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On Dialogues, Predication and Elementary Sentences
Abstract
”. S. Rahman/N. Clerbout. Vol. II, pp. 1-38. .(CIELO)
En el presente trabajo confrontamos el análisis del axioma de elección de Martin - Löf con la posición de J. Hintikka respecto de este axioma. Hintikka afirma que su Semántica Teorética de Juegos (STJ) para una Lógica de la Independencia Amigable (Lógica IA), justifica el axioma de elección de Zermelo en un sentido de primer orden perfectamente aceptable para los constructivistas. De hecho, los resultados de Martin - Löf conducen a las siguientes consideraciones:La versión preferida de Hintikka del axioma de elección, es ciertamente aceptable para los constructivistas y su significado no implica una lógica de orden superior.Sin embargo, la versión aceptable para los constructivistas se basa en una consideración intensional sobre las funciones. La extensionalidad es el corazón de la comprensión clásica del axioma de Zermelo y esta es la razón real tras el rechazo constructivista de éste.En general, las características de dependencia e independencia que motivan la Lógica IA, pueden formularse en el marco de la Teoría de Tipos Constructiva ( TTC ) sin tener que pagar el precio de un sistema que no es ni axiomatizable ni tiene una teoría subyacente de la inferencia – la lógica trata sobre la inferencia después de todo.Concluimos señalando que los recientes desarrollos en lógica dialógica muestran que el enfoque TTC hacia el significado, en general, y hacia el axioma de elección, en particular, es connatural al enfoque de la teorética de juegos, donde las características metalógicas (standard) se despliegan explícitamente a nivel del lenguaje - objeto. Por tanto, de algún modo, esto justifica, aunque de una manera bastante diferente, la exhortación de Hintikka por la fecundidad de la Semántica Teorética de Juegos en el contexto de los fundamentos de las matemáticas.
In the present paper we confront Martin- Lof’s analysis of the axiom of choice with J. Hintikka’s standing on this axiom. Hintikka claims that his game theoretical semantics (GTS) for Independence Friendly Logic (IF logic) justifies Zermelo’s axiom of choice in a first-order way perfectly acceptable for the constructivists. In fact, Martin- Lof’s results lead to the following considerations: Hintikka preferred version of the axiom of choice is indeed acceptable for the constructivists and its meaning does not involve higher order logic. However, the version acceptable for constructivists is based on an intensional take on functions. Extensionality is the heart of the classical understanding of Zermelo’s axiom and this is the real reason behind the constructivist rejection of it. More generally, dependence and independence features that motivate IF-Logic, can be formulated within the frame of constructive type theory (CTT) without paying the price of a system that is neither axiomatizable nor has an underlying theory of inference – logic is about inference after all. We conclude pointing out that recent developments in dialogical logic show that the CTT approach to meaning in general and to the axiom of choice in particular is very natural to game theoretical approaches where (standard) metalogical features are explicitly displayed at the object language-level. Thus, in some way, this vindicates, albeit in quite of a different manner, Hintikka’s plea for the fruitfulness of game-theoretical semantics in the context of the foundations of mathematics.
It is our main claim that the time is ripe to link the dynamic turn launched by game-theoretical approaches to meaning with P. Martin-Löf's Constructive Type Theory (CTT). Furthermore, we also claim that the dialogical framework provides the appropriate means to develop such a link. We will restrict our study to the discussion of two paradigmatic cases of dependences triggered by quantifiers, namely the case of the Axiom of Choice and the study of anaphora, that are by the way two of the most cherished examples of Hintikka.
France) du projet ANR Franco Allemand Lille (MESHS)/Konstanz): Théorie du Droit et Logique/Jurisprudenz und Logik (2012-15). Membre du équipe du projet ANR SÊMAINÔ, porté par L. Gazziero Co-fondateur et co-responsable du programme Langage, argumentation et cognitions dans les traditions orales (LACTO), porté par le Projet ADA et coordonné par l'université Lille 3 en partenariat avec huit universités africaines (
The aim of the paper is to develop a new constructivist approach to the game theoretical interpretation of AC based on the CTT-proof of Per Martin-Löf (1980). More precisely, Clerbout and Rahman showed that the CTT-understanding of AC, that stresses the type dependence involved by the function that constitutes the proof-object of the antecedent, can be seen as the result of both an " outside-inside " approach to meaning. It is this approach to meaning, so we claim, that provides a natural dialogical interpretation to AC, where the (inten-sional) function involved — understood as rules of correspondence produced by the players' interaction — constitutes a play object for the (first-order) universal quantifier that occurs in the antecedent of the formal expression of this axiom.
