Cofinal stable logics

We generalize the (∧,∨)-canonical formulas to (∧,∨)-canonical rules, and prove that each intuitionistic multi-conclusion consequence relation is axiomatizable by (∧,∨)-canonical rules. This yields a convenient characterization of stable superintuitionistic logics. The (∧,∨)-canonical formulas are analogues of the (∧,→)-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 (∧,∨,¬)-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.