Cofinal stable logics
Guram Bezhanishvili, Nick Bezhanishvili, Julia Ilin
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.
Keywords: intuitionistic logic, intuitionistic multi-conclusion consequence relation, axiomatization, Heyting algebra, variety, universal class