Through a phenomenological study of horse-human relations, this article explores the ways in which, as embodied beings, we live relationally, rather than as separate human identities. Conceptually this challenges oppositional logic and humanist assumptions, but where poststructuralist treatments of these issues tend to remain abstract, this article is concerned with an embodied demonstration of the ways in which we experience a relational or in-between logic in our everyday lives.

Abstract In this article we develop a relational understanding of sociality, that is, an account of social life that takes relation as primary. This stands in contrast to the common assumption that relations arise when subjects interact, an account that gives logical priority to separation. We will develop this relational understanding through a reading of the work of Martin Buber, a social philosopher primarily interested in dialogue, meeting, relationship, and the irreducibility and incomparability of reality. In particular, the article (...) contrasts Buber’s work with that of poststructuralist theorists who take as their starting point the deconstruction of the Hegelian logic of binary oppositions. Deconstruction understands difference as the excess that undoes the binary, but Buber, we argue, shows how difference derives from the primacy and ontological undefinability of relation. Relational logic does not exclude the logic of separations and oppositions: relation is the primal ground that makes separations possible. Content Type Journal Article Pages 1-13 DOI 10.1007/s11841-011-0278-9 Authors Andrew Metcalfe, School of Social Sciences and International Studies, University of New South Wales, Sydney, NSW 2052, Australia Ann Game, School of Social Sciences and International Studies, University of New South Wales, Sydney, NSW 2052, Australia Journal Sophia Online ISSN 1873-930X Print ISSN 0038-1527. (shrink)

Game logics describe general games through powers of players for forcing outcomes. In particular, they encode an algebra of sequential game operations such as choice, dual and composition. Logic games are special games for specific purposes such as proof or semantical evaluation for first-order or modal languages. We show that the general algebra of game operations coincides with that over just logical evaluation games, whence the latter are quite general after all. The main tool in (...) proving this is a representation of arbitrary games as modal or first-order evaluation games. We probe how far our analysis extends to product operations on games. We also discuss some more general consequences of this new perspective for standard logic. (shrink)

This paper introduces The Logic Game, a two-player strategy game designed to help students in introductory logic classes learn the truth conditions for the logical operators. The game materials can be printed using an ordinary printer on ordinary paper, takes 10-15 minutes to play, and the rules are fairly easy to learn. This paper includes a complete set of rules, a URL for a website hosting all of the game materials, and the results of a study of the effectiveness (...) of the Logic Game in three introductory logic classes. (shrink)

This paper is based on tutorials on 'Logic and Games' at the 7th Asian Logic Conference in Hsi-Tou, Taiwan, 1999, and until 2002 in Siena, Stuttgart, Trento, Udine, and Utrecht. We present logic games as a topic per se, giving models for dynamic interaction between agents. First, we survey some basic logic games. Then we show how their common properties raise general issues of game structure and 'game logics'. Next, we review logic games in the light (...) of general game logic. Finally, we discuss more 'realistic' influences from game theory into logic games, including players' preferences, and imperfect information. (shrink)

I argue that the medieval form of dialectical disputation known as obligationes can be viewed as a logical game of consistency maintenance. The game has two participants, Opponent and Respondent. Opponent puts forward a proposition P; Respondent must concede, deny or doubt, on the basis of inferential relations between P and previously accepted or denied propositions, or, in case there is none, on the basis of the common set of beliefs. Respondent loses the game if he concedes a contradictory (...) set of propositions. Opponent loses the game if Respondent is able to maintain consistency during the stipulated period of time. The obligational rules are here formalised by means of familiar notational devices, and the application of some game-theoretical concepts, such as (winning) strategy, moves, motivation, allows for an analysis of some crucial properties of the game. In particular, the primacy of inferential (syntactic) relations over semantic aspects and the dynamic character of obligations are outlined. (shrink)

