Abstract
We generalize the \({(\wedge, \vee)}\)-canonical formulas to \({(\wedge, \vee)}\)-canonical rules, and prove that each intuitionistic multi-conclusion consequence relation is axiomatizable by \({(\wedge, \vee)}\)-canonical rules. This yields a convenient characterization of stable superintuitionistic logics. The \({(\wedge, \vee)}\)-canonical formulas are analogues of the \({(\wedge,\to)}\)-canonical formulas, which are the algebraic counterpart of Zakharyaschev’s canonical formulas for superintuitionistic logics (si-logics for short). Consequently, stable si-logics are analogues of subframe si-logics. We introduce cofinal stable intuitionistic multi-conclusion consequence relations and cofinal stable si-logics, thus answering the question of what the analogues of cofinal subframe logics should be. This is done by utilizing the \({(\wedge,\vee,\neg)}\)-reduct of Heyting algebras. We prove that every cofinal stable si-logic has the finite model property, and that there are continuum many cofinal stable si-logics that are not stable. We conclude with several examples showing the similarities and differences between the classes of stable, cofinal stable, subframe, and cofinal subframe si-logics.
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Bezhanishvili, G., Bezhanishvili, N. & Ilin, J. Cofinal Stable Logics. Stud Logica 104, 1287–1317 (2016). https://doi.org/10.1007/s11225-016-9677-9
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DOI: https://doi.org/10.1007/s11225-016-9677-9
Keywords
- Intuitionistic logic
- Intuitionistic multi-conclusion consequence relation
- Axiomatization
- Heyting algebra
- Variety
- Universal class