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A SUBSTRUCTURAL GENTZEN CALCULUS FOR ORTHOMODULAR QUANTUM LOGIC

Part of: General logic Algebraic logic

Published online by Cambridge University Press:  27 January 2022

DAVIDE FAZIO ,
ANTONIO LEDDA ,
FRANCESCO PAOLI  and
GAVIN ST. JOHN Open the ORCID record for GAVIN ST. JOHN [Opens in a new window]
DAVIDE FAZIO
Affiliation:
DIPARTIMENTO DI PEDAGOGIA, PSICOLOGIA, FILOSOFIA UNIVERSITÀ DI CAGLIARI CAGLIARI, ITALY E-mail: dav.faz@hotmail.it E-mail: antonio.ledda@unica.it E-mail: gavinstjohn@gmail.com
ANTONIO LEDDA
Affiliation:
DIPARTIMENTO DI PEDAGOGIA, PSICOLOGIA, FILOSOFIA UNIVERSITÀ DI CAGLIARI CAGLIARI, ITALY E-mail: dav.faz@hotmail.it E-mail: antonio.ledda@unica.it E-mail: gavinstjohn@gmail.com
FRANCESCO PAOLI*
Affiliation:
DIPARTIMENTO DI PEDAGOGIA, PSICOLOGIA, FILOSOFIA UNIVERSITÀ DI CAGLIARI CAGLIARI, ITALY E-mail: dav.faz@hotmail.it E-mail: antonio.ledda@unica.it E-mail: gavinstjohn@gmail.com
GAVIN ST. JOHN
Affiliation:
DIPARTIMENTO DI PEDAGOGIA, PSICOLOGIA, FILOSOFIA UNIVERSITÀ DI CAGLIARI CAGLIARI, ITALY E-mail: dav.faz@hotmail.it E-mail: antonio.ledda@unica.it E-mail: gavinstjohn@gmail.com
*
E-mail: paoli@unica.it
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Abstract

We introduce a sequent system which is Gentzen algebraisable with orthomodular lattices as equivalent algebraic semantics, and therefore can be viewed as a calculus for orthomodular quantum logic. Its sequents are pairs of non-associative structures, formed via a structural connective whose algebraic interpretation is the Sasaki product on the left-hand side and its De Morgan dual on the right-hand side. It is a substructural calculus, because some of the standard structural sequent rules are restricted—by lifting all such restrictions, one recovers a calculus for classical logic.

Keywords

orthomodular latticequantum logicGentzen systemsubstructural logicleft residuated ℓ-groupoid

MSC classification

Primary: 03G12: Quantum logic
Secondary: 03B47: Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
Type
Research Article
Information
The Review of Symbolic Logic , Volume 16 , Issue 4 , December 2023 , pp. 1177 - 1198
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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