Here I describe a game that I use in my logic classes once we begin derivations. The game can help improve class dynamics, help struggling students recognizes they are not alone, open lines of communication between students, and help students of all levels prepare for exams. The game can provide struggling students with more practice with the fundamental rules of a logical system while also challenging students who excel at derivations. If students are struggling with particular rules or strategies (...) in the system, the game can be tailored to address them. I explain how the game has evolved since I started using it, highlighting the pedagogical benefits of the changes I have made, and I provide examples of the handouts I distribute and a “checklist” to use before, during, and after the game. (shrink)

In this article, I present a dynamic logic game for defeasible reasoning. I argue that, as far as defeasible reasoning is concerned, one should distinguish between practical and ideal rationality. Starting from the adaptive logic framework, I formalize both rationality notions by means of logic games. The presented adaptive logic games are based on (i) standard logic games on the one hand and (ii) dynamic proof procedures for adaptive logic on the other hand. The games are (...) similar to standard logic games, but have the extra property that some moves are revisable. This is handled by means of a main control game, which starts different standard logic games. I argue that the adaptive logic games form intuitive reasoning models for rationality in defeasible reasoning contexts. Moreover, I will also demonstrate that the games give a good insight in the computational complexity of defeasible reasoning forms. (shrink)

We develop a multi-agent deontic action logic to study the logical behaviour of two types of deontic conditionals: (1) conditional obligations, having the form "If group H were to perform action aH, then, in group F's interest, group G ought to perform action aG" and (2) conditional permissions, having the form "If group H were to perform action aH, then, in group F's interest, group G may perform action aG". First, we define a formal language for multi-agent deontic action (...) logic and a class of consequentialist models to interpret the formulas of the language. Second, we define a transformation that converts any strategic game into a consequentialist model. Third, we show that an outcome a* is a Nash equilibrium of a strategic game if and only if a conjunction of certain conditional permissions is true in the consequentialist model that results from the transformation of that strategic game. (shrink)

In [Dutilh Novaes, Medieval-obligations as logical Games of Consistency maintenance, synthese, (2004)], I proposed a reconstruction of Walter Burley’s theory of obligationes, based on the idea that Burley’s theory of obligationes could be seen as a logical game of consistency maintenance. In the present paper, I intend to test the game hypothesis on another important theory of obligationes, namely Roger Swyneshed’s theory. In his treatise on obligationes [edited by P.V. Spade, cf. Spade History and philosophy of Logic (...) 3(1982) 1-32], Swyneshed introduced significant modifications to the general framework of obligationes. To compare the two theories, I apply the same formal apparatus used in the previous paper. It will become patent that Swyneshed’s theory is considerably different from Burley’s, among other reasons because the dynamic aspects that play a major role in the latter are simply not present in the former. My conclusion is that Swyneshed’s version of obligationes is not directed towards consistency maintenance, but rather towards inference recognition, and that it is, from a game-theoretical perspective, less interesting a theory than Burley’s. (shrink)

In [Dutilh Novaes, Medieval-obligations as logical Games of Consistency maintenance, synthese, (2004)], I proposed a reconstruction of Walter Burley’s theory of obligationes, based on the idea that Burley’s theory of obligationes could be seen as a logical game of consistency maintenance. In the present paper, I intend to test the game hypothesis on another important theory of obligationes, namely Roger Swyneshed’s theory. In his treatise on obligationes [edited by P.V. Spade, cf. Spade History and philosophy of Logic (...) 3(1982) 1-32], Swyneshed introduced significant modifications to the general framework of obligationes. To compare the two theories, I apply the same formal apparatus used in the previous paper. It will become patent that Swyneshed’s theory is considerably different from Burley’s, among other reasons because the dynamic aspects that play a major role in the latter are simply not present in the former. My conclusion is that Swyneshed’s version of obligationes is not directed towards consistency maintenance, but rather towards inference recognition, and that it is, from a game-theoretical perspective, less interesting a theory than Burley’s. (shrink)

A systematic introduction suitable for readers who have little familiarity with logic. Provides numerous examples and complete proofs.

