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A SUBSTRUCTURAL GENTZEN CALCULUS FOR ORTHOMODULAR QUANTUM LOGIC

Published online by Cambridge University Press:  27 January 2022

DAVIDE FAZIO
Affiliation:
DIPARTIMENTO DI PEDAGOGIA, PSICOLOGIA, FILOSOFIA UNIVERSITÀ DI CAGLIARICAGLIARI, ITALYE-mail: dav.faz@hotmail.itE-mail: antonio.ledda@unica.itE-mail: gavinstjohn@gmail.com
ANTONIO LEDDA
Affiliation:
DIPARTIMENTO DI PEDAGOGIA, PSICOLOGIA, FILOSOFIA UNIVERSITÀ DI CAGLIARICAGLIARI, ITALYE-mail: dav.faz@hotmail.itE-mail: antonio.ledda@unica.itE-mail: gavinstjohn@gmail.com
FRANCESCO PAOLI*
Affiliation:
DIPARTIMENTO DI PEDAGOGIA, PSICOLOGIA, FILOSOFIA UNIVERSITÀ DI CAGLIARICAGLIARI, ITALYE-mail: dav.faz@hotmail.itE-mail: antonio.ledda@unica.itE-mail: gavinstjohn@gmail.com
GAVIN ST. JOHN
Affiliation:
DIPARTIMENTO DI PEDAGOGIA, PSICOLOGIA, FILOSOFIA UNIVERSITÀ DI CAGLIARICAGLIARI, ITALYE-mail: dav.faz@hotmail.itE-mail: antonio.ledda@unica.itE-mail: gavinstjohn@gmail.com
*

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.

Type
Research Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Association for Symbolic Logic

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