Open AccessSequent Calculi for Orthologic with Strict Implication
Author(s) -
Tomoaki Kawano
Publication year - 2021
Publication title -
bulletin of the section of logic
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.225
H-Index - 13
eISSN - 2449-836X
pISSN - 0138-0680
DOI - 10.18778/0138-0680.2021.22
Subject(s) - sequent , sequent calculus , completeness (order theory) , decidability , mathematics , calculus (dental) , discrete mathematics , cut elimination theorem , pure mathematics , combinatorics , natural deduction , proof calculus , mathematical analysis , mathematical proof , medicine , geometry , dentistry
In this study, new sequent calculi for a minimal quantum logic (\(\bf MQL\)) are discussed that involve an implication. The sequent calculus \(\bf GO\) for \(\bf MQL\) was established by Nishimura, and it is complete with respect to ortho-models (O-models). As \(\bf GO\) does not contain implications, this study adopts the strict implication and constructs two new sequent calculi \(\mathbf{GOI}_1\) and \(\mathbf{GOI}_2\) as the expansions of \(\bf GO\). Both \(\mathbf{GOI}_1\) and \(\mathbf{GOI}_2\) 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.