Non-contractive logics, Paradoxes, and Multiplicative Quantifiers

TL;DR

This paper explores non-contractive logical systems with additive and multiplicative quantifiers, analyzing their consistency and paradox-avoidance, and introduces a proof-theoretic approach to infinitary logic with multiplicative quantifiers.

Contribution

It provides a proof-theoretic analysis of non-contractive logics with multiplicative quantifiers, demonstrating their consistency and comparing them with additive systems and set theories.

Findings

Additive quantifier systems can be consistent with disquotational truth.

Multiplicative quantifiers can be simulated within a truth-free fragment.

An infinitary cut-elimination procedure ensures consistency of multiplicative quantifier logic.

Abstract

The paper investigates from a proof-theoretic perspective various non-contractive logical systems circumventing logical and semantic paradoxes. Until recently, such systems only displayed additive quantifiers (Gri\v{s}in, Cantini). Systems with multiplicative quantifers have also been 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 we compare it with Gri\v{s}in set theory. We then analyze the reasons behind the inconsistency phenomenon affecting multiplicative quantifers: 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 prove that the logic of these multiplicative quantifiers (but without disquotational truth) is consistent, by showing 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.