Sequent Calculi for Orthologic with Strict Implication

Tomoaki Kawano

doi:10.18778/0138-0680.2021.22

Authors

Tomoaki Kawano Tokyo Institute of Technology, School of Computing, Department of Mathematical and Computing Science, 2-12-1 Okayama, Meguro-ku, Tokyo, Japan https://orcid.org/0000-0002-0524-7614

DOI:

https://doi.org/10.18778/0138-0680.2021.22

Keywords:

Quantum logic, sequent calculus, completeness theorem, implication, orthologic

Abstract

In this study, new sequent calculi for a minimal quantum logic ( $M Q L$ ) are discussed that involve an implication. The sequent calculus $G O$ for $M Q L$ was established by Nishimura, and it is complete with respect to ortho-models (O-models). As $G O$ does not contain implications, this study adopts the strict implication and constructs two new sequent calculi ${GOI}_{1}$ and ${GOI}_{2}$ as the expansions of $G O$ . Both ${GOI}_{1}$ and ${GOI}_{2}$ are complete with respect to the O-models. In this study, the completeness and decidability theorems for these new systems are proven. Furthermore, some details pertaining to new rules and the strict implication are discussed.

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