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Co-constructive Logics for Proofs and Refutations

100%

Trafford J.

nr 4

22-40

This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we do not want to equate falsity with the non-existence of a proof since this would render a statement such as “pi is transcendental” false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for statements for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorov's logic of problems.

The Case of Dialetheism

86%

Tkaczyk M.

nr 2

203-224

The concept of dialetheia and the claim of dialetheism has been examined and compared to such related concept as contradiction, antinomy, consistency and paraconsistency. Dialetheia is a true contradiction and dialetheism is the claim that there exists at least one dialetheia. It has been observed that dialetheism is equivalent to the negation of the traditional principle of contradiction. Hence, dialetheism itself is no new idea in whatsoever. The novelty of dialetheism consists in the arguments delivered for its case. Key justification the partisans deliver for dialetheism has been examined and evaluated: antinomies, an alleged Gödel’s paradox, and existence of limits of thought. The structure of those arguments has been analyzed. It has been claimed that they share one and the same simple structure which may be called reverse paradox. The vital content dialetheists add to the traditional paradoxes is only the thesis of reliability of the vernacular prima facie knowledge. Three objections have been raised against the justification of dialetheism: firstly, it has been claimed that exactly the same argument supports principle of contradiction, secondly, it has been questioned whether the arguments preserve their value when logic is subject to revision, and thirdly, it has been claimed that the underlying logic of dialetheism is classical.