. It is argued that the meaning of the modal connectives must be given inferentially, by the rules for the assertion of formu-lae containing them, and not semantically by reference to possible worlds. Further, harmony confers transparency on the inferentialist account of meaning, when the introduction-rule specifies both neces-sary and sufficient conditions for assertion, and the elimination-rule does no more than exhibit the consequences of the meaning so con-ferred. Hence, harmony is not to be identified with normalization, since the standard modal natural deduction rules, though normaliz-able, are not in this sense harmonious. Harmonious rules for modality have lately been formulated, using labelled deductive systems.
The dialogical approach to logic has well known re- lations with other proof methods, like sequent calculi and semantic tableaux. In this paper we explore the connexion between dialogues and natural deduction, an area that is much less understood. We define a system of dialogical games that captures the notion of valid- ity of Johansson's minimal logic, and show how to extend the system for intuitionistic and classical logic. Then we describe an algorithm that transforms a dialogical proof into a Fitch-style natural deduction proof, and discuss the relation between the two approaches.
It is now 10 years ago that two of us took the train to Stockholm to meet Per Martin-Löf and discuss his ideas on the connection between type theory and computing science. This book describes different type theories (theories of types, polymorphic and monomorphic sets, and subsets) from a computing science perspective. It is intended for researchers and graduate students with an interest in the foundations of computing science, and it is mathematically self-contained. We started writing this book about six years ago. One reason for this long time is that our increasing experience in using type theory has made several changes of the theory necessary. We are still in this process, but have nevertheless decided to publish the book now. We are, of course, greatly indebted to Per Martin-L¨of; not only for creating the subject of this book, but also for all the discussions we have had with him. Beside Martin-Löf, we have discussed type theory with many people and we in particular want to thank Samson Abramsky, Peter Aczel, Stuart Anderson, Roland Backhouse, Bror Bjerner, Robert Constable, Thierry Coquand, Peter Dybjer, Roy Dyckhoff, Gerard Huet, Larry Paulson, Christine Paulin-Mohring, Anne Salvesen, Björn von Sydow, and Dan Synek. Thanks to Dan Synek also for his co-authorship of the report which the chapter on trees is based on. Finally, we would like to thank STU, the National Swedish Board For Technical Development, for financial support.
In this article two game-theoretically flavored approaches to logic are systematically compared: dialogical logic founded by Paul Lorenzen and Kuno Lorenz, and the game-theoretical semantics of Jaakko Hintikka. For classical Propositional logic and for classical First-order logic, an exact connection between 'intuitionistic dialogues with hypotheses' and semantical games is established. Various questions of a philosophical nature are also shown to arise as a result of the comparison, among them the relation between the model-theoretical and proof-theoretical approaches to the philosophy of logic and mathematics.
We will take as one of the main issues of this paper the challenge which the dialogical approaches offer to the relation of semantics and pragmatics concerning the concept of proof (strategy) and proposition (game). While our aim here will be to present the main technical and philosophical features of what can be seen as the dialogical approach to logic, illustrated through both very well known and new dialogics, we would also like to delineate the common pragmatic attitude which constitutes the cohesive force within the dialogical universe.
At the end of the 19th century Hugh MacColl (1837-1909), the father of pluralism in formal logic, attempted in the north of France (Boulogne sur mer) to formulate a modal logic which would challenge the semantics of material implication of the post-Boolean wave. It seems that in some of his various attempts MacColl suggested some systems where the rule of necessitation fails. Moreover, the idea that no logical necessity has universal scope-or that no logic could be applied to any argumentative context-seems to be akin and perhaps even central to his pluralistic philosophy of logic. Some years later Clarence Irwin Lewis furnished the axiomatics for several of these logics and since then the critics on the material implication have shown an increasing interest in these modal logics called "non-normal". When Saul Kripke studied their semantics of "impossible worlds" as a way to distinguish between "necessity" and "validity" these logics reached a status of some respectability. As is well known, around the 70s non-normal logics were associated with the problem of omniscience in the epistemic interpretation of modal logic, specially in the work of Jaakko Hintikka and Veikko Rantala. Actually impossible worlds received a intensive study and development too in the context of relevant and paraconsistent logics-specially within the "Saint-Andrews-Australasian connection" in the work of such people as Graham Priest, Stephen Read, Greg Restall and Richard Routley-Sylvan. Nowadays, though the association with omniscience seems to have faded out, the study of non-normal logics has received a new impulse motivated through the study of counterlogicals. The aim of the paper is to offer a dialogical interpretation of non-normal modal dialogics which will suggest some explorations beyond the concept of non-normality. This interpretation will be connected to the discussion of two issues, namely: 1 Counterlogicals as a minimalist defense of logical pluralism (pluralism for a monist) following the path prefigured by MacColl and 2 The difficulties involved in the application of the so-called Hintikka strategy and hybrid languages while constructing tableau systems for nonnormal modal logics.