This paper looks to Bernard Suits’s analysis of games and game playing for at least a partial answer to the question in its title. It applies Suits’s analysis to Sudoku, a popular logic puzzle, and to Ana-lytical Reasoning, a question type in standardized assessments. The purpose is both to test Suits’s analysis in a novel domain and to give educators and test developers useful insight into the relationship between logic exercises and games.

This volume contains papers presented at the Poznań Reasoning Week multi-conference held in Poznań in September 11-15, 2018. PRW aims at bringing together experts whose research offers a broad range of perspectives on systematic analyses of reasoning processes and their formal modelling. The 2018 edition consisted of three conferences, which addressed the following topics: (i) games in reasoning research, (ii) the interplay of logic and cognition, and (iii) refutation systems. The papers collected in this volume address all these topics.

I LOGIC IN PHILOSOPHY— PHILOSOPHY OF LOGIC i. On the relation of logic to philosophy I n this book, the consequences of certain logical insights for ...

The first learning game to be developed to help students to develop and hone skills in constructing proofs in both the propositional and first-order predicate calculi. It comprises an autotelic (self-motivating) learning approach to assist students in developing skills and strategies of proof in the propositional and predicate calculus. The text of VALIDITY consists of a general introduction that describes earlier studies made of autotelic learning games, paying particular attention to work done at the Law School of Yale University, (...) called the ALL Project (Accelerated Learning of Logic). Following the introduction, the game of VALIDITY is described, first with reference to the propositional calculus, and then in connection with the first-order predicate calculus with identity. Sections in the text are devoted to discussions of the various rules of derivation employed in both calculi. Three appendices follow the main text; these provide a catalogue of sequents and theorems that have been proved for the propositional calculus and for the predicate calculus, and include suggestions for the classroom use of VALIDITY in university-level courses in mathematical logic. (shrink)

A comprehensive examination of the interfaces of logic, computer science, and game theory, drawing on twenty years of research on logic and games.

This introduction to modern work in analytic philosophy uses the example of the proper name to give a clear explanation of the logical theories of Gottlob Frege, and explain the application of his ideas to ordinary language. McCulloch then shows how meaning is rooted in the philosophy of mind and the question of intentionality, and looks at the ways in which thought can be "about" individual material objects.

Game Logic is a modal logic which extends Propositional Dynamic Logic by generalising its semantics and adding a new operator to the language. The logic can be used to reason about determined 2-player games. We present an overview of meta-theoretic results regarding this logic, also covering the algebraic version of the logic known as Game Algebra.

In the literature there are at least two main formal structures to deal with situations of interactive epistemology: Kripke models and type spaces. As shown in many papers :149–225, 1999; Battigalli and Siniscalchi in J Econ Theory 106:356–391, 2002; Klein and Pacuit in Stud Log 102:297–319, 2014; Lorini in J Philos Log 42:863–904, 2013), both these frameworks can be used to express epistemic conditions for solution concepts in game theory. The main result of this paper is a formal comparison between (...) the two and a statement of semantic equivalence with respect to two different logical systems: a doxastic logic for belief and an epistemic–doxastic logic for belief and knowledge. Moreover, a sound and complete axiomatization of these logics with respect to the two equivalent Kripke semantics and type spaces semantics is provided. Finally, a probabilistic extension of the result is also presented. A further result of the paper is a study of the relationship between the epistemic–doxastic logic for belief and knowledge and the logic STIT by Belnap and colleagues. (shrink)

We discuss games of both perfect and imperfect information at two levels of structural detail: players’ local actions, and their global powers for determining outcomes of the game. We propose matching logical languages for both. In particular, at the ‘action level’, imperfect information games naturally model a combined ‘dynamic-epistemic language’ – and we find correspondences between special axioms and particular modes of playing games with their information dynamics. At the ‘outcome level’, we present suitable notions of (...) game equivalence, plus some simple representation results. (shrink)

