Stable Canonical Rules

Abstract

Abstract. We introduce stable canonical rules and prove that each normal modal multi-conclusion consequence relation is axiomatizable by stable canonical rules. We apply these results to construct finite refutation patterns for modal formulas, and prove that each normal modal logic is axiomatizable by stable canonical rules. We also define stable multi-conclusion consequence relations and stable logics and prove that these systems have the finite model property. We conclude the paper with a number of examples of stable and nonstable systems, and show how to axiomatize them. §1. Introduction. It is a well-known result of Zakharyaschev [38] that each normal extension of K4 is axiomatizable by canonical formulas. This result was generalized in two directions. In [2] it was generalized to all normal extensions of wK4, and in [21] Zakharyaschev's canonical formulas were generalized to multi-conclusion canonical rules and it was proved that each normal modal multi-conclusion consequence relation over K4 is axiomatizable by canonical rules.The key ingredients of Zakharyaschev's technique include the concepts of subreduction, closed domain condition, and selective filtration. While selective filtration is very effective in the transitive case [15], and also generalizes to the weakly transitive case [2,6], it is less effective for K. This is one of the reasons why canonical formulas and rules do not work well for K [15,21]. In [3] a different approach to canonical formulas for intuitionistic logic was developed that uses the technique of filtration instead of selective filtration. The new canonical formulas were called stable canonical formulas, and it was shown that each superintuitionistic logic is axiomatizable by stable canonical formulas.In this paper we generalize the technique of [3] to the modal setting. Since the technique of filtration works well for K, we show that this new technique is effective in the nontransitive case as well. We give an algebraic account of filtration, introduce stable canonical rules, and prove that each normal modal multi-conclusion consequence relation is axiomatizable by stable canonical rules. This allows us to construct finite refutation patterns for modal formulas, and to show that each normal modal logic is axiomatizable by stable canonical rules. For normal extensions of K4 we prove that stable canonical rules can be replaced by stable canonical formulas, thus providing an alternative to Zakharyaschev's axiomatization [38]. This approach also yields a new class of multi-conclusion consequence relations and logics. Following [3], we call these systems stable. We show that every stable multi-conclusion consequence relation and every stable logic has the finite model property. We also give a number of examples of stable and nonstable logics, and show how to axiomatize them. For more in-depth development of the theory of stable modal systems see [5]. The theory of stable superintuitionistic logics and stable intuitionistic multi-conclusion consequence relations is developed in [3,4]...

Cofinal Stable Logics

“…This provides an intuitionistic analogue of similar results in [6]. Our proofs are modifications of those in [6,Sec. 5] and generalize those in [5,Sec.…”

“…The (∧, ∨)-canonical rules are the intuitionistic counterpart of the stable canonical rules of [6], and are an alternative of Jeřábek's canonical multi-conclusion rules [21]. We also indicate how to generalize (∧, →)-canonical formulas to (∧, →)-canonical rules, which provide another uniform axiomatization of all intuitionistic multi-conclusion consequence relations.…”

Angluin Learning via Logic

“…e.g. [25,26]) but modal logicians usually focus on properties of filtrations that do not play a role in learning. Closely related to this question is a more diagrammatic understanding of termination and correctness of our algorithm: we believe that all essential ingredients for this are contained in our work but we need to understand filtrations on a more abstract, categorical level.…”

Admissible Bases Via Stable Canonical Rules

“…Stable homomorphisms were considered in [3] under the name of semi-homomorphisms and in [13] under the name of continuous morphisms. We now come to stable canonical rules: It was proved in [1] that every multi-conclusion consequence relation above K is axiomatizable by stable canonical rules (relative to arbitrary finite modal algebras -not only to those validating K4-axiom). The same proof can easily be extended to our multi-conclusion consequence relations above K4.…”

“…Our goal is to establish the same property for stable multi-conclusion canonical rules for IPC, K4, and S4. These rules were recently introduced in [1], where it was shown that each normal modal multi-conclusion consequence relation is axiomatizable by stable multi-conclusion canonical rules. The same result for intuitionistic multi-conclusion consequence relations was established in [2].…”