Gabriel Sandu

It is far from clear what is meant by logic or what should be meant by it. It is nevertheless reasonable to identify logic as the study of inferences and inferential relations. The obvious practical use of logic is in any case to help us to reason well, to draw good inferences. And the typical form the theory of any part of logic seems to be a set of rules of inference. This answer already introduces some structure into a discussion of the nature of logic, for in an inference we can distinguish the input called a premise or premises from the output known as the conclusion. 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. Conceiving of logic as the study of inference is nevertheless only the first approximation to the title question, in that it prompts more questions than it answers. It is not clear what counts as an inference or what a theory of such inferences might look like. What are the rules of inference based on? Where do we find them? The ultimate end

Tarski has exerted enormous influence not only on the development of mathematical logic, but on twentieth-century philosophy and philosophical analysis. This influence has been twofold, with the two components pulling in a sense in opposite directions. A comparison with the influence of the Vienna Circle provides an instructive vantage point in viewing Tarski’s influence. On the one hand, Tarski has provided powerful tools for logical analysis in philosophy. His first and most important contribution was to show that — and how — the crucial semantical concept of truth can be defined for various formal languages. They include the most central types of language used and studied by twentieth century logicians and philosophers. The subsequent work by Tarski and his school in creating contemporary model theory in its narrow sense as a branch of technical logic has likewise enriched tremendously the conceptual arsenal of philosophers. The extent and depth of Tarski’s influence is in fact easily underestimated. In comparison, Carnap’s well-known books in logical syntax and logical semantics did not in the same way contribute to the logical techniques that philosophers might find useful. A large part of Tarski’s influence lies in facilitating a switch of emphasis from purely syntactical to model-theoretical, semantical methods. It was Gödel’s and Tarski’s early work that forced logicians and philosophers for the first time to take seriously the distinction between semantical and syntactical concepts and methods. The general theoretical implications of the distinction were not realized at once. In the light of hindsight, it is for instance quite striking that Tarski still in his classical monograph repeatedly assimilates to each other the task of defining truth for some part of mathematics or logic and axiomatizing its truths in a sense that involves explicit syntactical rules of inference. It was nevertheless Tarski more than anyone else who began to develop and exploit systematically model-theoretical concepts in logical theory. In comparison, Gödel never developed systematically the model-theoretical aspects of logic. Others, for instance Quine, stubbornly remained within the syntactical tradition. Carnap, who did switch his attention from the syntactical to a semantical approach, was not powerful enough a logician to create a sufficiently rich store of model-theoretical techniques and ideas to be equally helpful in philosophical analysis. In contrast, several major developments in analytic philosophy have had their starting points in Tarski’s ideas. Well-known cases in point are Davidson’s truth-conditional semantics and the possible worlds semantics developed by Tarski’s favorite former student Richard Montague

We consider two versions of truth as grounded in verification procedures: Dummett's notion of proof as an effective way to establish the truth of a statement and Hintikka's GTS notion of truth as given by the existence of a winning strategy for the game associated with a statement. Hintikka has argued that the two notions should be effective and that one should thus restrict one's attention to recursive winning strategies. In the context of arithmetic, we show that the two notions do not coincide: on the one hand, proofs in PA do not yield recursive winning strategies for the associated game; on the other hand, there is no sound and effective proof procedure that captures recursive GTS truths. We then consider a generalized version of Game Theoretical Semantics by introducing games with backward moves. In this setting, a connection is made between proofs and recursive winning strategies. We then apply this distinction between two kinds of verificationist procedures to a recent debate about how we recognize the truth of Gödelian sentences.