We present a game semantics for Linear Logic, in which formulas denote games and proofs denote winning strategies. We show that our semantics yields a categorical model of Linear Logic and prove full completeness for Multiplicative Linear Logic with the MIX rule: every winning strategy is the denotation of a unique cut-free proof net. A key role is played by the notion of history-free strategy; strong connections are made between history-free strategies and the Geometry of Interaction. Our semantics incorporates (...) a natural notion of polarity, leading to a refined treatment of the additives. We make comparisons with related work by Joyal, Blass, et al. (shrink)

Inquisitive first-order logic, InqBQ, is a system which extends classical first-order logic with formulas expressing questions. From a mathematical point of view, formulas in this logic express properties of sets of relational structures. This paper makes two contributions to the study of this logic. First, we describe an Ehrenfeucht–Fraïssé game for InqBQ and show that it characterizes the distinguishing power of the logic. Second, we use the game to study cardinality quantifiers in the inquisitive setting. That is, we study what (...) statements and questions can be expressed in InqBQ about the number of individuals satisfying a given predicate. As special cases, we show that several variants of the question how many individuals satisfy α(x) are not expressible in InqBQ, both in the general case and in restriction to finite models. (shrink)

We present a game semantics in the style of Lorenzen for Girard's linear logic . Lorenzen suggested that the meaning of a proposition should be specified by telling how to conduct a debate between a proponent P who asserts and an opponent O who denies . Thus propositions are interpreted as games, connectives as operations on games, and validity as existence of a winning strategy for P. We propose that the connectives of linear logic can be naturally interpreted (...) as the operations on games introduced for entirely different purposes by Blass . 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 , which was connected to linear logic by de Paiva , fits with game semantics. (shrink)

This collection of papers in epistemic logic is oriented towards applications to game theory and individual decision theory. Most of these papers were presented at the inaugural conference of the LOFT (Logic for the Theory and Games and Decisions) conference series, which took place in 1994 in Marseille. Among the notions dealt with are those of common knowledge and common belief, infinite hierarchies of beliefs and belief spaces, logical omniscience, positive and negative introspection, backward induction and rationalizable equilibria (...) in game theory. (shrink)

Logics of joint strategic ability have recently received attention, with arguably the most influential being those in a family that includes Coalition Logic (CL) and Alternating-time Temporal Logic (ATL). Notably, both CL and ATL bypass the epistemic issues that underpin Schelling-type coordination problems, by apparently relying on the meta-level assumption of (perfectly reliable) communication between cooperating rational agents. Yet such epistemic issues arise naturally in settings relevant to ATL and CL: these logics are standardly interpreted on structures where agents move (...) simultaneously, opening the possibility that an agent cannot foresee the concurrent choices of other agents. In this paper we introduce a variant of CL we call Two-Player Strategic Coordination Logic (SCL2). The key novelty of this framework is an operator for capturing coalitional ability when the cooperating agents cannot share strategic information. We identify significant differences in the expressive power and validities of SCL2 and CL2, and present a sound and complete axiomatization for SCL2. We briefly address conceptual challenges when shifting attention to games with more than two players and stronger notions of rationality. (shrink)

This paper addresses the theoretical notion of a game as it arisesacross scientific inquiries, exploring its uses as a technical andformal asset in logic and science versus an explanatory mechanism. Whilegames comprise a widely used method in a broad intellectual realm(including, but not limited to, philosophy, logic, mathematics,cognitive science, artificial intelligence, computation, linguistics,physics, economics), each discipline advocates its own methodology and aunified understanding is lacking. In the first part of this paper, anumber of game theories in formal studies are critically (...) surveyed. Inthe second part, the doctrine of games as explanations for logic isassessed, and the relevance of a conceptual analysis of games tocognition discussed. It is suggested that the notion of evolution playsa part in the game-theoretic concept of meaning. (shrink)

Mathematical game theory has been embraced by a variety of scholars: social scientists, biologists, linguists, and now, increasingly, logicians. This volume illustrates the recent advances of game theory in the field. Logicians benefit from things like game theory's ability to explain informational independence between connectives; meanwhile, game theorists have even begun to benefit from logical epistemic analyses of game states. In concert with such pioneering work, this volume also present surprising developments in classical fields, including first-order logic and set (...) theory. (shrink)

