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- Title
NON-CONTRACTIVE LOGICS, PARADOXES, AND MULTIPLICATIVE QUANTIFIERS.
- Authors
NICOLAI, CARLO; PIAZZA, MARIO; TESI, MATTEO
- Abstract
The paper investigates from a proof-theoretic perspective various non-contractive logical systems, which circumvent logical and semantic paradoxes. Until recently, such systems only displayed additive quantifiers (Grišin and Cantini). Systems with multiplicative quantifiers were proposed in the 2010s (Zardini), but they turned out to be inconsistent with the naive rules for truth or comprehension. We start by presenting a first-order system for disquotational truth with additive quantifiers and compare it with Grišin set theory. We then analyze the reasons behind the inconsistency phenomenon affecting multiplicative quantifiers. After interpreting the exponentials in affine logic as vacuous quantifiers, we show how such a logic can be simulated within a truth-free fragment of a system with multiplicative quantifiers. Finally, we establish that the logic for these multiplicative quantifiers (but without disquotational truth) is consistent, by proving that an infinitary version of the cut rule can be eliminated. This paves the way to a syntactic approach to the proof theory of infinitary logic with infinite sequents.
- Subjects
PARADOX; SET theory; DISPLAY systems; LOGIC; ADDITIVES
- Publication
Review of Symbolic Logic, 2024, Vol 17, Issue 4, p996
- ISSN
1755-0203
- Publication type
Academic Journal
- DOI
10.1017/S1755020323000138