Notre Dame Journal of Formal Logic

Noncontractive Classical Logic

Lucas Rosenblatt

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Abstract

One of the most fruitful applications of substructural logics stems from their capacity to deal with self-referential paradoxes, especially truth-theoretic paradoxes. Both the structural rules of contraction and the rule of cut play a crucial role in typical paradoxical arguments. In this paper I address a number of difficulties affecting noncontractive approaches to paradox that have been discussed in the recent literature. The situation was roughly this: if you decide to go substructural, the nontransitive approach to truth offers a lot of benefits that are not available in the noncontractive account. I sketch a noncontractive theory of truth that has these benefits. In particular, it has both a proof- and a model-theoretic presentation, it can be extended to a first-order language, and it retains every classically valid inference.

Article information

Source
Notre Dame J. Formal Logic, Volume 60, Number 4 (2019), 559-585.

Dates
Received: 6 July 2017
Accepted: 24 March 2018
First available in Project Euclid: 6 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1567735227

Digital Object Identifier
doi:10.1215/00294527-2019-0020

Mathematical Reviews number (MathSciNet)
MR4019862

Zentralblatt MATH identifier
07167758

Subjects
Primary: 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30}
Secondary: 03B47: Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) {For proof-theoretic aspects see 03F52}

Keywords
substructural logic truth paradoxes

Citation

Rosenblatt, Lucas. Noncontractive Classical Logic. Notre Dame J. Formal Logic 60 (2019), no. 4, 559--585. doi:10.1215/00294527-2019-0020. https://projecteuclid.org/euclid.ndjfl/1567735227


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