In this paper, a game-theoretical semantics is developed for the so-called alpha part of Charles S. Peirce's System of Existential Graphs of 1896. This alpha part is that portion of Peirce's graphs that corresponds to propositional logic. The paper both expounds a game-theoretical semantics for the graphs that seems close to Peirce's own intentions and proves for the alpha part of the graphs that this semantics is adequate.

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)

The concept of an infinite game played on a finite graph is perhaps novel in the context of an rather extensive recent literature in which infinite games are generally played on an infinite game tree. We claim two advantages for our model, which is admittedly more restrictive. First, our games have a more apparent resemblance to ordinary parlor games in spite of their infinite duration. Second, by distinguishing those nodes of the graph that determine the (...) winning and losing of the game , we are able to offer a complexity analysis that is useful in computer science applications. (shrink)

Formal learning theory formalizes the process of inferring a general result from examples, as in the case of inferring grammars from sentences when learning a language. In this work, we develop a general framework—the supervised learning game—to investigate the interaction between Teacher and Learner. In particular, our proposal highlights several interesting features of the agents: on the one hand, Learner may make mistakes in the learning process, and she may also ignore the potential relation between different hypotheses; on the other (...) hand, Teacher is able to correct Learner’s mistakes, eliminate potential mistakes and point out the facts ignored by Learner. To reason about strategies in this game, we develop a modal logic of supervised learning and study its properties. Broadly, this work takes a small step towards studying the interaction between graph games, logics and formal learning theory. (shrink)

The purpose of this paper is to work toward an explicit logic and semantics for a game theoretically inspired theory of action. The purpose of the logic is to explicate the conceptual machinery implicit in the dialogue-game model of rational discourse developed in Carlson (1983).A variety of ideas and techniques of modal and philosophical logic are used to define a model structure that generalizes the game theoretical notion of a game in extensive form (von Neumann and Morgenstern, 1944). Relative to (...) this model structure, semantic characterizations are given to the action-theoretic notions oftime, possibility, belief, preference, ability, intention, action, andrationality. The unification of these characterizations under the game-theoretical paradigm leads to insights about the logical interdependences between these concepts. (shrink)

The descriptive complexity of a problem is the complexity of describing the problem in some logical formalism. One of the few techniques for proving separation results in descriptive complexity is to make use of games on graphs played between two players, called the spoiler and the duplicator. There are two types of these games, which differ in the order in which the spoiler and duplicator make various moves. In one of these games, the rules seem to be (...) tilted towards favoring the duplicator. These seemingly more favorable rules make it easier to prove separation results, since separation results are proven by showing that the duplicator has a winning strategy. In this paper, the relationship between these games is investigated. It is shown that in one sense, the two games are equivalent. Specifically, each family of graphs used in one game to prove a separation result can in principle be used in the other game to prove the same result. This answers an open question of Ajtai and the author from 1989. It is also shown that in another sense, the games are not equivalent, in that there are situations where the spoiler requires strictly more resources to win one game than the other game. This makes formal the informal statement that one game is easier for the duplicator to win. (shrink)

When seeking to coordinate in a game with imperfect information, it is often relevant for a player to know what other players know. Keeping track of the information acquired in a play of infinite duration may, however, lead to infinite hierarchies of higher-order knowledge. We present a construction that makes explicit which higher-order knowledge is relevant in a game and allows us to describe a class of games that admit coordinated winning strategies with finite memory.

Several results about the game of cops and robbers on infinite graphs are analyzed from the perspective of computability theory. Computable robber-win graphs are constructed with the property that no computable robber strategy is a winning strategy, and such that for an arbitrary computable ordinal \, any winning strategy has complexity at least \}\). Symmetrically, computable cop-win graphs are constructed with the property that no computable cop strategy is a winning strategy. Locally finite infinite trees and graphs are explored. The (...) Turing computability of a binary relation used to classify cop-win graphs is studied, and the computational difficulty of determining the winner for locally finite computable graphs is discussed. (shrink)

We investigate the modal logic of stepwise removal of objects, both for its intrinsic interest as a logic of quantification without replacement, and as a pilot study to better understand the complexity jumps between dynamic epistemic logics of model transformations and logics of freely chosen graph changes that get registered in a growing memory. After introducing this logic (MLSR) and its corresponding removal modality, we analyze its expressive power and prove a bisimulation characterization theorem. We then provide (...) a complete Hillbert-style axiomatization for the logic of stepwise removal in a hybrid language enriched with nominals and public announcement operators. Next, we show that model-checking for MLSR is PSPACE-complete, while its satisfiability problem is undecidable. Lastly, we consider an issue of fine-structure: the expressive power gained by adding the stepwise removal modality to fragments of first-order logic. (shrink)

