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$\gamma $-ADMISSIBILITY IN FIRST-ORDER RELEVANT LOGICS: PROOF USING NORMAL MODELS IN THE MARES–GOLDBLATT SETTING

Part of: General logic

Published online by Cambridge University Press:  12 December 2024

NICHOLAS FERENZ Open the ORCID record for NICHOLAS FERENZ [Opens in a new window]  and
THOMAS MACAULAY FERGUSON Open the ORCID record for THOMAS MACAULAY FERGUSON [Opens in a new window]
NICHOLAS FERENZ*
Affiliation:
CENTER OF PHILOSOPHY, SCHOOL OF ARTS AND HUMANITIES UNIVERSITY OF LISBON CIDADE UNIVERSITÁRIA, ALAMEDA DA UNIVERSIDADE 1649-004, LISBON PORTUGAL
THOMAS MACAULAY FERGUSON
Affiliation:
DEPARTMENT OF COGNITIVE SCIENCE RENSSELAER POLYTECHNIC INSTITUTE 110 8TH STREET, TROY, NY 12180 USA E-mail: tferguson@gradcenter.cuny.edu
*
E-mail: nicholas.ferenz@gmail.com
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Abstract

For relevant logics, the admissibility of the rule of proof $\gamma $ has played a significant historical role in the development of relevant logics. For first-order logics, however, there have been only a handful of $\gamma $-admissibility proofs for a select few logics. Here we show that, for each logic L of a wide range of propositional relevant logics for which excluded middle is valid (with fusion and the Ackermann truth constant), the first-order extensions QL and LQ admit $\gamma $. Specifically, these are particular “conventionally normal” extensions of the logic $\mathbf {G}^{g,d}$, which is the least propositional relevant logic (with the usual relational semantics) that admits $\gamma $ by the method of normal models. We also note the circumstances in which our results apply to logics without fusion and the Ackermann truth constant.

Keywords

relevant logicgamma admissibilityfirst-order relevant logic

MSC classification

Primary: 03B47: Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)

Information

Type
Research Article
Information
The Review of Symbolic Logic , Volume 18 , Issue 2 , June 2025 , pp. 398 - 419
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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