Kripke semantics for the logic of problems and propositions

Abstract

In this paper we study the propositional fragment of the joint logic of problems and propositions introduced by Melikhov. We provide Kripke semantics for this logic and show that is complete with respect to those models and has the finite model property. We consider examples of the use of -models usage. In particular, we prove that is a conservative extension of the logic . We also show that the logic is complete with respect to Kripke frames with sets of audit worlds introduced by Artemov and Protopopescu (who called them audit set models). Bibliography: 31 titles.

The predicate version of the joint logic of problems and propositions

“…In this paper we introduce Kripke semantics for the predicate logic QHC and show this logic to be sound and complete with respect to this semantics. We also prove the conservativity of QHC over the intuitionistic modal predicate logic QH4 (the analogous result for the propositional fragments of these logics was established in [3]). We show that the logic QHC has the disjunction and existence properties, namely, if an intuitionistic formula α ∨ β is provable in QHC, then either α or β is provable in QHC; if an intuitionistic formula ∃ x α(x) is provable in QHC, then α(c) is provable in QHC for some constant symbol c (provided that the language contains at least one constant symbol).…”

“…Kripke models with audit worlds for the logic IEL + , which form a basis for the semantics of QHC in this paper, were introduced in [4]. The author proved in [3] that the propositional logic HC is a conservative extension of PLL + . Melikhov denoted the logic PLL + by H4, and we use this notation in what follows.…”

“…to a problem α yields the proposition ?α denoting 'the problem α has a solution'. The semantics of the propositional fragment of the logic QHC, the logic HC, was studied in detail by this author in [3]. In that work we introduced Lindenbaum HC-algebras, Kripke-style semantics with two sets of worlds and also Kripke-style semantics with audit worlds.…”

“…with the appropriate restrictions on their arity, while atomic formulae are defined as in the standard predicate logic. As for the propositional fragment of QHC, the logic HC (see [3]), connectives and quantifiers do not change the sort of a formula, while modal operators do. The classical truth and falsity are denoted by 1 and 0, and the intuitionistic truth and falsity are denoted by ⊤ and ⊥, respectively.…”

A.N. Kolmogorov’s Calculus of Problems and Homotopy Type Theory

“…brings a version of the Intuitionistic Epistemic Logic studied by Artemov and Protopopescu [4] (as well as the Lax Logic of Fairtlough and Walton [17] and the Russell-Prawitz modal logic of Aczel [1], see [65, p. 22] for further details). QHC also admits a topological semantics [66, section 4] as well as a Kripkestyle semantics described by Anastasia Onoprienko, which she used for establishing the completeness of QHC with respect to this semantics [73], [74].…”