Showing papers on "Normal modal logic published in 1984"

Basic Modal Logic

Abstract: It is popular practice to borrow metaphors between different fields of thought. When it comes to evaluating modal logic it is tempting to borrow from the anthropologists who seem to agree that our civilization has lived through two great waves of change in the past, the Agricultural Revolution and the Industrial Revolution. Where we stand today, where the world is going, is difficult to say. If there is a deeper pattern fitting all that is happening today, then many of us do not see it. All we know, really, is that history is pushing on.

Quantification in Modal Logic

Abstract: The novice may wonder why quantified modal logic (QML) is considered difficult. QML would seem to be easy: simply add the principles of first-order logic to propositional modal logic. Unfortunately, this choice does not correspond to a intuitively satisfying semantics. From the semantical point of view, we are confronted with a number of decisions concerning the quantifiers, and these in turn prompt new questions about the semantics of identity, terms, and predicates. Since most of the choices can be made independently, the number of interesting quantified modal logics seems bewilderingly large.

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.

Automata theoretic techniques for modal logics of programs: (Extended abstract)

Abstract: We present a new technique for obtaining decision procedures for modal logics of programs. The technique centers around a new class of finite automata on infinite trees for which the emptiness problem can be solved in polynomial time. The decision procedures then consist of constructing an automaton Af for a given formula f, such that Af accepts some tree if and only if f is satisfiable. We illustrate our technique by giving an exponential decision procedure for deterministic propositional dynamic logic and a variant of the μ-calculus of Kozen.

The predicate modal logic of provability.

Abstract: On montre que des points fixes, quand ils existent sont uniques jusqu'a l'equivalence prouvable

The preservation of coherence

Abstract: It is argued that the preservation of truth by an inference relation is of little interest when premiss sets are contradictory. The notion of a level of coherence is introduced and the utility of modal logics in the semantic representation of sets of higher coherence levels is noted. It is shown that this representative role cannot be transferred to first order logic via frame theory since the modal formulae expressing coherence level restrictions are not first order definable. Finally, an inference relation, calledyielding, is introduced which is intermediate between the coherence preservingforcing relation introduced elsewhere by the authors and the coherence destroying, inference relation of classical logic.

Modalities and quantum mechanics

Abstract: Three approaches concerning the usage of modalities in the language of quantum mechanics were considered; Mittelstaedt and I built up a dialog semantics for modalities on a metalinguistic level, and a calculus of quantum modal logic is known that is complete and sound with respect to this dialogic semantics Van Fraassen replaced the usual interpretation of quantum mechanics (with the projection postulate) by his “modal interpretation” based on a modal object language Dalla Chiara translated a nonmodal object language for quantum mechanics and the appropriate quantum logic into a modal language Specifically we are interested in the similarities and the differences of these three approaches

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

Modal logic and model theory

Abstract: We propose a first order modal logic, theQS4E-logic, obtained by adding to the well-known first order modal logicQS4 arigidity axiom schemas:A → □A, whereA denotes a basic formula. In this logic, thepossibility entails the possibility of extending a given classical first order model. This allows us to express some important concepts of classical model theory, such as existential completeness and the state of being infinitely generic, that are not expressibile in classical first order logic. Since they can be expressed in\(L_{\omega _1 \omega } \)-logic, we are also induced to compare the expressive powers ofQS4E and\(L_{\omega _1 \omega } \). Some questions concerning the power of rigidity axiom are also examined.

Linear Reasoning in Modal Logic

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Conjunctive normal forms and weak modal logics without the axiom of necessity

Abstract: On definit les logiques modales faibles. Definitions et caracterisations des L-tautologie. Demonstrations de completude. Applications