Showing papers on "Normal modal logic published in 1989"

Duality between modal algebras and neighbourhood frames

Abstract: This paper presents duality results between categories of neighbourhood frames for modal logic and categories of modal algebras (i.e. Boolean algebras with an additional unary operation). These results extend results of Goldblatt and Thomason about categories of relational frames for modal logic.

Notes on conditional logic

Abstract: This paper consists of some lecture notes in which conditional logic is treated as an extension of modal logic. Completeness and filtration theorems are provided for some basis systems.

Modal Logic and Self-Reference

Abstract: Ever since Epimenides made his startling confession, philosophers and mathematicians have been fascinated by self-reference Of course, mathematicians are not free to admit this

An algebraic approach to intuitionistic modal logics in connection with intermediate predicate logics

Abstract: Modal counterparts of intermediate predicate logics will be studied by means of algebraic devise. Our main tool will be a construction of algebraic semantics for modal logics from algebraic frames for predicate logics. Uncountably many examples of modal counterparts of intermediate predicate logics will be given.

Is Alethic Modal Logic Possible

Abstract: The title of my paper may appear paradoxical, misplaced, or even worse, out of date. The possibility of a reasonable modal logic was denied by Quine on philosophical grounds, but his objections have been dead for a while, even though they have not yet been completely buried.1 What has made a crucial difference is the development of what has generally been taken to be a viable semantics (model theory) for modal logic.2 This semantics has provided a basis from which Quine’s objections can apparently be answered satisfactorily and which yields a solid foundation for the different axiom systems for modal logic. Thus the question of the possibility of modal logic has apparently been disposed of for good, and my title question accordingly may seem pointless.

Quantified Modal Logic and the Plural De Re

Abstract: within a language whose only modal operators are the box and the diamond; other modal idioms cannot be expressed within such a language at all. Nonetheless, quantified modal logic has enjoyed considerable success in uncovering and explaining ambiguities in modal sentences and fallacies in modal reasoning. A prime example of this success is the now standard analysis of the distinction between modality de dido and modality de re. The analysis has been applied first and foremost to modal sentences containing definite descriptions. Such sentences are often ambiguous between an interpretation de dicto, according to which a modal property is attributed to a proposition (or, on some views, a sentence), and an interpretation de re, according to which a modal property is attributed to an individual. When these sentences are translated into the language of quantified modal logic, the de dicfo/de re ambiguity turns out to involve an ambiguity of scope. If the definite description is within the scope of the modal operator, then the operator attaches to a complete sentence, and the resulting sentence is de dicto. If the definite description is outside the scope of the modal operator, then the operator attaches to a predicate to form a modal predicate, and the resulting sentence is de re. Quantified modal logic has the resources to clarify and disambiguate English modal sentences containing definite descriptions. In this paper, I explore to what extent the analysis in terms of scope can be applied to modal sentences containing denoting phrases other than definite descriptions, phrases such as ‘some F’ and ‘every F.1 I will focus upon categorical modal sentences of the following two forms: mo s a1 idioms must be artificially restructured if they are to be expressed

Modal theorem proving: an equational viewpoint

Abstract: We propose a new method for automated theorem proving in first order modal logic. Essentially, the method consists in a translation of modal logic into a specially designed typed first order logic cal led Path Logic, such that classical modal systems (first order Q, T, 04, S4, S5) can be characterized by sets of equations. The question of modal theorem proving then amounts to classical theorem proving in some equational theories. Different methods can be investigated and in this paper we cons ider Resolution. We may use Resolution with Paramodulation, or a combination of Resolution and Rewriting techniques. In both cases, known results provide "free of charge" a framework immediately applicable to Path Logic, with completeness theorems. Considering efficiency, the Rewriting method seems better and we present here in details its application to Path Logic. In particular we show how it is possible to define a special kind of skolemisation and design a unification algorithm which insures that two clauses will always have a finite set of resolvents.

A new proof of the fixed-point theorem of provability logic.

Abstract: We give a simple semantic proof of the fixed-point theorem of the modal system G (also known as GL, PRL, L, and K4W). It is modeled after a syntactic proof of Sambin, yet is simpler than his, due to our taking advantage of the Kripke semantics for G. The advantages of this particular version are that it is les complicated and the fixed-point so obtained has the same general «appearance» as the original formula

The Equivalence of the Disjunction and Existence Properties for Modal Arithmetic

Abstract: In a modal system of arithmetic, a theory S has the modal disjunction property if whenever S ⊢ □φ ∨ □ψ, either S ⊢ □φ or S ⊢ □ψ. S has the modal numerical existence property if whenever S ⊢ ∃x □φ(x), there is some natural number n such that S ⊢ □φ(n). Under certain broadly applicable assumptions, these two properties are equivalent.

Medium modal logic-formal system and semantics

Abstract: The authors construct a medium modal logic based on medium logic. They present three systems, MT, MS/sub 4/, and MS/sub 5/, which are, respectively, extensions of classical modal logic systems T, S/sub 4/, and S/sub 5/. The authors take medium set theory as a metalanguage to study semantics, proof soundness, and completeness. >

A Standard System of Deontic Logic

Abstract: Not much is certain in deontic logic. There are not many theorems, in any system, which are undisputed, i.e. with regard to which one or more authors have not stated that they cannot be accepted as a rational reconstruction of normative reasoning. There is, nevertheless, a formal system on which, although it has been disputed as a whole as well as with regard to its theorems, several other systems are founded, which other systems can be regarded as extensions of the first-mentioned system. One may therefore to a certain extent rightly speak of a ‘standard system of deontic logic’. This is even more justified by the fact that alternative systems have often been developed as a reaction to this system. Every deontic logician has to determine, in one way or another, his attitude towards this standard system.