Preference is a basic notion in human behaviour, underlying such varied phenomena as individual rationality in the philosophy of action and game theory, obligations in deontic logic (we should aim for the best of all possible worlds), or collective decisions in social choice theory. Also, in a more abstract sense, preference orderings are used in conditional logic or non-monotonic reasoning as a way of arranging worlds into more or less plausible ones. The field of preference logic (cf. Hansson [10]) studies (...) formal systems that can express and analyze notions of preference between various sorts of entities: worlds, actions, or propositions. The art is of course to design a language that combines perspicuity and low complexity with reasonable expressive power. In this paper, we take a particularly simple approach. As preferences are binary relations between worlds, they naturally support standard unary modalities. In particular, our key modality ♦ϕ will just say that is ϕ true in some world which is at least as good as the current one. Of course, this notion can also be indexed to separate agents. The essence of this language is already in [4], but our semantics is more general, and so are our applications and later language extensions. Our modal language can express a variety of preference notions between propositions. Moreover, as already suggested in [9], it can “deconstruct” standard conditionals, providing an embedding of conditional logic into more standard modal logics. Next, we take the language to the analysis of games, where some sort of preference logic is evidently needed ([23] has a binary modal analysis different from ours). We show how a qualitative unary preference modality suffices for defining Nash Equilibrium in strategic games, and also the Backward Induction solution for finite extensive games. Finally, from a technical perspective, our treatment adds a new twist. Each application considered in this paper suggests the need for some additional access to worlds before the preference modality can unfold its true power.. (shrink)

We associate the semantic game with chance moves conceived by Blinov with Blamey’s partial logic. We give some equivalent alternatives to the semantic game, some of which are with a third player, borrowing the idea of introducing the pseudo-player called Nature in game theory. We observe that IF propositional logic proposed by Sandu and Pietarinen can be equivalently translated to partial logic, which implies that imperfect information may not be necessary for IF propositional logic. We also indicate that some independent (...) quantifiers can be regarded as dependent quantifiers of indeterminate sequence, using the interjunction connective in partial logic. We conclude our paper by indicating some further research in a more general setting. (shrink)

This paper provides a Genzten style formulation of the game logic framework GLm (0 m ), and proves the cut-elimination theorem for GLm. As its application, we prove the term existence theorem for GL used in Part I.

An introduction to the special issue on epistemic logic and the foundations of game theory edited by Michael Bacharach and Philippe Mongin. Contributors are Michael Bacharach, Robert Stalnaker, Salvatore Modica and Aldo Rustichini, Luc Lismont and Philippe Mongin, and Hyun-Song Shin and Timothy Williamson.

This Is A New Release Of The Original 1897 Edition.

This paper provides a logic framework for investigations of game theoretical problems. We adopt an infinitary extension of classical predicate logic as the base logic of the framework. The reason for an infinitary extension is to express the common knowledge concept explicitly. Depending upon the choice of axioms on the knowledge operators, there is a hierarchy of logics. The limit case is an infinitary predicate extension of modal propositional logic KD4, and is of special interest in applications. In Part I, (...) we develop the basic framework, and show some applications: an epistemic axiomatization of Nash equilibrium and formal undecidability on the playability of a game. To show the formal undecidability, we use a term existence theorem, which will be proved in Part II. (shrink)

A general game player automatically learns to play arbitrary new games solely by being told their rules. For this purpose games are specified in the general Game Description Language (GDL), a variant of Datalog with function symbols that uses a few game-specific keywords. A recent extension of basic GDL allows the description of nondeterministic games with any number of players who may have incomplete, asymmetric information. In this paper, we analyse the epistemic structure and expressiveness of this (...) language in terms of modal epistemic logic and prove two main results: (1) The operational semantics of GDL entails that the situation at any stage of a game can be characterised by a multi-agent epistemic (i.e., S5-) model; (2) GDL is sufficiently expressive to model any situation that can be described by a (finite) multi-agent epistemic model. (shrink)