We investigate the definability in monadic ∑11 and monadic Π11 of the problems REGk, of whether there is a regular subgraph of degree k in some given graph, and XREGk, of whether, for a given rooted graph, there is a regular subgraph of degree k in which the root has degree k, and their restrictions to graphs in which every vertex has degree at most k, namely REGkk and XREGkk, respectively, for k ≥ 2 . Our motivation partly (...) stems from the fact that REGkk and XREGkk are logspace equivalent to CONN and REACH, respectively, for k ≥ 3, where CONN is the problem of whether a given graph is connected and REACH is the problem of whether a given graph has a path joining two given vertices. We use monadic first - order reductions, monadic ∑11 games and a recent technique due to Fagin, Stockmeyer and Vardi to almost completely classify whether these problems are definable in monadic ∑11 and monadic Π11, and we compare the definability of these problems. (shrink)

Although it is known that reachability in undirected finite graphs can be expressed by an existential monadic second-order sentence, our main result is that this is not the case for directed finite graphs (even in the presence of certain "built-in" relations, such as the successor relation). The proof makes use of Ehrenfeucht-Fraisse games, along with probabilistic arguments. However, we show that for directed finite graphs with degree at most k, reachability is expressible by an existential monadic second-order sentence.

As they are conventionally formulated, Boolean games assume that players make their choices in ignorance of the choices being made by other players – they are games of simultaneous moves. For many settings, this is clearly unrealistic. In this paper, we show how Boolean games can be enriched by dependency graphs which explicitly represent the informational dependencies between variables in a game. More precisely, dependency graphs play two roles. First, when we say that variable x depends on (...) variable y, then we mean that when a strategy assigns a value to variable x, it can be informed by the value that has been assigned to y. Second, and as a consequence of the first property, they capture a richer and more plausible model of concurrency than the simultaneous-action model implicit in conventional Boolean games. Dependency graphs implicitly define a partial ordering of the run-time events in a game: if x is dependent on y, then the assignment of a value to y must precede the assignment of a value to x; if x and y are independent, however, then we can say nothing about the ordering of assignments to these variables—the assignments may occur concurrently. We refer to Boolean games with dependency graphs as partial-order Boolean games. After motivating and presenting the partial-order Boolean games model, we explore its properties. We show that while some problems associated with our new games have the same complexity as in conventional Boolean games, for others the complexity blows up dramatically. We also show that the concurrency in partial-order Boolean games can be modelled using a closure-operator semantics, and conclude by considering the relationship of our model to Independence-Friendly logic. (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)

We introduce a graph-theoretical representation of proofs of multiplicative linear logic which yields both a denotational semantics and a notion of truth. For this, we use a locative approach related to game semantics and the Danos–Regnier interpretation of GoI operators as paths in proof nets . We show how we can retrieve from this locative framework both a categorical semantics for Multiplicative Linear Logic with distinct units and a notion of truth. Moreover, we show how a restricted version of (...) our model can be reformulated in the exact same terms as Girardʼs geometry of interaction . This shows that this restriction of our framework gives a combinatorial approach to J.-Y. Girardʼs geometry of interaction in the hyperfinite factor, while using only graph-theoretical notions. (shrink)

In this paper I present a dynamic-epistemic hybrid logic for reasoning about information and intention changes in situations of strategic interaction. I provide a complete axiomatization for this logic, and then use it to study intentions-based transformations of decision problems.

The papers collected in this volume exemplify some of the trends in current approaches to logic, language and computation. Written by authors with varied academic backgrounds, the contributions are intended for an interdisciplinary audience. The first part of this volume addresses issues relevant for multi-agent systems: reasoning with incomplete information, reasoning about knowledge and beliefs, and reasoning about games. Proofs as formal objects form the subject of Part II. Topics covered include: contributions on logical frameworks, linear logic, and different (...) approaches to formalized reasoning. Part III focuses on representations and formal methods in linguistic theory, addressing the areas of comparative and temporal expressions, modal subordination, and compositionality. (shrink)

