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 thede re reference, and a mistaken assimilation of direct reference to ostensive (perspectival) identification. Thede dicto vs.de re contrast does not involve direct reference, being merely a matter of rule-ordering (scope).The New Theorists' thesis of the necessity of identities of directly referred-to individuals is a consequence of an unmotivated and arbitrary restriction they tacitly impose on the identification of individuals. (shrink)

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.

Part I of Frege’s Grundgesetze is devoted to the “exposition [Darlegung]” of his formal system.

The paper attempts to give a solution to the Fitch's paradox though the strategy of the reformulation of the paradox in temporal logic, and a notion of knowledge which is a kind of ceteris paribus modality. An analogous solution has been offered in a different context to solve the problem of metaphysical determinism.

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.

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. (shrink)

Both Frege's Grundgesetze, and Lagrange's treatises on analytical functions pursue a foundational purpose. Still, the former's program is not only crucially different from the latter's. It also depends on a different idea of what foundation of mathematics should be like . Despite this contrast, the notion of function plays similar roles in their respective programs. The purpose of my paper is emphasising this similarity. In doing it, I hope to contribute to a better understanding of Frege's logicism, especially in relation (...) to its crucial differences with a set-theoretic foundational perspective. This should also spread some light on a question arisen by J. Hintikka and G. Sandu in a widely discussed paper, namely whether Frege should or should not be credited with the notion of arbitrary function underlying our standard interpretation of second-order logic. In section 1, I account for Lagrange's notion of function. In section 2, I advance some remarks on connected historical matters. This will provide an appropriate framework for discussing the role played by the notion of function in Frege's Grundgesetze. Section 3 is devoted to this. Some concluding remarks will close the paper. (shrink)

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. (shrink)

Henkin quantifiers have been introduced in Henkin (1961). Walkoe (1970) studied basic model-theoretical properties of an extension $L_{*}^{1}$ (H) of ordinary first-order languages in which every sentence is a first-order sentence prefixed with a Henkin quantifier. In this paper we consider a generalization of Walkoe's languages: we close $L_{*}^{1}$ (H) with respect to Boolean operations, and obtain the language L¹(H). At the next level, we consider an extension $L_{*}^{2}$ (H) of L¹(H) in which every sentence is an L¹(H)-sentence prefixed with (...) a Henkin quantifier. We repeat this construction to infinity. Using the (un)definability of truth - in - N for these languages, we show that this hierarchy does not collapse. In addition, we compare some of the present results to the ones obtained by Kripke (1975), McGee (1991), and Hintikka (1996). (shrink)

In this paper we show that first-order languages extended with partially ordered connectives and partially ordered quantifiers define, under a certain interpretation, their own truth-predicate. The interpretation in question is in terms of games of imperfect information. This result is compared with those of Kripke and Feferman.

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 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. (shrink)

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. (shrink)

Independence‐Friendly logic introduced by Hintikka and Sandu studies patterns of dependence and independence of quantifiers which exceed those found in ordinary first‐order logic. The present survey focuses on the game‐theoretical interpretation of IF‐logic, including connections to solution concepts in classical game theory, but we shall also present its compositional interpretation together with its connections to notions of dependence and dependence between terms.

This chapter begins with a discussion of the three phases of the interaction between logic and linguistics on the nature of universal grammar. It then attempts to reconstruct the dynamics and interactions between these approaches in logic and in linguistic theory, which represent the major landmarks in the quest for the individuation of the universal structure of language.

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. (shrink)

Frege has one magnificent achievement to his credit, viz. the creation of modern formal logic. As a philosopher and as a theoretical logician, he was nevertheless as parochial as he was, geographically speaking. Hence Frege’s concepts and problems offer singularly unfortunate starting points for constructive work in the foundations of logic and mathematics. Even if he is right in some of his views, they depend on severely restrictive assumptions that have to be noted and eliminated. These restrictive assumptions have not (...) been sufficiently acknowledged in most of the recent discussions of Frege’s ideas. This has given rise to wild overstatements of Frege’s achievements. To give only two examples, Frege is sometimes said to have had within his reach “the concepts necessary to frame the notion of the completeness of a formalization of logic as well as soundness.” Or, he is credited with the creation of mathematical logic “issuing in a formal system logic incorporating propositional calculus, first- and second-order quantification theory, and a theory of sets developed within second-order quantification theory.” We will soon see that both these claims are not only severely exaggerated but that they miss some of the most characteristic features of Frege’s entire way of thinking about logical and semantical matters. (shrink)

The idea behind these games is to obtain an alternative characterization of logical notions cherished by logicians such as truth in a model, or provability. We offer a quick survey of Hintikka's evaluation games, which offer an alternative notion of truth in a model for first-order langauges. These are win-lose, extensive games of perfect information. We then consider a variation of these games, IF games, which are win-lose extensive games of imperfect information. Both games presuppose that the meaning of the (...) basic vocabulary of the language is given. To give an account of the linguistic conventions which settle the meaning of the basic vocabulary, we consider signaling games, inspired by Lewis' work. We close with IF probabilistic games, a strategic variant of IF games which combines semantical games with von Neumann's minimax theorem. (shrink)

