What is modal logic

13 EADERS of Mr. Bertrand Russell's philosophical £v works know that one of his pupils before the outbreak of the war, an Austrian, Mr. Ludwig Wittgenstein, caused him to change his views in some important particulars. Curiosity can now be satisfied. The “Tractatus Logico-Philosophicus “which Mr. Ogden has included in his new library of philosophy is a remarkable and strikingly original work. It is published in German and English in parallel pages. It is difficult to appreciate the reason for this, seeing that the author is evidently familiar with our language and has himself carefully revised the proofs of the translation. Also we should have liked to have the Tractatus without Mr. Russell's Introduction, not, we hasten to add, on account of any fault or shortcoming in that introduction, which is highly appreciative and in part a defence of himself, in part explanatory of the author, but for the reason that good wine needs no bush and that Mr. Russell's bush has the unfortunate effect of dulling the palate instead of whetting the appetite. In his penultimate sentence Mr. Russell says; “To have constructed a theory of logic which is not at any point obviously wrong is to have achieved a work of extraordinary difficulty and importance.” We agree, but how uninspiring when compared with Mr. Wittgenstein's own statement of aim: “What can be said at all can be said clearly, and whereof one cannot speak, thereof one must be silent.” Tractatus Logico-Philosophicus. By Ludwig Wittgenstein. (International Library of Psychology, Philosophy and Scientific Method.) Pp. 189. (London: Kegan Paul and Co., Ltd.; New York: Harcourt, Brace and Co., Inc., 1922.) 10s. 6d. net.

Tractatus Logico-Philosophicus

Convention: a philosophical study

This chapter is devoted to a more detailed examination of game theory. Game theory is an important tool for analyzing strategic behavior, is concerned with how individuals make decisions when they recognize that their actions affect, and are affected by, the actions of other individuals or groups. Strategic behavior recognizes that the decision-making process is frequently mutually interdependent. Game theory is the study of the strategic behavior involving the interaction of two or more individuals, teams, or firms, usually referred to as players. Two game theoretic scenarios were examined in this chapter: Simultaneous-move and multi-stage games. In simultaneous-move games the players effectively move at the same time. A normal-form game summarizes the players, possible strategies and payoffs from alternative strategies in a simultaneous-move game. Simultaneous-move games may be either noncooperative or cooperative. In contrast to noncooperative games, players of cooperative games engage in collusive behavior. A Nash equilibrium, which is a solution to a problem in game theory, occurs when the players’ payoffs cannot be improved by changing strategies. Simultaneous-move games may be either one-shot or repeated games. One-shot games are played only once. Repeated games are games that are played more than once. Infinitely-repeated games are played over and over again without end. Finitely-repeated games are played a limited number of times. Finitely-repeated games have certain or uncertain ends.

An Introduction to Game Theory

Logical Foundations of Probability By Rudolf Carnap Pp xvii + 607 (London: Routledge and Kegan Paul, Ltd, 1951) 42s net

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Logical Foundations of Probability

Game theory is a mathematical tool to study the behavior of independent agents in strategic interaction. Reasoning and communication have an essentially strategic aspect. Game theoretic is thus a suitable tool to illuminate the interactive aspects of logic and language.

Game-Theoretical Semantics

Bringing together over twenty years of research, this book gives a complete overview of independence-friendly logic. It emphasizes the game-theoretical approach to logic, according to which logical concepts such as truth and falsity are best understood via the notion of semantic games. The book pushes the paradigm of game-theoretical semantics further than the current literature by showing how mixed strategies and equilibria can be used to analyze independence-friendly formulas on finite models. The book is suitable for graduate students and advanced undergraduates who have taken a course on first-order logic. It contains a primer of the necessary background in game theory, numerous examples and full proofs.

Independence-friendly logic : a game-theoretic approach

Publisher Summary This chapter provides an overview of the concept of logic. Logic can be identified as the study of inferences and inferential relations. The practical use of logic is in any case to reason well, and to draw good inferences. The transition from a premise or a number of premises to the conclusion is governed by a rule of inference. If the inference is in accordance with the appropriate rule, it is called valid. Rules of inference are often thought of as the alpha and omega of logic. Inferences can be either deductive, that is, necessarily truth preserving, or ampliative, that is, not necessarily truth preserving. Philosophers generally consider deductive reasoning as the paradigmatic type of inference. However, there is a widespread view among nonphilosophers that might be called the Sherlock Holmes view. According to it all, good reasoning including ampliative reasoning, turns on “deductions” and “logic.” Concepts related to logic and inference, deductive inference and information, analytic inferences, and ampliative inference are discussed. Details of logical and extralogical systematization are provided. It also discusses concepts related to syntax and semantics.

What is Logic

We shall introduce in this paper a language whose formulas will be interpreted by games of imperfect information. Such games will be defined in the same way as the games for first-order formulas except that the players do not have complete information of the earlier course of the game. Some simple logical properties of these games will be stated together with the relation of such games of imperfect information to higher-order logic. Finally, a set of applications will be outlined.

On the logic of informational independence and its applications

We study partiality in propositional logics containing formulas with either undefined or over-defined truth-values. Undefined values are created by adding a four-place connective W termed transjunction to complete models which, together with the usual Boolean connectives is shown to be functionally complete for all partial functions. Transjunction is seen to be motivated from a game-theoretic perspective, emerging from a two-stage extensive form semantic game of imperfect information between two players. This game-theoretic approach yields an interpretation where partiality is generated as a property of non-determinacy of games. Over-defined values are produced by adding a weak, contradictory negation or, alternatively, by relaxing the assumption that games are strictly competitive. In general, particular forms of extensive imperfect information games give rise to a generalised propositional logic where various forms of informational dependencies and independencies of connectives can be studied.

Gabriel Sandu

Papers

Game-Theoretical Semantics

Independence-friendly logic : a game-theoretic approach

What is Logic

On the logic of informational independence and its applications

Partiality and games: propositional logic