Gabriel Sandu

Game-Theoretical Semantics

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.

Independence-Friendly Logic: A Game-Theoretic Approach

Preface 1. Introduction 2. Game theory 3. First-order logic 4. Independence-friendly (IF) logic 5. Properties of IF logic 6. Expressive power of IF logic 7. Probabilistic IF logic 8. Further topics References Index.

On the logic of informational independence and its applications

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.

Partiality and games: propositional logic

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.

PARTIALLY ORDERED CONNECTIVES

We show that a coherent theory of partially ordered connectives can be developed along the same line as partially ordered quantification. We estimate the expressive power of various partially ordered connectives and use methods like Ehrenfeucht games and infinitary logic to get various undefinability results.

The fallacies of the new theory of reference

The so-called New Theory of Reference (Marcus. Kripke etc.) is inspired by the insight that in modal and intensional contexts quantifiers presuppose nondescriptive unanalyzable identity criteria which do not reduce to any descriptive conditions. From this valid insight the New Theorists fallaciously move to the idea that free singular terms can exhibit a built-in direct reference and that there is even a special class of singular terms (proper names) necessarily exhibiting direct reference. This fallacious move has been encouraged by a mistaken belief in the substitutional interpretation of quantifiers. by the myth of the de re reference, and a mistaken assimilation of “direct reference” to ostensive (perspectival) identification. The de ditto vs. de re contrast does not involve direct reference, being merely a matter of rule-ordering (“scope”).

Aspects of Compositionality

We introduce several senses of the principle ofcompositionality. We illustrate the difference between them with thehelp of some recent results obtained by Cameron and Hodges oncompositional semantics for languages of imperfect information.

Equilibrium semantics of languages of imperfect information

Abstract In this paper, we introduce a new approach to independent quantifiers, as originally introduced in Informational independence as a semantic phenomenon by Hintikka and Sandu (1989) [9] under the header of independence-friendly (IF) languages. Unlike other approaches, which rely heavily on compositional methods, we shall analyze independent quantifiers via equilibriums in strategic games. In this approach, coined equilibrium semantics, the value of an IF sentence on a particular structure is determined by the expected utility of the existential player in any of the game’s equilibriums. This approach was suggested in Henkin quantifiers and complete problems by Blass and Gurevich (1986) [2] but has not been taken up before. We prove that each rational number can be realized by an IF sentence. We also give a lower and upper bound on the expressive power of IF logic under equilibrium semantics.

Partially ordered connectives and finite graphs

We prove that connectivity of finite graphs is not expressible in the extension of first-order logic by any set of unary generalized quantifiers. On the other hand, we show that connectivity is definable by the simplest partially ordered connective D 1,1. As a consequence, D 1,1 is not definable in terms of unary quantifiers.

What is the Logic of Parallel Processing

We can associate with each consistent formula F of first-order logic a computing device as its representation. This computing device is one which will calculate the Skolem functions of F (for a denumerable domain). When two such devices are operating in parallel, the resulting architecture does not necessarily represent any ordinary first-order formula, but it will represent a formula in independence-friendly (IF) logic, which hence can be considered as a true logic of parallel processing. In order to preserve representability by a digital automaton (Turing machine), a nonstandard (constructivistic) interpretation of the logic in question has to be adopted. It is obtained by restricting the Skolem functions available to verify a formula F to recursive ones, as in the Godel’s Dialectica interpretation.