We consider Effort Games, a game-theoretic model of cooperation in open environments, which is a variant of the principal-agent problem from economic theory. In our multiagent domain, a common project depends on various tasks; carrying out certain subsets of the tasks completes the project successfully, while carrying out other subsets does not. The probability of carrying out a task is higher when the agent in charge of it exerts effort, at a certain cost for that agent. A central authority, (...) called the principal, attempts to incentivize agents to exert effort, but can only reward agents based on the success of the entire project.We model this domain as a normal form game, where the payoffs for each strategy profile are defined based on the different probabilities of carrying out each task and on the boolean function that defines which task subsets complete the project, and which do not. We view this boolean function as a simple coalitional game, and call this game the underlying coalitional game. We suggest the Price of Myopia as a measure of the influence the model of rationality has on the minimal payments the principal has to make in order to motivate the agents in such a domain to exert effort.We consider the computational complexity of testing whether exerting effort is a dominant strategy for an agent, and of finding a reward strategy for this domain, using either a dominant strategy equilibrium or using iterated elimination of dominated strategies. We show these problems are generally #P-hard, and that they are at least as computationally hard as calculating the Banzhaf power index in the underlying coalitional game. We also show that in a certain restricted domain, where the underlying coalitional game is a weighted voting game with certain properties, it is possible to solve all of the above problems in polynomial time. We give bounds on PoM in weighted voting effort games, and provide simulation results regarding PoM in another restricted class of effort games, namely effort games played over Series-Parallel Graphs. (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)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Signs of Logic: Peircean Themes on the Philosophy of Language, Games, and CommunicationRobert W. BurchAhti-Veikko Pietarinen Signs of Logic: Peircean Themes on the Philosophy of Language, Games, and Communication Dordrecht, The Netherlands: Springer, 2006. xiv + 496 pp.This compendious volume of fourteen of Pietarinen's essays on Peirce, plus a three-page set of "Final Words" relating to the work of Robert Aumann, is a "must-have" for both (...) the Peirce scholar and any other philosopher who wishes to relate Peirce's thinking on language and logic to other major thinkers and logical themes of the twentieth century (and beyond). The essays extend or complement discussions of Peirce's logic and philosophy of language that have come to be expected from the "Finnish school" of philosophers, which includes Hintikka and Hilpinen, as well as Pietarinen.Nine of the essays (Chapters 3, 4, 5, 7, 8, 10, 12, 13, and 14), comprising about two-thirds of the entire book, are revised and expanded versions of essays that Pietarinen has previously published. The remaining one-third (approximately) of the book is brand new. Chapters 1 and 2 serve as general introductions to the book's major interests. Chapter 6 is on evidence for constructivism in Peirce, as well as several other topics. Chapter 9 compares Peirce's ideas with Wilfred Hodges's "dialogue foundations" and contains a small philosophical dialogue, which has Peirce, Wittgenstein, and Hodges as its characters. Chapter 11 concerns evolutionary semantics and language [End Page 577] games. And the cursory Chapter 15 of "Final Words" predicts that the game-theoretical approach to semantics and Peirce scholarship has an exploding future, like an erupting volcano.From the foregoing one can gather, correctly, that each essay in the book is separately conceived and that the book is not a single continuous line of argumentative development of one or even of a few themes. Still, the essays are similar to one another in two ways, which ways are closely related to each other. First, they all concern the Peircean doctrine that semeiotic and semeiosis constitute the heartland of logic, and that this heartland is vastly richer than the sort of pure mathematical calculi that might mistakenly be taken for logic itself (and were in fact so taken for much of the twentieth century). Second, they all deal with the Peircean insistence that semeiosis is inherently a form of Praxis in which interchanges occur between or among various "agents" or "actors," one of which might be nature itself; so that at the very least logic is dialogic, and so that more generally one may justifiably speak of semeiosis as involving something like a "game." These lines of exploration will surprise no seasoned Peirce scholar, but the special genius of the Finnish "school" of Peirce interpretation from Hintikka forward is to have spotlighted in dramatic fashion the game-theoretical nature of many elements in Peirce's semeiotic.In these essays Pietarinen amply exhibits two attainments of mind that are indispensable for the serious Peirce scholar. One is the object of what we might call "The Eisele Imperative" after the vehement urgings of Carolyn Eisele: this is the possession of some significant degree of sophistication in mathematics. Pietarinen's mathematical skills are plainly sufficient for his discussions of logic and game theory. For example, his Chapter 5 is an excellent technical discussion of Peirce's existential graphs, and his Chapter 7 is an admirable technical discussion of certain aspects of game theory and generalized quantifiers. Another crucial skill needed by the serious Peirce scholar is the object of what we might call "The Pietarinen Corrective" after the practices exemplified in these essays: this is the possession of extremely broad knowledge of recent and contemporary philosophy, especially philosophy of language, logic, metaphysics, epistemology, and philosophy of science.Indeed, the breadth of Pietarinen's acquaintance with twentieth century logic and philosophy of language, both "analytic" and "continental," is breathtaking. This allows him scope for comparisons of Peirce's views with an extraordinarily wide, and not always easily anticipated, range of philosophers and philosophic traditions. Pietarinen's range of reference is so extensive that it would be pointless to attempt to discuss in... (shrink)

