Showing papers on "Normal modal logic published in 1973"

Model Existence Theorems for Modal and Intuitionistic Logics

Abstract: In classical logic a collection of sets of statements (or equivalently, a property of sets of statements) is called a consistency property if it meets certain simple closure conditions (a definition is given in §2). The simplest example of a consistency property is the collection of all consistent sets in some formal system for classical logic. The Model Existence Theorem then says that any member of a consistency property is satisfiable in a countable domain. From this theorem many basic results of classical logic follow rather simply: completeness theorems, the compactness theorem, the Lowenheim-Skolem theorem, and the Craig interpolation lemma among others. The central position of the theorem in classical logic is obvious. For the infinitary logic the Model Existence Theorem is even more basic as the compactness theorem is not available; [8] is largely based on it. In this paper we define appropriate notions of consistency properties for the first-order modal logics S 4, T and K (without the Barcan formula) and for intuitionistic logic. Indeed we define two versions for intuitionistic logic, one deriving from the work of Gentzen, one from Beth; both have their uses. Model Existence Theorems are proved, from which the usual known basic results follow. We remark that Craig interpolation lemmas have been proved model theoretically for these logics by Gabbay ([5], [6]) using ultraproducts. The existence of both ultra-product and consistency property proofs of the same result is a common phenomena in classical and infinitary logic. We also present extremely simple tableau proof systems for S 4, T, K and intuitionistic logics, systems whose completeness is an easy consequence of the Model Existence Theorems.

A Truth Value Semantics for Modal Logic1

Abstract: Publisher Summary This chapter discusses the truth value semantics for modal logic. The apodictic judgment differs from the assertory in that it suggests the existence of universal judgments from which the proposition can be inferred, while in the case of the assertory one, such a suggestion is lacking. If a proposition is advanced as possible, either the speaker is suspending judgment by suggesting that he or she knows no laws from which the negation of the proposition would follow or he says that the generalization of the negation is false. The chapter also discusses the canonical completeness theorem and the supposition that there are no irreducible modal facts.

A Generalization of Intuitionistic and Modal Logics

Abstract: Publisher Summary This chapter describes a generalization of intuitionistic and modal logics. It describes the way in which completeness theorems for intuitionistic and modal logics can be obtained as a special case of the completeness theorem of classic frameworks. Only two non-classic applications of structures—intuitionistic and S 4 -Modal logics—are considered in the chapter. Constant-domain intuitionistic and S 4 -Modal logics can be obtained merely by allowing all integers to belong to all prefixes.

A Survey of Decidability Results for Modal, Tense and Intermediate Logics1

Abstract: Publisher Summary This chapter discusses survey of decidability results for modal, tense, and intermediate logics. The chapter considers first-order theories based on these logics, in particular theories based on the intuitionistic predicate logic. Modal and tense logics are concerned with the addition of one or two additional unary connectives to the language of classical logic. The language of intuitionistic logic and its extensions is, of course, the same as that of classical logic. The main methods that can be employed in solving the decision problem of a given system are presented. Some of the main modal systems are discussed in the chapter. The algebraic models corresponding to intuitionistic logic are the well known Heyting (or Brouwerian) algebras.

Logical Consequence in Modal Logic: Alternative Semantic Systems for Normal Modal Logics

Abstract: Publisher Summary This chapter discusses the alternative semantic systems of normal modal logics. The novelty of the approach discussed in the chapter is twofold: first, to treat modal logics as consequence systems rather than logistic systems; and second, to view modal logics as extensions of non-modal ones. This second feature leads to the realization that certain usually desired properties of modal logics (e.g., compactness, weak completeness, and strong completeness) are at least in part determined by properties of the semantic systems of their non-modal “base.”

On Evaluating Deontic Logics

Abstract: Professor van Fraassen, in his paper ‘The Logic of Conditional Obligation’, argues that certain problems, raised for the minimal deontic logic D by contrary-to-duty imperatives, by the Good Samaritan paradox, and by Powers’ John and Suzy paradox, can be handled by his conditional logic of obligation CD. In my comments I shall for the most part neither attack nor support this claim: rather, I shall point out some reasons why I find it difficult to evaluate.

Descriptions, essences and quantified modal logic

Abstract: It is true, then, that a generalized statement of Arnauldic essentialism is equivalent to a qmlwff. But no known objection to quantified modal logic rightly or even reasonably dispatches any qmlwff to which essentialism commits us. This is scarcely a surprising state of affairs. For, the heart (I shall not say ‘essence’) of these complaints against quantified modal logic is that quantified modal logic cannot be given a coherent interpretation that preserves the distinction between $$\begin{gathered}\user1{ }\phi \hfill \\{\text{and}} \hfill \\\user1{ }\square \user1{ }\phi . \hfill \\\end{gathered}$$ Now when a logician charges that a system of logic cannot coherently be given its intended interpretation, and another logician demonstrably provides for that system a formal semantics that is complete and consistent and consonant with the underlying intuitions of the system, then it is universally concluded that the original objection has altogether satisfactorily been disposed of. Kripke and others have furnished, for systems of quantified modal logic, such complete and consistent semantics. Yet skepticism concerning quantified modal logic continues in some quarters vigorously to be pressed. The question is: what really is the basis of such skepticism?, for it is not, as all the world should know, that quantified modal logic cannot be interpreted. Whatever the answer may be, we should at least be prepared to consider the possibility that some of the apparent trouble for quantified modal logic proceeds from unintended and undetected breaches of such principles as semantic constancy.