In classical India, Jain philosophers developed a theory of viewpoints (naya-vāda) according to which any statement is always performed within and dependent upon a given epistemic perspective or viewpoint.
The Jainas furnished this epistemology with an (epistemic) theory of disputation that takes into account the viewpoint in
which the main thesis has been stated. The main aim of our paper is to delve into the Jain notion of viewpoint-contextualisation
and to develop the elements of a suitable logical system that should offer a reconstruction of the Jainas’ epistemic theory
of disputation. A crucial step of our project is to approach the Jain theory of disputation with the help of a theory of meaning
for logical constants based on argumentative practices called dialogical logic. Since in the dialogical framework the meaning of the logical constants is given by the norms or rules for their use in a
debate, it provides a meaning theory closer to the Jain context-sensitive disputation theory than the main-stream formal model-theoretic
semantics.
KeywordsJain logic–Jain theory of viewpoints–Jain epistemology–Universals–Particulars–Argumentation–Dialogical logic
In the present paper I wish to regard constructive logic as a self-contained system for the treatment of epistemological issues; the explanations of the constructivist logical notions are cast in an epistemological mold already from the outset. The discussion offered here intends to make explicit this implicit epistemic character of constructivism. Particular attention will be given to the intended interpretation laid down by Heyting. This interpretation, especially as refined in the type-theoretical work of Per Martin-Löf, puts the system on par with the early efforts of Frege and Whitehead-Russell. This quite recent work, however, has proved valuable not only in the philosophy and foundations of mathematics, but has also found practical application in computer science, where the language of constructivism serves as an implementable programming language, and within the philosophy of language. My presentation will be carried out through a contrast with standard metamathematical work. In the course of the development I have occasion to offer some novel considerations (in Sections 6 and 8) on the nature of proof and inference(-acts).
Nous montrons dans cet article comment les approches dynamiques en logique contemporaine ont retrouvé, quoique sous d'autres attendus, une configuration théorique qu'on peut attribuer à Aristote au moment de la rédaction des Topiques. Dans cette configuration, la logique et la rhétorique – ou au moins la dialectique – se complètent dans le cadre conceptuel homogène offert par la dialectique, entendue comme une certaine forme relativement codifiée de débat critique. L'idée principale est qu'aujourd'hui, de même qu'alors, la dialectique est le cadre dans lequel il est possible de penser le rapport entre la logique et les formes générales de l'argumentation d'une part, et entre la logique et la théorie de la connaissance scientifique d'autre part.
My original training as a philosopher, at Uppsala and at Oxford, was ruggedly analytical. Also the notion of an analytic judgement, or ‘proposition’, or ‘sentence’, or ‘statement’, (one did not overly distinguish these notions) was repeatedly treated of by excellent teachers and colleagues. There were aficionados of Quine and experts on Kant among them, but no names, no pack-drill! If there was one central topic in traditional epistemology on which I felt philosophically at ease, it was that of analyticity. In the early 1980s, I entered for the first time a pluralist philosophical environment in the Philosophy Department of the Catholic University at Nijmegen, with ample representation in phenomenology, Hegelian idealism, and (neo)Thomism. To my considerable surprise, I discovered that it could be enjoyable as well as instructive talking to such rare birds in the philosophical aviary. A colleague drew my attention to Thomas Aquinas’ Five Ways, which I had never read, having adopted, from the exposition in Anders Wedberg’s History of Philosophy, the opinion that, like Kant’s transcendental deduction, Aquinas’ demonstrations were ‘worthless’. However, the Summa Theologica was readily available on open shelves in the library at Nijmegen, and my curiosity got the better of me. Upon consultation of its second question, my shock was great. In a discussion of whether the judgement Deus est admits of demonstration, Aquinas introduces the notion of a propositio per se nota, that is, an S is P judgement known in, or—perhaps better—from itself: The explanation offered is that the predicate P is included, or contained, in the notion (= concept) of the subject S. Needless to say, in view of my previous deep and thorough (as I misguidedly thought) exposure to analyticity, I had a powerful déjà lu experience, pertaining to Kant, four centuries later. Clearly, I had been choused. What was the hidden tale behind this, and why had my eminent teachers not told me that the notion of an analytic judgement was known long before Kant?