This chapter explores logical semantics, that is, the structural meaning of logical expressions like connectives, quantifiers, and modalities. It focuses on truth-theoretical semantics for formalized languages, a tradition emerging from Carnap's and Tarski's work in the first half of the last century that specifies the meaning of these expressions in terms of the truth-conditions of the sentences in which they occur. It considers Tarski-style definitions of the semantics of a given language in a stronger metalanguage, Tarski's impossibility results, and attempts (...) to overcome them in the post-Tarskian tradition. (shrink)

Linguistique et philosophie logique du langage : deux traditions de pensée que bien des choses opposent. La première est plutôt mentaliste, et orientée vers l’étude de la syntaxe; la seconde, plus préoccupée de sémantique, cherche volontiers le sens dans les conditions de vérité des phrases. Ce portrait n’est pas faux, mais il est incomplet : entre logique et linguistique, les relations n’ont pas été, ne sont pas que d’opposition.Dans cet ouvrage, les auteurs proposent une sorte d’histoire conceptuelle des interactions fécondes (...) entre les deux disciplines au cours du XXe siècle. La première partie, consacrée à la notion de catégorie sémantique et/ou syntaxique, raconte comment les théories a priori de la signification ont progressivement donné lieu au programme des grammaires catégorielles, d’inspiration plus descriptive et empirique.La deuxième partie traite d’un autre épisode, datant des années cinquante à soixante-dix, et lié à la naissance des grammaires génératives : celui au cours duquel l’opposition entre la thèse avancée par Chomsky de l’autonomie de la syntaxe, et l’idée de la priorité conceptuelle de la sémantique, soutenue par des logiciens comme Montague, vient au premier plan.Enfin la troisième partie traite des recherches tout à fait contemporaines concernant l’étude des expressions indéfinies et des relations anaphoriques qu’elles soutiennent, thème où se dessinent des convergences nouvelles entre l’analyse logique et l’analyse linguistique : la compréhension des rapports entre généralité et référence dans les langages naturels y gagnera certainement en finesse et en adéquation empirique. (shrink)

In his article The Foundations of Mathematics (1925) Ramsey was concerned with the nature of the statements of 'pure mathematics' and the way these statements differ from those in empirical sciences. He thought that the answer given to these questions by Hilbert and the formalist school according to which mathematical statements are meaningless formulas, is unsatisfactory for several reasons, which will not be discussed here. He also expressed serious doubts about the intuitionist program developed by Brouwer and Weyl. It is (...) the logicist school of Frege, Russell and Whitehead, which attempted to reduce mathematics to few logical concepts, that came closest to Ramsey’s views, although he was not completely satisfied with it, either. (shrink)

This paper is a short survey of different languages with imperfect information introduced in. The imperfect information concerns both quantifiers and connectives. At the end, I will sketch a connection between these languages and linear 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. (shrink)

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. (shrink)

Team semantics is a highly general framework for logics which describe dependencies and independencies among variables. Typically, the dependencies considered in this context are properties of sets of configurations or data records. We show how team semantics can be further generalized to support languages for the discussion of interventionist counterfactuals and causal dependencies, such as those that arise in manipulationist theories of causation. We show that the “causal teams” we introduce in the present paper can be used for modelling some (...) classical counterfactual scenarios which are not captured by the usual causal models. We then analyse the basic properties of our counterfactual languages and discuss extensively the differences with respect to the Lewisian tradition. (shrink)

In this paper we consider truth as a vague predicate and inquire into the relation between truth and definite truth. We use some tools from modal logic to clarify this distinction, as done in McGee . Finally, we consider the question whether some of the results given by McGee can be transferred to the case in which the underlying logic is stronger than first-order logic. The result will be seen to be negative.

This is a collection of studies by contemporary Romanian philosophers of science. As it happens, it ruffled a lot of feathers, to the extent that the book is largely ignored in Romania, although it does quite well internationally, with some really good citations. Check it out!

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. (shrink)

Our aim in this paper is to provide a referential account of functional anaphora within a Skolem functions framework. We will give an interpretation of indefinite NPs as Skolem terms in order to show that the referential link established between an anaphoric pronoun and its antecedent is a descriptive one. Then we will argue that functional anaphora can be understood as a particular kind of E-type pronouns, in the sense that, for a large corpus, the pronoun can be replaced by (...) a descriptive expression of a first-order language. It will next be shown that the functional framework can be fruitfully applied to other kinds of sentences involving pronominal anaphora,especially in modal contexts. (shrink)

The article presents Erik Stenius' conception of logical constants and compares it with the standard approach.