Co-constructive Logics for Proofs and Refutations

Abstract

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.

Affine Logic for Constructive Mathematics

“…If intuitionistic logic is the logic of affirmative propositions, and co-intuitionistic logic [Tra15,Shr05] is the logic of refutative propositions, then we can view affine logic as a logic of propositions that are subject to either affirmation or refutation.…”

Media Ontology and Transcendental Instrumentality

“…Proof is meant to demonstrate the premises of the reasoning by which truth can be determined. However, proof has been understood both in terms of ‘actualism’ in reference to an actually existing proof, and ‘potentialism’ involving how a hypothesis is potentially provable (Trafford, 2014: 23). Whilst potentialism seems to re-inscribe a Platonism in logical thinking, in terms of a tenseless time, actualism focuses on the act of proving and not on the proof as an object (2014: 23).…”