PP201508: 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 multiconclusion 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 (silogics for short). Consequently, stable silogics are analogues of subframe silogics. We introduce cofinal stable intuitionistic multiconclusion consequence relations and cofinal stable silogics, 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 silogic has the finite model property, and that there are continuum many cofinal stable silogics 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 silogics.
Item Type:  Report 

Report Nr:  PP201508 
Series Name:  Prepublication (PP) Series 
Year:  2015 
Uncontrolled Keywords:  intuitionistic logic, intuitionistic multiconclusion 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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