Showing papers on "Modal operator published in 1984"

Models for normal intuitionistic modal logics

Abstract: Kripke-style models with two accessibility relations, one intuitionistic and the other modal, are given for analogues of the modal systemK based on Heyting's prepositional logic. It is shown that these two relations can combine with each other in various ways. Soundness and completeness are proved for systems with only the necessity operator, or only the possibility operator, or both. Embeddings in modal systems with several modal operators, based on classical propositional logic, are also considered. This paper lays the ground for an investigation of intuitionistic analogues of systems stronger thanK. A brief survey is given of the existing literature on intuitionistic modal logic.

Likelihood, probability, and knowledge

Abstract: The modal logic LL was introduced by Halpern and Rabin [HR] as a means of doing qualitative reasoning about likelihood. Here the relationship between LL and probability theory is examined. It is shown that there is a way of translating probability assertions into LL in a sound manner, so that LL in some sense can capture the probabilistic interpretation of likelihood. However, the translation is subtle; several more obvious attempts are shown to lead to inconsistencies. We also extend LL by adding modal operators for knowledge. The propositional version of the resulting logic LLK is shown to have a complete axiomatization and to be decidable in exponential time, provably the best possible.

An Extended Joint Consistency Theorem for a Family of Free Modal Logics with Equality

Abstract: ?0. Introduction. In this paper, we establish an extended joint consistency theorem for an infinite family of free modal logics with equality. The extended joint consistency theorem incorporates the Craig and Lyndon interpolation lemmas and the Robinson joint consistency theorem. In part, the theorem states that two theories which are jointly unsatisfiable are separated by a sentence in the vocabulary common to both theories. Our family of free modal logics includes the free versions of I, M, and S4 studied by Leblanc [5, Chapters 8 and 9], supplemented with equality as in [3]. In the relational semantics for these logics, there is no restriction on the accessibility relation in I, while in M(S4) the restriction is reflexivity (reflexivity and transitivity). We say that a restriction on the accessibility relation countenances backward-looping if it implies a sentence of the form Vx1 ... xn(xRx2 &... &xn Rxn D xkRxj) (1 < j < k < n ? 2), where the xi (1 < i < n) are distinct individual variables. Just as reflexivity and transitivity do not countenance backward-looping, neither do any of the restrictions in our family of free modal logics. (The above terminology is derived from the effect of such restrictions on Kripke tableaux constructions.) The Barcan formula, its converse, the Fitch formula, and the formula T T'D DIT : T' do not hold in our logics.2

A Symmetric Approach to Axiomatizing Quantifiers and Modalities

Abstract: We present an axiomatization of several of the basic modal logics, with the idea of giving the two modal operators □ and ◊ equal weight as far as possible. Then we present a parallel axiomatization of classical quantification theory, working our way up through a sequence of rather curious subsystems. It will be clear at the end that the essential difference between quantifiers and modalities is amusing in a vacuous sort of way. Finally we sketch tableau proof systems for the various logics we have introduced along the way. Also, the “natural” model theory for the subsystems of quantification theory that come up is somewhat curious. In a sense, it amounts to a “stretching out” of the Henkin-style completeness proof, severing the maximal consistent part of the construction quite thoroughly from the part of the construction that takes care of existential-quantifier instances.

A Modal Argument

Abstract: The following is a version of a familiar type of modal argument for God’s existence: (a) Since the concept of God is the concept of a being than which no greater being is logically possible, the concept of God is such that it is true in each possible world that if God exists in that world, then it is logically impossible for him to fail to exist there, i.e., his existence in that world is logically necessary. (b) There is a possible world, W, in which God exists.