -A — anyone who has contradictory feelings about his

mother’s affection — also, implicitly, holds that 2+2=5. For from "It is true that my mother loves

me" he may deduce "Either it is true that my mother loves me, or else 2+2=5." And from "Either

it is true that my mother loves me, or else 2+2=5" and "It is not true that my mother loves me,"

he may deduce "2+2=5."

So: Boolean logic is fine for mathematics, but common sense tells us that human minds

contain numerous contradictions. Does a human mind really use a deductive system that implies

everything? It appears that somehow we keep our contradictions under control. For example, a

person may contradict himself regarding abortion rights or the honesty of his wife or the ultimate

meaning of life — and yet, when he thinks about theoretical physics or parking his car, hemay

reason deductively to one particular conclusion, finding any contradictory conclusion ridiculous.

It might be that, although we do use the "contradiction-sensitive" deduction system of standard

mathematical logic, we carefully distinguish deductions in one sphere from deductions in

another. That is, for example, it might be that we have separate deductive systems for dealing

with physics, car parking, domestic relations, philosophy, etc. — so that we never, in practice,

reason "A implies A+B", unless A and B are closely related. If this were the case, a contradiction

in one realm would destroy only reasoning in that realm. So if we contradicted ourselves when

thinking about the meaning of life, then this might give us the ability to deduce any statement

whatsoever about other philosophical issues — but not about physics or everyday life.

In his Ph.D. dissertation, daCosta (1984) conceived the idea of a paraconsistent logic, one in

which a single contradiction in I does not imply that D(I,T)=S. Others have extended this idea in

various ways. Most recently, Avram (1990) has constructed a paraconsistent logic which

incorporates the "relevance logic" discussed in the previous paragraph. Propositions are divided

into classes and the inference from A to A+B is allowed only when A and B are in the same

class.

I suggest that Boolean logic is indeed adequate for the purpose of common-sense deduction.

My defense of this position comes in two parts. First, I believe that Avron is basically right in

saying that contradictions are almost always localized. To be precise, I hypothesize that a mind

does not tend to form the disjunction A+B unless %%[(St(A%v)-St(v)]-[St(B%w)-St(w)]%% is

small for some (v,w).

I do not think it is justified to partition propositions into disjoint sets and claim that each entity

is relevant only to those entities in the same set as it. This yields an elegant formal system, but of

course in any categorization there will be borderline cases, and it is unacceptable to simply

ignore them away. My approach is to define relevance not by a partition into classes but rather

using the theory of structure. What the formulation of the previous paragraph says is that two

completely unrelated entities will only rarely be combined in one logical formula.

However, there is always the possibility that, by a fluke, two completely unrelated entities will

be combined in some formula, say A+B. In this case a contradiction could spread from one

context to another. I suspect that this is an actual danger to thought processes, although certainly

a rare one. It is tempting to speculate that this is one possible route to insanity: a person could

start out contradicting themselves only in one context, and gradually sink into insanity by

contradicting themselves in more and more different contexts.

This brings us to the second part of the argument in favor of Boolean logic. What happens

when contradictions do arise? If a contradiction arises in a highly specific context, does it remain

there forever, thus invalidating all future reasoning in that context? I suspect that this is possible.

But, as will be elaborated in later chapters, I suggest that this is rendered unlikely by the

overallarchitecture of the mind. It is an error to suppose that the mind has only one center for

logical deduction. For all we know, there may be tens of thousands of different deductive

systems operating in different parts of the brain, sometimes perhaps more than one devoted to the

same specialized context. And perhaps then, as Edelman (1987) has proposed in the context of

perception and motor control, those systems which fail to perform a useful function will

eventually be destroyed and replaced. If a deductive system has the habit of generating arbitrary

propositions, it will not be of much use and will not last. This idea is related to the automata

networks discussed in the final chapter.

One thing which is absolutely clear from all this is the following: if the mind does use

Boolean logic, and it does harbor the occasional contradiction, then the fact that it does not

generate arbitrary statements has nothing to do with deductive logic. This is one important sense

in which deduction is dependent upon general structure of the mind, and hence implicitly on

other forms of logic such as analogy and induction.

Kaynak: A New Mathematical Model of Mind

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