We present here a modal version of the structure seeking dialogues (SSD) that were introduced in [Rahman & Keiff 2004]. For this purpose, we use a semantics that corresponds to the semantics of non normal modal systems.
The relation between logic and knowledge has been at the heart of a lively debate since the 1960s. On the one hand, the epistemic approaches based their formal arguments in the mathematics of Brouwer and intuitionistic logic. Following Michael Dummett, they started to call themselves 'antirealists'. Others persisted with the formal background of the Frege-Tarski tradition, where Cantorian set theory is linked via model theory to classical logic. Jaakko Hintikka tried to unify both traditions by means of what is now known as 'explicit epistemic logic'. Under this view, epistemic contents are introduced into the object language as operators yielding propositions from propositions, rather than as metalogical constraints on the notion of inference. The Realism-Antirealism debate has thus had three players: classical logicians, intuitionists and explicit epistemic logicians. The editors of the present volume believe that in the age of Alternative Logics, where manifold developments in logic happen at a breathtaking pace, this debate should be revisited. Contributors to this volume happily took on this challenge and responded with new approaches to the debate from both the explicit and the implicit epistemic point of view.
Formal calculi of deduction have proved useful in logic and in the foundations of mathematics, otas well as in metamathematics. Examples of some of these uses are:
This chapter tells how, within a century, the notions of judgment and inference were driven out of logical theory and replaced by propositions and (logical) consequence. Systematic considerations guide the treatment. The history is unashamedly Whiggish: the current position is shown as the outcome, or even culmination, of a historical development.
The principal content of this article is a (new) foundation for intuitionistic logic, based on an analysis of argumentative processes as codified in the concepts of a dialogue and a strategy for dialogues. This work is presented in Section 3. A general historical introduction is given in Section2. Since already there the reader will need to know exactly what a dialogue and a strategy shall be, these basic concepts are defined in the (purely technical) Section 1.
The aim of this paper is to present the dialogical approach to logic as an interesting alternative to the model-theoretic approach. In the introduction I plead for pluralism concerning logical systems and logical methodology before giving a short outline of Dialogical Logic. Then, I discuss and reject several prejudices against Dialogical Logic. I present three conceptual distinctions that are characteristic of the dialogical approach to logic (namely, formal vs. general truth, level of games vs. level of strategies and particle rules vs. structural rules) and their fruitfulness for logical research is demonstrated at the hand of some examples.
Without violating the spirit of Game-Theoretical semantics, its results can be re-worked in Martin-Lf''s Constructive Type Theory by interpreting games as types of Myself''s winning strategies. The philosophical ideas behind Game-Theoretical Semantics in fact highly recommend restricting strategies to effective ones, which is the only controversial step in our interpretation. What is gained, then, is a direct connection between linguistic semantics and computer programming.
The extensive research in logic conducted by using concepts and methods of game theory as documented in this collection of papers, allows to see dialogue logic in a number of new perspectives. This situation may gain further clarity by looking back to the inception of dialogue logic in the late fifties and early sixties.
In the present paper Hintikka’s game-theoretical semantics and the dialogical logic of Lorenzen and Lorenz are discussed and
compared from the viewpoint of their underlying philosophical meaning theories. The question of whether the proposed meaning
theories can be claimed to suffer from circularity is taken up. The relations of the two frameworks to verificationist and
anti-realist ideas are considered. Finally, van Heijenoort’s concept of ‘logic as calculus’ generalized by Hintikka to the
idea of ‘language as calculus’ will be reformulated as a view we label ‘anti-universalism.’ We discuss briefly the fourfold
division of foundational views obtained by relating a distinction between ‘universalism’ and ‘anti-universalism’ to the distinction
between ‘realism’ and ‘anti-realism.’