The paper argues that there are two main kinds of joint action, direct joint bringing about (or performing) something (expressed in terms of a DO-operator) and jointly seeing to it that something is the case (expressed in terms of a Stit-operator). The former kind of joint action contains conjunctive, disjunctive and sequential action and its central subkinds. While joint seeing to it that something is the case is argued to be necessarily intentional, direct joint performance can also be nonintentional. Actions performed by social groups are analyzed in terms of the notions of joint action (basically DO and Stit).A precise semantical analysis of the aforementioned kinds of joint action is given in terms of time-trees. With each participant a tree is connected, and the trees are joined defining joint possible worlds in terms of state-expressing nodes from the trees. Sentences containing DO and Stit are semantically evaluated with respect to such joint possible worlds. Intentional joint actions are characterized in terms of the notion of we-intention (joint intention), characterized formally by means of a special operator.

In this paper we relax the assumption that the logical constants of ordinary first-order logic be linearly ordered. As a consequence, we shall have formulas involving not only partially ordered quantifiers, but also partially ordered connectives. The resulting language, called the language of informational independence will be given an interpretation in terms of games of imperfect information. The II-logic will be seen to have some interesting properties: It is very natural to define in this logic two negations, weak negation as failure to verify a sentence, and strong negation as the existence of a falsifying strategy: One can express in this logic complete problems of finite structures, like the non-connectedness and 3-colorability of finite graphs, the satisfiability problem for Boolean circuits built up from NAND-gates, etc

Deflationists have argued that truth is an ontologically thin property which has only an expressive function to perform, that is, it makes possible to express semantic generalizations like 'All the theorems are true', 'Everything Peter said is true', etc. Some of the deflationists have also argued that although truth is ontologically thin, it suffices in conjunctions with other facts not involving truth to explain all the facts about truth. The purpose of this paper is to show that in the case of arithmetic, it is difficult to combine the expressive with the explanatory function of truth if the latter is understood in a deflationist way. We will make our point by investigating several logical systems: first-order logic, full second-order logic, and existential second-order (?11-) logic.

I introduce a formal language called the language of informational independence (IL-language, for short) that extends an ordinary first-order language in a natural way. This language is interpreted in terms of semantical games of imperfect information. In this language, one can define two negations: (i) strong or dual negation, and (ii) weak or contradictory negation. The latter negation, unlike the former, can occur only sentence-initially. Then I argue that, to a certain extent, the two negations match the distinction existing in natural languages between sentential and constituent negation. As a corollary, I derive the fact that there are no mechanical rules for forming the contradictory negation of an English sentence.

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

Linguistique et philosophie logique du langage: deux traditions de pensee que bien des choses opposent. La premiere est plutot mentaliste, et orientee vers l'etude de la syntaxe; la seconde, plus preoccupee de semantique, cherche volontiers le sens dans les conditions de verite des phrases. Ce portrait n'est pas faux, mais il est incomplet: entre logique et linguistique, les relations n'ont pas ete, ne sont pas que d'opposition. Dans cet ouvrage, les auteurs proposent une sorte d'histoire conceptuelle des interactions fecondes entre les deux disciplines au cours du XXe siecle. La premiere partie, consacree a la notion de categorie semantique et/ou syntaxique, raconte comment les theories a priori de la signification (Husserl, Frege, Russell) ont progressivement donne lieu au programme des grammaires categorielles, d'inspiration plus descriptive et empirique. La deuxieme partie traite d'un autre episode, datant des annees cinquante a soixante-dix, et lie a la naissance des grammaires generatives: celui au cours duquel l'opposition entre la these avancee par Chomsky de l'autonomie de la syntaxe, et l'idee de la priorite conceptuelle de la semantique, soutenue par des logiciens comme Montague, vient au premier plan. Enfin la troisieme partie traite des recherches tout a fait contemporaines concernant l'etude des expressions indefinies et des relations anaphoriques qu'elles soutiennent, theme ou se dessinent des convergences nouvelles entre l'analyse logique et l'analyse linguistique: la comprehension des rapports entre generalite et reference dans les langages naturels y gagnera certainement en finesse et en adequation empirique.

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 [9] under the header of independence-friendly 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 [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