On Zardini’s Rules for Multiplicative Quantification as the Source of Contra(di)ctions

Petersen, Uwe

doi:10.1017/s1755020322000284

The Review of Symbolic Logic

2022

DOI: 10.1017/s1755020322000284

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On Zardini’s Rules for Multiplicative Quantification as the Source of Contra(di)ctions

Uwe Petersen¹

Abstract: Certain instances of contraction are provable in Zardini’s system $\mathbf {IK}^\omega $ which causes triviality once a truth predicate and suitable fixed points are available.

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2023

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Cited by 2 publications

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Systems for Non-Reflexive Consequence

Nicolai,

Rossi

2023

Stud Logica

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Substructural logics and their application to logical and semantic paradoxes have been extensively studied. In the paper, we study theories of naïve consequence and truth based on a non-reflexive logic. We start by investigating the semantics and the proof-theory of a system based on schematic rules for object-linguistic consequence. We then develop a fully compositional theory of truth and consequence in our non-reflexive framework.

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Systems for Non-Reflexive Consequence

Nicolai,

Rossi

2023

Stud Logica

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Non-Contractive Logics, Paradoxes, and Multiplicative Quantifiers

Nicolai¹,

Piazza²,

Tesi³

2023

The Review of Symbolic Logic

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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.

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On Zardini’s Rules for Multiplicative Quantification as the Source of Contra(di)ctions

Abstract: Certain instances of contraction are provable in Zardini’s system $\mathbf {IK}^\omega $ which causes triviality once a truth predicate and suitable fixed points are available.

Cited by 2 publications

References 9 publications

Systems for Non-Reflexive Consequence

Systems for Non-Reflexive Consequence

Non-Contractive Logics, Paradoxes, and Multiplicative Quantifiers

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