Showing papers on "Normal modal logic published in 1972"

The semantics of entailment — III

Abstract: Publisher Summary This chapter discusses the semantics of entailment. Earlier, modal logics had no semantics. Bearing a real world G, a set of worlds K, and a relation R of relative possibility between worlds, Saul Kripke beheld this situation and saw that it was formally explicable and made model structures. It came to pass that soon everyone was making model structures, and some were deontic, some were temporal, and some were epistemic, according to the conditions on the binary relation R. The models made by Kripke, Hintikka, and Thomason were, however, not relevant. Central to the semantics being developed is a ternary relation R that takes the place for the relevant logics of the Kripke binary relation for standard modal and intuitionistic logics.

Tableau methods of proof for modal logics.

Abstract: 1 Introduction: In [1] Fitch proposed a new proof proceedure for several standard modal logics. The chief characteristic of this was the inclusion in the object language, of symbols representing worlds in Kripke models. In this paper we incorporate the device into a tableau proof system and it is seen that the resulting (propositional) proof system is highly analogous to a classical first order tableau system, with the modal operators behaving like quantifiers. Exploiting this similarity, a tableau completeness proof for first order logic directly becomes a Kripke completeness proof for modal logic, and Smullyan's fundamental theorem of quantification theory (a Herbrand-like theorem) [7] has its analog. Indeed, more than analogy is at work here; from an appropriate abstract point of view certain modal logics, first order classical and intuitionistic logic, and various infinitary logics may be treated simultaneously, an approach due to R. Smullyan and developed in a forthcoming monograph (see [8] for a preliminary version). We will treat only tableau proof systems and some familiarity with [7] is presumed. In addition to being metatheoretically interesting, specific tableau systems we give for S5, S4, T, B, DS4, DT, and K are quite simple to use. The extension of these systems to first order systems is straightforward , and is discussed briefly in the last section.

On the Usefulness of Modal Logic in Axiomatizations of Physics

Abstract: Natural sciences such as geology, geography, and astronomy are mainly interested in describing features of the real world. Other sciences such as physics and chemistry deal with certain classes of possible phenomena, no matter whether they really taken place or not. The former sciences are based on the latter, so that all are interested in possible phenomena or worlds.

Post completeness in modal logic

Abstract: Let ⊥, →, and □ be primitive, and let us have a countable supply of propositional letters. By a ( modal) logic we understand a proper subset of the set of all formulas containing every tautology and being closed under modus ponens and substitution. A logic is regular if it contains every instance of □A ∧ □B ↔ □(A ∧ B) and is closed under the rule A regular logic is normal if it contains □⊤. The smallest regular logic we denote by C (the same as Lemmon's C2 ), the smallest normal one by K . If L and L ' are logics and L ⊆ L ′, then L is a sublogic of L ', and L ' is an extension of L; properly so if L ≠ L '. A logic is quasi-regular (respectively, quasi-normal ) if it is an extension of C (respectively, K ). A logic is Post complete if it has no proper extension. The Post number , denoted by p ( L ), is the number of Post complete extensions of L. Thanks to Lindenbaum, we know that There is an obvious upper bound, too: Furthermore, .

$\varepsilon$-calculus based axiom systems for some propositional modal logics.

Abstract: 1 Introduction: Instead of considering propositional S4 (say) as a theory built on top of classical propositional logic, we can consider it as the classical first order theory of its Kripke models. This first order theory can be formulated in any of the ways first order theories usually are: tableau, Gentzen system, natural deduction, conventional axiom system, ε-calculus. If the first order theory of Kripke S4 models can be given in a formulation which is technically easy to use, the result is a convenient S4 proof system. Thus, in [2] we gave a tableau formulation of the S4 model theory which dealt automatically with the peculiarities of Kripke models, and produced a proof system for S4 (and similarly for other modal logics) which is simple to apply. In this paper we give what is essentially an ε-calculus formulation of the Kripke model theory of S4 (and S5, T, B, and also first order versions) which also automatically treats the peculiarities of the Kripke models. It is less convenient in use than the tableau system of [2] but still has a curious intrinsic interest.