PP-2015-08: Bezhanishvili, Guram and Bezhanishvili, Nick and Ilin, Julia (2015) Cofinal stable logics. [Report]
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Abstract
We generalize the $(\wedge, \vee)$-canonical formulas of Bezhanishvili & Bezhanishvili (to appear) to $(\wedge, \vee)$-canonical rules, and prove that each intuitionistic multi-conclusion consequence relation is axiomatizable by $(\wedge, \vee)$-canonical rules. This provides an intuitionistic analogue of Bezhanishvili, Bezhanishvili & Iemhoff (submitted), and is an alternative of Jeřábek (2009). It also yields a convenient characterization of stable superintuitionistic logics introduced in Bezhanishvili & Bezhanishvili (to appear).
The $(\wedge, \vee)$-canonical formulas are analogues of the $(\wedge,\to)$-canonical formulas of Bezhanishvili & Bezhanishvili (2009), 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.
| Item Type: | Report |
|---|---|
| Report Nr: | PP-2015-08 |
| Series Name: | Prepublication (PP) Series |
| Year: | 2015 |
| Uncontrolled Keywords: | intuitionistic logic, intuitionistic multi-conclusion consequence relation, axiomatization, Heyting algebra, variety, universal class |
| Subjects: | Logic |
| Depositing User: | Julia Ilin |
| Date Deposited: | 12 Oct 2016 14:37 |
| Last Modified: | 12 Oct 2016 14:37 |
| URI: | https://eprints.illc.uva.nl/id/eprint/519 |
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