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

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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 Published by: Springer Stable URL: http://www.jstor.org/stable/20116214 . Accessed: 18/08/2013 15:25 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. . Springer is collaborating with JSTOR to digitize, preserve and extend access to Synthese. http://www.jstor.org This content downloaded from 192.236.36.29 on Sun, 18 Aug 2013 15:25:41 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/action/showPublisher?publisherCode=springer http://www.jstor.org/stable/20116214?origin=JSTOR-pdf http://www.jstor.org/page/info/about/policies/terms.jsp http://www.jstor.org/page/info/about/policies/terms.jsp 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 This content downloaded from 192.236.36.29 on Sun, 18 Aug 2013 15:25:41 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 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 This content downloaded from 192.236.36.29 on Sun, 18 Aug 2013 15:25:41 PM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 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