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MANY-VALUED LOGIC OF INFORMAL PROVABILITY: A NON-DETERMINISTIC STRATEGY

Published online by Cambridge University Press:  13 June 2018

PAWEL PAWLOWSKI  and
RAFAL URBANIAK
PAWEL PAWLOWSKI*
Affiliation:
Centre for Logic and Philosophy of Science, Ghent University
RAFAL URBANIAK*
Affiliation:
Centre for Logic and Philosophy of Science, Ghent University and Institute of Philosophy, Sociology and Journalism, University of Gdansk
*
*CENTRE FOR LOGIC AND PHILOSOPHY OF SCIENCE GHENT UNIVERSITY GHENT, BELGIUM E-mail: [email protected]
†CENTRE FOR LOGIC AND PHILOSOPHY OF SCIENCE GHENT UNIVERSITY GHENT, BELGIUM and INSTITUTE OF PHILOSOPHY, SOCIOLOGY AND JOURNALISM UNIVERSITY OF GDANSK GDANSK, POLAND E-mail: [email protected]
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Abstract

Mathematicians prove theorems in a semi-formal setting, providing what we’ll call informal proofs. There are various philosophical reasons not to reduce informal provability to formal provability within some appropriate axiomatic theory (Leitgeb, 2009; Marfori, 2010; Tanswell, 2015), but the main worry is that we seem committed to all instances of the so-called reflection schema: B(φ) → φ (where B stands for the informal provability predicate). Yet, adding all its instances to any theory for which Löb’s theorem for B holds leads to inconsistency.

Currently existing approaches (Shapiro, 1985; Horsten, 1996, 1998) to formalizing the properties of informal provability avoid contradiction at a rather high price. They either drop one of the Hilbert-Bernays conditions for the provability predicate, or use a provability operator that cannot consistently be treated as a predicate.

Inspired by (Kripke, 1975), we investigate the strategy which changes the underlying logic and treats informal provability as a partial notion. We use non-deterministic matrices to develop a three-valued logic of informal provability, which avoids some of the above mentioned problems.

Keywords

03A0500A30informal provabilitymany-valued logicnon-deterministic semanticsLöb’s theoremparadoxes of provability
Type
Research Article
Information
The Review of Symbolic Logic , Volume 11 , Issue 2 , June 2018 , pp. 207 - 223
Copyright
Copyright © Association for Symbolic Logic 2018 

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