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

  • PAWEL PAWLOWSKI (a1) and RAFAL URBANIAK (a2)
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.

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Corresponding author
*CENTRE FOR LOGIC AND PHILOSOPHY OF SCIENCE GHENT UNIVERSITY GHENT, BELGIUM E-mail: haptism89@gmail.com
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: rfl.urbaniak@gmail.com
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The Review of Symbolic Logic
  • ISSN: 1755-0203
  • EISSN: 1755-0211
  • URL: /core/journals/review-of-symbolic-logic
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