This anthology is a thorough introduction to classic literature for those who have not yet experienced these literary masterworks. For those who have known and loved these works in the past, this is an invitation to reunite with old friends in a fresh new format. From Shakespeare's finesse to Oscar Wilde's wit, this unique collection brings together works as diverse and influential as The Pilgrim's Progress and Othello. As an anthology that invites readers to immerse themselves in the masterpieces of (...) the literary giants, it is must-have addition to any library. (shrink)

An overview of different versions and applications of Lorenzen’s dialogue game approach to the foundations of logic, here largely restricted to the realm of manyvalued logics, is presented. Among the reviewed concepts and results are Giles’s characterization of Łukasiewicz logic and some of its generalizations to other fuzzy logics, including interval based logics, a parallel version of Lorenzen’s game for intuitionistic logic that is adequate for finite- and infinite-valued Gödel logics, and a truth comparison game for infinite-valued Gödel logic.

Background Logic, Critical Thinking, and Irrational Language Games.

Wittgenstein’s language games can be put into a wider service by virtue of elements they share with some contemporary opinions concerning logic and the semantics of computation. I will give two examples: manifestations of language games and their possible variations in logical studies, and their role in some of the recent developments in computer science. It turns out that the current paradigm of computation that Girard termed Ludics bears a striking resemblance to members of language games. (...) Moreover, the kind of interrelations that are emerging could be scrutinised from the viewpoint of logic that virtually necessitates game-theoretic conceptualisations, demonstrating the fact that the meaning of utterances may, in many situations, be understood as Wittgenstein’s language games of ‘showing or telling what one sees’. This provides motivation for the use of games in relation to logic and formal semantics that some commentators have called for. Many of the ideas can be traced to C.S. Peirce, for whom signs were vehicles of strategic communication. The conclusion about Wittgenstein is that the notions of saying and showing converge in his late philosophy. (shrink)

Qualitative Coalitional Games are a variant of coalitional games in which an agent's desires are represented as goals that are either satisfied or unsatisfied, and each choice available to a coalition is a set of goals, which would be jointly satisfied if the coalition made that choice. A coalition in a QCG will typically form in order to bring about a set of goals that will satisfy all members of the coalition. Our goal in this paper is to (...) develop and study logics for reasoning about QCGs. We begin by introducing a logic for reasoning about “static” QCGs, where participants play a single game, and we then introduce and study Temporal QCGs , i.e., games in which a sequence of QCGs is played. In order to represent and reason about such games, we introduce a linear time temporal logic of QCGs, called ℒ. We give a complete axiomatisation of ℒ, use it to investigate the properties of TQCGs, identify its expressive power, establish its complexity, characterise classes of TQGCs with formulas from our logical language, and use it to formulate several solution concepts for TQCGs. (shrink)

Paraconsistent logics are the formal systems in which absurdities do not trivialise the logic. In this paper, we give Hintikka-style game theoretical semantics for a variety of paraconsistent and non-classical logics. For this purpose, we consider Priest’s Logic of Paradox, Dunn’s First-Degree Entailment, Routleys’ Relevant Logics, McCall’s Connexive Logic and Belnap’s four-valued logic. We also present a game theoretical characterisation of a translation between Logic of Paradox/Kleene’s K3 and S5. We underline how non-classical logics require different verification games and (...) prove the correctness theorems of their respective game theoretical semantics. This allows us to observe that paraconsistent logics break the classical bidirectional connection between winning strategies and truth values. (shrink)

We investigate the expressive power of Parikh's Game Logic interpreted in Kripke structures, and show that the syntactical alternation hierarchy of this logic is strict. This is done by encoding the winning condition for parity games of rank n. It follows that Game Logic is not captured by any finite level of the modal -calculus alternation hierarchy. Moreover, we can conclude that model checking for the -calculus is efficiently solvable iff this is possible for Game Logic.