We give a sufficient condition for the inexpressibility of the k-th extended vectorization of a generalized quantifier $\sf Q$ in ${\rm FO}({\vec Q}_k)$ , the extension of first-order logic by all k-ary quantifiers. The condition is based on a model construction which, given two ${\rm FO}({\vec Q}_1)$ -equivalent models with certain additional structure, yields a pair of ${\rm FO}({\vec Q}_k)$ -equivalent models. We also consider some applications of this condition to quantifiers that correspond to graph properties, such as connectivity (...) and planarity. (shrink)

We investigate Maker–Breaker games on graphs of size $\aleph _1$ in which Maker’s goal is to build a copy of the host graph. We establish a firm dependence of the outcome of the game on the axiomatic framework. Relating to this, we prove that there is a winning strategy for Maker in the $K_{\omega,\omega _1}$ -game under ZFC+MA+ $\neg $ CH and a winning strategy for Breaker under ZFC+CH. We prove a similar result for the $K_{\omega _1}$ -game. (...) Here, Maker has a winning strategy under ZF+DC+AD, while Breaker has one under ZFC+CH again. (shrink)

The article studies properties of functional dependencies between strategies of players in Nash equilibria of multiplayer strategic games. The main focus is on the properties of functional dependencies in the context of a fixed dependency graph for payoff functions. A logical system describing properties of functional dependence for any given graph is proposed and is proven to be complete.

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)

In _Modal Logic for Open Minds,_ Johan van Benthem provides an up-to-date introduction to the field of modal logic, outlining its major ideas and exploring the numerous ways in which various academic fields have adopted it. Van Benthem begins with the basic theories of modal logic, semantics, bisimulation, and axiomatics, and also covers more advanced topics, such as expressive power and computational complexity. The book then moves to a wide range of applications, including new developments in information flow, intelligent agency, (...) and games. Taken together, the chapters show modal logic at the crossroads of philosophy, mathematics, linguistics, computer science, and economics. Most of the chapters are followed by exercises, making this volume ideal for undergraduate and graduate students in philosophy, computer science, symbolic systems, cognitive science, and linguistics. (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.

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 μ-calculus alternation hierarchy. Moreover, we can conclude that model checking for the μ-calculus is efficiently solvable iff this is possible for Game Logic.

In _Modal Logic for Open Minds,_ Johan van Benthem provides an up-to-date introduction to the field of modal logic, outlining its major ideas and exploring the numerous ways in which various academic fields have adopted it. Van Benthem begins with the basic theories of modal logic, semantics, bisimulation, and axiomatics, and also covers more advanced topics, such as expressive power and computational complexity. The book then moves to a wide range of applications, including new developments in information flow, intelligent agency, (...) and games. Taken together, the chapters show modal logic at the crossroads of philosophy, mathematics, linguistics, computer science, and economics. Most of the chapters are followed by exercises, making this volume ideal for undergraduate and graduate students in philosophy, computer science, symbolic systems, cognitive science, and linguistics. (shrink)

Peirce believed that his pragmaticism can be conclusively proven. Beginning in 1903, he drafted several attempts, ending by 1908 with a semeiotic proof. Around 1905, he exposes the proof using the theory of Existential Graphs . This paper modernizes the semantics Peirce proposed for EGs in terms of game-theoretic semantics . Peirce's 1905 proof is then reconstructed in three parts, by relating pragmaticism to the GTS conception of meaning, showing that Peirce's proof is an argument for a relational structure of (...) the meaning of intellectual signs that our interpretative and strategic practices give rise to, and bringing out the key links between EGs and pragmaticism. (shrink)

When no single universal model for a set of structures exists at a given cardinal, then one may ask in which models of set theory does there exist a small family which embeds the rest. We show that for λ+-graphs (λ regular) omitting cliques of some finite or uncountable cardinality, it is consistent that there are small universal families and 2λ > λ+. In particular, we get such a result for triangle-free graphs.

If P is a hereditary property then we show that, for the existence of a perfect f-factor, P is a sufficient condition for countable graphs and yields a sufficient condition for graphs of size ℵ1. Further we give two examples of a hereditary property which is even necessary for the existence of a perfect f-factor. We also discuss the ℵ2-case.

