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Extending paraconsistent quantum logic: a single‐antecedent/succedent system approach
Author(s) -
Kamide Norihiro
Publication year - 2018
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201700012
Subject(s) - sequent , sequent calculus , mathematics , paraconsistent logic , decidability , discrete mathematics , modal logic , embedding , cut elimination theorem , negation , calculus (dental) , pure mathematics , mathematical proof , modal , higher order logic , computer science , theoretical computer science , description logic , medicine , chemistry , geometry , dentistry , artificial intelligence , polymer chemistry , programming language
In this study, some conservative extensions of paraconsistent quantum logic, such as Nelsonian, modal, infinitary and temporal, are investigated by extending a single‐antecedent/succedent sequent calculus PQL for paraconsistent quantum logic. A sequent calculus NQL , which is obtained from PQL by adding implication and co‐implication, is introduced as a variant of Nelson's paraconsistent four‐valued logic. Sequent calculi MPQL , IPQL and TPQL are introduced, respectively, as modal, infinitary and temporal extensions of PQL . The cut‐elimination and duality theorems for these calculi are proved, and some extended calculi including NQL and MPQL , as well as their fragments, are shown to be decidable. A theorem for embedding NQL into its negation‐free fragment and a theorem for embedding TPQL into IPQL are proved.