Co-constructive logics for proofs and refutations

Trafford, James (2015) Co-constructive logics for proofs and refutations. Studia Humana, 3 (4). pp. 22-40. ISSN 2299-0518

Creators:
Trafford, James
Description/Abstract:

This paper considers logics which are formally dual to intuition- istic logic in order to investigate a co-constructive logic for proofs and refu- tations. 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 state- ments 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.

Item Type:
Article
Official URL:
http://studiahumana.com
Date:
January 2015
Identification Number:
10.1515/sh-2015-0004
Additional Information:

"The non-commercial use of the article will be governed by the Creative Commons Attribution-NonCommercial-NoDerivs license as currently displayed on http://creativecommons.org/licenses/by-nc-nd/3.0/"

"Authors retain the following rights....the right to self-archive the article." - http://studiahumana.com/for-authors.html (accessed 9/6/2015).

Uncontrolled Keywords or tags :
constructivism, paraconsistency, intuitionistic logic, co-intuitionistic logic, negation
Schools:
School of Communications
Depositing User:
Dr James Trafford
Date Deposited:
17 Dec 2014 11:39
Last Modified:
05 Mar 2022 04:31
Trafford, James:
ORCID iD: https://orcid.org/0000-0002-6762-2918
URI:
https://research.uca.ac.uk/id/eprint/1993

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