G H Von Wright - Finnish-Soviet Logic Colloquium __ Truth, Negation, and Contradiction (1986) [10 2307_20116214]

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
Published by: Springer
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G. H. VON WRIGHT 
TRUTH, NEGATION, AND CONTRADICTION 
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 
traditionally several ways 
- 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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TRUTH, NEGATION, AND CONTRADICTION 5 
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