Three-Valued Temporal Logic <Emphasis Type="Italic">Q</Emphasis><Subscript><Emphasis Type="Italic">t</Emphasis></Subscript> and Future Contingents

Prior’s three-valued modal logic Q was developed as a philosophically interesting modal logic. Thus, we should be able to modify Q as a temporal logic. Although a temporal version of Q was suggested by Prior, the subject has not been fully explored in the literature. In this paper, we develop a three-valued temporal logic Q _t and give its axiomatization and semantics. We also argue that Q _t provides a smooth solution to the problem of future contingents. Presented by Daniele Mundici     

Giles’s Game and the Proof Theory of Łukasiewicz Logic

In the 1970s, Robin Giles introduced a game combining Lorenzen-style dialogue rules with a simple scheme for betting on the truth of atomic statements, and showed that the existence of winning strategies for the game corresponds to the validity of formulas in Łukasiewicz logic. In this paper, it is shown that ‘disjunctive strategies’ for Giles’s game, combining ordinary strategies for all instances of the game played on the same formula, may be interpreted as derivations in a corresponding proof system. In particular, such strategies mirror derivations in a hypersequent calculus developed in recent work on the proof theory of Łukasiewicz logic. Presented by Daniele Mundici     

The Tense Logic for Master Argument in Prior’s Reconstruction

In this paper we examine Prior’s reconstruction of Master Argument [4] in some modal-tense logic. This logic consists of a purely tense part and Diodorean definitions of modal alethic operators. Next we study this tense logic in the pure tense language. It is the logic K _t 4 plus a new axiom (P): ‘p Λ G p ⊃ P G p’. This formula was used by Prior in his original analysis of Master Argument. (P) is usually added as an extra axiom to an axiomatization of the logic of linear time. In that case the set of moments is a total order and must be left-discrete without the least moment. However, the logic of Master Argument does not require linear time. We show what properties of the set of moments are exactly forced by (P) in the reconstruction of Prior. We make also some philosophical remarks on the analyzed reconstruction. Presented by Jacek Malinowski     

The Logic of Location

I consider the idea of a propositional logic of location based on the following semantic framework, derived from ideas of Prior. We have a collection L of locations and a collection S of statements such that a statement may be evaluated for truth at each location. Typically one and the same statement may be true at one location and false at another. Given this semantic framework we may proceed in two ways: introducing names for locations, predicates for the relations among them and an “at” preposition to express the value of statements at locations; or introduce statement operators which do not name locations but whose truth-conditional effect depends on the truth or falsity of embedded statements at various locations. The latter is akin to Prior’s approach to tense logic. In any logic of location there will be some basic operators which we can define. By ringing the changes on the topology of locations, different logical systems may be generated, and the challenge for the logician is then in each case to find operators, axioms and rules yielding a proof theory adequate to the semantics. The generality of the approach is illustrated with familiar and not so familiar examples from modal, tense and place logic, mathematics, and even the logic of games.

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The “Reversal of Generations” Phenomenon as Illustrated by the Lives of John Henry Newman and Abraham Lincoln

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Taking Degrees of Truth Seriously

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Paths to Late Midlife Well-Being for Women and Men: The Importance of Identity Development and Social Role Quality

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Arthur Prior and medieval logic

Though Arthur Prior is now best known for his founding of modern temporal logic and hybrid logic, much of his early philosophical career was devoted to history of logic and historical logic. This interest laid the foundations for both of his ground-breaking innovations in the 1950s and 1960s. Because of the important r?le played by Prior??s research in ancient and medieval logic in his development of temporal and hybrid logic, any student of Prior, temporal logic, or hybrid logic should be familiar with the medieval logicians and their work. In this article we give an overview of Prior??s work in ancient and medieval logic.     

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ALTERNATIVE AXIOMATICS AND COMPLEXITY OF DELIBERATIVE STIT THEORIES

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On Reduction Rules,Meaning-as-use,and Proof-theoretic Semantics

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Actuality, Necessity, and Logical Truth

The traditional view that all logical truths are metaphysically necessary has come under attack in recent years. The contrary claim is prominent in David Kaplan’s work on demonstratives, and Edward Zalta has argued that logical truths that are not necessary appear in modal languages supplemented only with some device for making reference to the actual world (and thus independently of whether demonstratives like ‘I’, ‘here’, and ‘now’ are present). If this latter claim can be sustained, it strikes close to the heart of the traditional view. I begin this paper by discussing and refuting Zalta’s argument in the context of a language for propositional modal logic with an actuality connective (section 1). This involves showing that his argument in favor of real world validity his preferred explication of logical truth, is fallacious. Next (section 2) I argue for an alternative explication of logical truth called general validity. Since the rule of necessitation preserves general validity, the argument of section 2 provides a reason for affirming the traditional view. Finally (section 3) I show that the intuitive idea behind the discredited notion of real world validity finds legitimate expression in an object language connective for deep necessity. Earlier versions of this paper were read at the universities of Graz, Maribor, and Salzburg, and at a workshop on the philosophy of logic at the National Autonomous University of Mexico (UNAM) in Mexico City. My thanks to those present at these events for many helpful suggestions. Thanks are also due to an anonymous referee for Philosophical Studies.     

Traditional Logic,Modern Logic and Natural Language

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Notes on Constructive Negation

We put together several observations on constructive negation. First, Russell anticipated intuitionistic logic by clearly distinguishing propositional principles implying the law of the excluded middle from remaining valid principles. He stated what was later called Peirce’s law. This is important in connection with the method used later by Heyting for developing his axiomatization of intuitionistic logic. Second, a work by Dragalin and his students provides easy embeddings of classical arithmetic and analysis into intuitionistic negationless systems. In the last section, we present in some detail a stepwise construction of negation which essentially concluded the formation of the logical base of the Russian constructivist school. Markov’s own proof of Markov’s principle (different from later proofs by Friedman and Dragalin) is described.