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ARTICLES:
Co-constructive Logics for Proofs and Refutations
Issue: 3:4 (The twelfth issue)
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.
Conditionals in Interaction
Issue: 6:1 (The twenty first issue)
There are several issues with the standard approach to the
relationship between conditionals and assertions, particularly when the
antecedent of a conditional is (or may be) false. One prominent alternative is to
say that conditionals do not express propositions, but rather make conditional
assertions that may generate categorical assertions of the consequent in certain
circumstances. However, this view has consequences that jar with standard
interpretations of the relationship between proofs and assertion. Here, I analyse
this relationship, and say that, on at least one understanding of proof,
conditional assertions may reflect the dynamics of proving, which (sometimes)
generate categorical assertions. In particular, when we think about the
relationship between assertion and proof as rooted in a dialogical approach to
both, the distinction between conditional and categorical assertions is quite
natural.
relationship between conditionals and assertions, particularly when the
antecedent of a conditional is (or may be) false. One prominent alternative is to
say that conditionals do not express propositions, but rather make conditional
assertions that may generate categorical assertions of the consequent in certain
circumstances. However, this view has consequences that jar with standard
interpretations of the relationship between proofs and assertion. Here, I analyse
this relationship, and say that, on at least one understanding of proof,
conditional assertions may reflect the dynamics of proving, which (sometimes)
generate categorical assertions. In particular, when we think about the
relationship between assertion and proof as rooted in a dialogical approach to
both, the distinction between conditional and categorical assertions is quite
natural.
