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```Truth, Negation, and Contradiction
Author(s): G. H. Von Wright
Source: Synthese, Vol. 66, No. 1, Finnish-Soviet Logic Colloquium (Jan., 1986), pp. 3-14
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G. H. VON WRIGHT
1.
The position of negation in "classical" logic is characterized by three
fundamental laws or principles: the Law of Excluded Middle, the Law
of (Non)Contradiction, and the Law of Double Negation. There are
- not necessarily all of them equivalent
- of
formulating these laws. One way is as follows:
"Every proposition is true or false", for the Law of Excluded Middle.
"No proposition is true and false", for the Law of Contradiction. And,
"a proposition is true if, and only if, it is false that it is false", for the
Law of Double Negation.
In the object language of the classical calculus the laws are some
times stated as follows: p v ~p,
~
(p & ~p), and p<->-p. Thus, for
example, the authors of Principia Mathematica say, in so many words,1
that the first of the three formulas is the Law of Excluded Middle, and
the second the Law of Contradiction. This, however, is a rather sloppy
mode of expression. What is meant is that (the propositions expressed
by) the formulas are logically true for all substitution instances of the
variable. Thus the first formula, corresponding to the Law of Excluded
Middle, would say that the disjunction of any proposition with its
negation is logically or necessarily true
- and the second, corresponding
to the Law of Contradiction, that the conjunction of any proposition
with its negation is logically or necessarily not true. Any given
proposition and its negation form a pair of jointly exhaustive and
mutually exclusive alternatives, one could also say.
2.
Although all three laws may be said to be "orthodox" and to have a
"received" status in logic, they have also been contested.
Aristotle, traditionally credited with the discovery of the first two
laws, seems to have had difficulties with the Law of Excluded Middle, as
Synthese 66 (1986) 3-14.
? 1986 by D. Reidel Publishing Company
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4 G. H. VON WRIGHT
witnessed by the famous ninth chapter of De Interpretatione. The same
difficulties were felt and discussed by logicians in the Middle Ages.
In modern times, the Law of Excluded Middle has been an object of
criticism from different quarters. -Lukasiewicz rejected it in the for
mulation "every proposition is true or false", which he distinguished
from the formulation which says that the disjunction of any proposition
with its negation is necessarily true.2 For the first formulation he coined
the useful name the Principle of Bivalence.3 -Lukasiewicz's criticism was
a starting point for the development of so-called many-valued or
polyvalent logics.
Another consequential criticism of the Law of Excluded Middle
stems from Brouwer. One target of Brouwer's criticism was the use of
inverse proof in mathematics. Therefore his criticism also hit the Law of
Double Negation. By disproving the antithesis of a given thesis one has
not necessarily proved the thesis. The negation of the negation is not
identical with the base.
Brouwer's criticism, too, became the starting point of new develop
ments in formal logic. In intuitionist logic neither pv ~p nor p<->-p
are valid formulas. This is sometimes expressed by saying that in
tuitionist negation is different from classical negation.
3.
The Law of Contradiction has been more immune than the other two to
doubt or criticism within the classical tradition. But some peculiarities
of contradictions have been noted - and startled logicians.
One such peculiarity is known as Duns Scotus's Law or Principle,
after the great medieval logician, the doctor subtilis of the Scholastics,
who seems to have been the first to draw attention to it. It is sometimes
identified, in the object language of the calculus, with the formula
~
P~~* (P-* 4) or with the, in the classical calculus, equivalent formula
p&~p->q. The first is sometimes read, rather inappropriately, "a
false proposition implies any proposition" and the second, not very
appropriately either, "a contradiction implies just any proposition".
The more serious interpretation of the formulas is as follows.
If a proposition and its negation can both be derived in a deductive
system, then one can, using the above formulas as principles of
inference or entailment, modo ponente derive in this system just any
proposition (which can be expressed in its language). The appearance
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of contradiction in a deductive system is a "catastrophe" since it
"explodes" and therefore "trivializes" the system. These observations
are the basis for the well-known definition of a consistent logico
deductive system or calculus as one in which not just any proposition is
derivable.
Scotus's principle, however, has also worried logicians. The idea that
a contradiction "entails" just any proposition may appear counterin
tuitive. Entailment or logical consequence seems to presuppose some
kind of "community of content" between the entailing and the entailed
propositions. A motive force behind so-called relevance logic is a desire
to circumvent the counterintuitive consequences of Scotus's law.
If Scotus's principle is not accepted as a logical law of entailment, the
appearance of contradictions in a system does not necessarily
"explode" the system and need therefore not be considered a catas
trophe. This attitude to contradictions underlies yet another recent
development in formal logic known as paraconsistent logic. With it the
study of logic in what may still be termed with some justification "the
classical tradition" has arrived in the neighbourhood of another,
eminently "nonclassical" tradition, viz., Hegelian or Dialectical Logic.4
In Dialectical Logic there is an operation called Dialectical Syn
thesis. It leads to something called the Unity of Opposites (coincidentia
oppositorum). It can be described, in outline, as follows:
A thesis is put forward, call it 0. It has an antithesis which is its
negation,
~ 6. It is then shown, one way or another, that the thesis is not
true. Thus we have ?To, where the symbol "T" stands for the phrase
"it is true that". It is also shown that the antithesis is not true,
~ T ~ 0.
Thus neither the thesis nor the antithesis is true. From this is concluded
that both the thesis and the antithesis are true, T6&T
~ 0. This is
called Dialectical```