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.
Highlights
Superintuitionistic logics are propositional logics extending the intuitionistic propositional calculus IPC
Consistent si-logics are known as intermediate logics as they are exactly the logics situated between IPC and the classical propositional calculus CPC
We indicate how to generalize (∧, →)-canonical formulas to (∧, →)-canonical rules, which provide another uniform axiomatization of all intuitionistic multi-conclusion consequence relations
Summary
Superintuitionistic logics (si-logics for short) are propositional logics extending the intuitionistic propositional calculus IPC. Another locally finite variety closely related to the variety of Heyting algebras is that of bounded distributive lattices This suggests a different approach to canonical formulas, which was developed in [5], where (∧, ∨)-canonical formulas were introduced. It is only natural to introduce cofinal stable si-logics as the logics axiomatized by (∧, ∨, ¬)-canonical formulas when D = ∅ For this we need to work with the pseudocomplemented lattice reduct of Heyting algebras, instead of the bounded lattice reduct like in the case of stable logics. We first generalize (∧, ∨)-canonical formulas to (∧, ∨)canonical rules, which axiomatize all intuitionistic multi-conclusion consequence relations. We indicate how to generalize (∧, →)-canonical formulas to (∧, →)-canonical rules, which provide another uniform axiomatization of all intuitionistic multi-conclusion consequence relations. We conclude the paper with several examples that indicate the similarities and differences between the classes of stable, cofinal stable, subframe, and cofinal subframe si-logics
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