Frame definability, canonicity and cut elimination in common sense modal predicate logics

The paper presents semantically complete Hilbert-style systems for some variants of common sense modal predicate logic proposed by van Benthem and further developed by Seligman. The paper also investigates frame definability in the logics and shows what axiom schema is canonical in the logics. In addition to these semantic investigations on the logics, the paper provides the sequent calculi for some of the logics which enjoy cut elimination theorem.

Modal logic

Modal logic, narrowly conceived, is the study of principles of reasoning involving necessity and possibility. More broadly, it encompasses a number of structurally similar inferential systems. In this sense, deontic logic (which concerns obligation, permission and related notions) and epistemic logic (which concerns knowledge and related notions) are branches of modal logic. Still more broadly, modal logic is the study of the class of all possible formal systems of this nature. It is customary to take the language of modal logic to be that obtained by adding one-place operators ‘□’ for necessity and ‘◇’ for possibility to the language of classical propositional or predicate logic. Necessity and possibility are interdefinable in the presence of negation: □A↔¬◊¬A and ◊A↔¬□¬A hold. A modal logic is a set of formulas of this language that contains these biconditionals and meets three additional conditions: it contains all instances of theorems of classical logic; it is closed under modus ponens (that is, if it contains A and A→B it also contains B); and it is closed under substitution (that is, if it contains A then it contains any substitution instance of A; any result of uniformly substituting formulas for sentence letters in A). To obtain a logic that adequately characterizes metaphysical necessity and possibility requires certain additional axiom and rule schemas: K □(A→B)→(□A→□B) T □A→A 5 ◊A→□◊A Necessitation A/□A. By adding these and one of the □–◇ biconditionals to a standard axiomatization of classical propositional logic one obtains an axiomatization of the most important modal logic, S5, so named because it is the logic generated by the fifth of the systems in Lewis and Langford’s Symbolic Logic (1932). S5 can be characterized more directly by possible-worlds models. Each such model specifies a set of possible worlds and assigns truth-values to atomic sentences relative to these worlds. Truth-values of classical compounds at a world w depend in the usual way on truth-values of their components. □A is true at w if A is true at all worlds of the model; ◇A, if A is true at some world of the model. S5 comprises the formulas true at all worlds in all such models. Many modal logics weaker than S5 can be characterized by models which specify, besides a set of possible worlds, a relation of ‘accessibility’ or relative possibility on this set. □A is true at a world w if A is true at all worlds accessible from w, that is, at all worlds that would be possible if w were actual. Of the schemas listed above, only K is true in all these models, but each of the others is true when accessibility meets an appropriate constraint. The addition of modal operators to predicate logic poses additional conceptual and mathematical difficulties. On one conception a model for quantified modal logic specifies, besides a set of worlds, the set Dw of individuals that exist in w, for each world w. For example, ∃x□A is true at w if there is some element of Dw that satisfies A in every possible world. If A is satisfied only by existent individuals in any given world ∃x□A thus implies that there are necessary individuals; individuals that exist in every accessible possible world. If A is satisfied by non-existents there can be models and assignments that satisfy A, but not ∃xA. Consequently, on this conception modal predicate logic is not an extension of its classical counterpart. The modern development of modal logic has been criticized on several grounds, and some philosophers have expressed scepticism about the intelligibility of the notion of necessity that it is supposed to describe.

THE QUANTIFIED ARGUMENT CALCULUS

I develop a formal logic in which quantified arguments occur in argument positions of predicates. This logic also incorporates negative predication, anaphora and converse relation terms, namely, additional syntactic features of natural language. In these and additional respects, it represents the logic of natural language more adequately than does any version of Frege’s Predicate Calculus. I first introduce the system’s main ideas and familiarize it by means of translations of natural language sentences. I then develop a formal system built on these principles, the Quantified Argument Calculus or Quarc. I provide a truth-value assignment semantics and a proof system for the Quarc. I next demonstrate the system’s power by a variety of proofs; I prove its soundness; and I comment on its completeness. I then extend the system to modal logic, again providing a proof system and a truth-value assignment semantics. I proceed to show how the Quarc versions of the Barcan formulas, of their converses and of necessary existence come out straightforwardly invalid, which I argue is an advantage of the modal Quarc over modal Predicate Logic as a system intended to capture the logic of natural language.

Zur Einheit der modalen Syllogistik des Aristoteles

On the unity of modal syllogistics in Aristotle. The goal of this paper is an interpretation of Aristotle’s modal syllogistics closely oriented on the text using the resources of modern modal predicate logic. Modern predicate logic was successfully able to interpret Aristotle’s assertoric syllogistics uniformly, that is, with one formula for universal premises. A corresponding uniform interpretation of modal syllogistics by means of modal predicate logic is not possible. This thesis does not imply that a uniform view is abandoned. However, it replaces the simple unity of the assertoric by the complex unity of the modal. The complexity results from the fact that though one formula for universal premises is used as the basis, it must be moderated if the text requires.Aristotle introduces his modal syllogistics by expanding his assertoric syllogistics with an axiom that links two apodictic premises to yield a single apodictic sentence. He thus defines a regular modern modal logic. By means of the regular modal logic that is thus defined, he is able to reduce the purely apodictic syllogistics to assertoric syllogistics. However, he goes beyond this simple structure when he looks at complicated inferences.In order to be able to link not only premises of the same modality, but also premises with different modalities, he introduces a second axiom, the T-axiom, which infers from necessity to reality or – equivalently – from reality to possibility. Together, the two axioms, the axiom of regularity and the T-axiom, define a regular T-logic. It plays an important role in modern logic. In order to be able to account for modal syllogistics adequately as a whole, another modern axiom is also required, the so-called B-axiom. It is very difficult to decide whether Aristotle had the B-axiom. Each of the two last named axioms is sufficient to achieve the required contextual moderation of the basic formula for universal propositions.

Constant Domain Quantified Modal Logics Without Boolean Negation

his paper provides a sound and complete axiomatisation for constant domain modal logics without Boolean negation. This is a simpler case of the difficult problem of providing a sound and complete axiomatisation for constant-domain quantified relevant logics, which can be seen as a kind of modal logic with a two-place modal operator, the relevant conditional. The completeness proof is adapted from a proof for classical modal predicate logic (I follow James Garson’s 1984 presentation of the completeness proof quite closely), but with an important twist, to do with the absence of Boolean negation.