8.2 Paraconsistency

    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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