We present a game (or dialogue) semantics in the style of Lorenzen (1959) for Girard's linear logic (1987). Lorenzen suggested that the (constructive) meaning of a proposition ϕ should be specified by telling how to conduct a debate between a proponent P who asserts ϕ and an opponent O who denies ϕ. Thus propositions are interpreted as games, connectives (almost) as operations on games, and validity as existence of a winning strategy for P. (The qualifier ‘almost’ will be discussed later when more details have been presented.) We propose that the connectives of linear logic can be naturally interpreted as the operations on games introduced for entirely different purposes by Blass (1972). We show that affine logic, i.e., linear logic plus the rule of weakening, is sound for this interpretation. We also obtain a completeness theorem for the additive fragment of affine logic, but we show that completeness fails for the multiplicative fragment. On the other hand, for the multiplicative fragment, we obtain a simple characterization of game-semantical validity in terms of classical tautologies. An analysis of the failure of completeness for the multiplicative fragment leads to the conclusion that the game interpretation of the connective ⊗ is weaker than the interpretation implicit in Girard's proof rules; we discuss the differences between the two interpretations and their relative advantages and disadvantages. Finally, we discuss how Gödel's Dialectica interpretation (1958), which was connected to linear logic by de Paiva (1989), fits with game semantics.
The meaning of a sentence determines how the truth of the proposition expressed by the sentence may be proved and hence one would expect proof theory to be influenced by meaning-theoretical considerations. In the present Chapter we consider a proposal that also reverses the above priorities and determines meaning in terms of proof. The proposal originates in the criticism that Michael Dummett has voiced against a realist, truth-theoretical, conception of meaning and has been developed largely by him and Dag Prawitz, whose normalization procedures in technical proof theory constitute the main technical basis of the proposal.
Dialogical logic is a game-theoretical approach to logic. Logic is studied with the help of certain games, which can be thought of as idealized argumentations. Two players, the Proponent, who puts forward the initial thesis and tries to defend it, and the Opponent, who tries to attack the Proponent’s thesis, alternately utter argumentative moves according to certain rules. For a long time the dialogical approach had been worked out only for classical and intuitionistic logic. The seven papers of this dissertation show that this narrowness was uncalled for. The initial paper presents an overview and serves as an introduction to the other papers. Those papers are related by one central theme. As each of them presents dialogical formulations of a different non-classical logic, they show that dialogical logic constitutes a powerful and flexible general framework for the development and study of various logical formalisms and combinations thereof. As such it is especially attractive to logical pluralists that reject the idea of “the single correct logic”. The collection contains treatments of free logic, modal logic, relevance logic, connexive logic, linear logic, and multi-valued logic.
In the present paper I wish to regard constructive logic as a self-contained system for the treatment of epistemological issues; the explanations of the constructivist logical notions are cast in an epistemological mold already from the outset. The discussion offered here intends to make explicit this implicit epistemic character of constructivism. Particular attention will be given to the intended interpretation laid down by Heyting. This interpretation, especially as refined in the type-theoretical work of Per Martin-Löf, puts the system on par with the early efforts of Frege and Whitehead-Russell. This quite recent work, however, has proved valuable not only in the philosophy and foundations of mathematics, but has also found practical application in computer science, where the language of constructivism serves as an implementable programming language, and within the philosophy of language. My presentation will be carried out through a contrast with standard metamafhematical work. In the course of the development I have occasion to offer some novel considerations (in Sections 6 and 8) on the nature of proof and inference(-acts).
Logique dynamique de la fiction
- J Redmond
Redmond, J. (2010). Logique dynamique de la fiction. Pour une approche dialogique,
London: College Publications, 2010.
The Conception of Validity in Dialogical Logic
- H Rückert
Rückert, H. (2011b). "The Conception of Validity in Dialogical Logic". Talk at the
workshop Proofs and Dialogues, Tübingen, 2011.
Elemente der Sprachkritik Eine Alternative zum Dogmatismus und Skeptizismus in der Analytischen Philosophie
- K Lorenz
Lorenz, K. (1970). Elemente der Sprachkritik Eine Alternative zum Dogmatismus und
Skeptizismus in der Analytischen Philosophie, Frankfurt:Suhrkamp Verlag, 1970.