In the 1890s, Peirce reformulated quantification theory by expressing it in a language of diagrams, called existential graphs. Peirce thought that the iconicity of his graphs made them suitable for analyzing logical reasoning. Iconic signs can be said to show their meaning, and this paper studies the ways in which graphs do this. Peirce's pragmatic analysis of propositions resembles game-theoretical semantics, and existential graphs show what they mean by displaying the structure of the semantic game for the proposition represented by (...) a graph. (shrink)

We consider systems of rational agents who act and interact in pursuit of their individual and collective objectives. We study and formalise the reasoning of an agent, or of an external observer, about the expected choices of action of the other agents based on their objectives, in order to assess the reasoner’s ability, or expectation, to achieve their own objective. To formalize such reasoning we extend Pauly’s Coalition Logic with three new modal operators of conditional strategic reasoning, thus introducing the (...) Logic for Local Conditional Strategic Reasoning \. We provide formal semantics for the new conditional strategic operators in concurrent game models, introduce the matching notion of bisimulation for each of them, prove bisimulation invariance and Hennessy–Milner property for each of them, and discuss and compare briefly their expressiveness. Finally, we also propose systems of axioms for each of the basic operators of \ and for the full logic. (shrink)

We present a setting in which the search for a proof of B or a refutation of B can be carried out simultaneously: in contrast, the usual approach in automated deduction views proving B or proving ¬B as two, possibly unrelated, activities. Our approach to proof and refutation is described as a two-player game in which each player follows the same rules. A winning strategy translates to a proof of the formula and a counter-winning strategy translates to a refutation of (...) the formula. The game is described for multiplicative and additive linear logic . A game theoretic treatment of the multiplicative connectives is intricate and our approach to it involves two important ingredients. First, labeled graph structures are used to represent positions in a game and, second, the game playing must deal with the failure of a given player and with an appropriate resumption of play. This latter ingredient accounts for the fact that neither player might win. (shrink)

A sound and complete axiomatization of two tabloid blogs is presented, Leiter Logic (KB) and Deontic Leiter Logic (KDB), the latter of which can be extended to Shame Game Logic for multiple agents. The (B) schema describes the mechanism behind this class of tabloids, and illustrates the perils of interpreting a provability operator as an epistemic modal. To mark this difference, and to avoid sullying Brouwer's good name, the (B) schema for epistemic modals should be called the Blog Schema.

In the most popular logics combining knowledge and awareness, it is not possible to express statements about knowledge of unawareness such as “Ann knows that Bill is aware of something Ann is not aware of”—without using a stronger statement such as “Ann knows that Bill is aware of \(p\) and Ann is not aware of \(p\) ”, for some particular \(p\) . In Halpern and Rêgo (Proceedings of KR 2006; Games Econ Behav 67(2):503–525, 2009b) Halpern and Rêgo introduced (...) a logic in which such statements about knowledge of unawareness can be expressed. The logic extends the traditional framework with quantification over formulae, and is thus very expressive. As a consequence, it is not decidable. In this paper we introduce a decidable logic which can be used to reason about certain types of unawareness. Our logic extends the traditional framework with an operator expressing full awareness, i.e., the fact that an agent is aware of everything, and another operator expressing relative awareness, the fact that one agent is aware of everything another agent is aware of. The logic is less expressive than Halpern’s and Rêgo’s logic. It is, however, expressive enough to express all of the motivating examples in Halpern and Rêgo (Proceedings of KR 2006; Games Econ Behav 67(2):503–525, 2009b). In addition to proving that the logic is decidable and that its satisfiability problem is PSPACE-complete, we present an axiomatisation which we show is sound and complete. (shrink)

We describe Peirce’s 1903 system of modal gamma graphs, its transformation rules of inference, and the interpretation of the broken-cut modal operator. We show that Peirce proposed the normality rule in his gamma system. We then show how various normal modal logics arise from Peirce’s assumptions concerning the broken-cut notation. By developing an algebraic semantics we establish the completeness of fifteen modal logics of gamma graphs. We show that, besides logical necessity and possibility, Peirce proposed an epistemic interpretation (...) of the broken-cut modality, and that he was led to analyze constructions of knowledge in the style of epistemic logic. (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)

We take a logical approach to threshold models, used to study the diffusion of e.g. new technologies or behaviors in social net-works. In short, threshold models consist of a network graph of agents connected by a social relationship and a threshold to adopt a possibly cascading behavior. Agents adopt new behavior when the proportion of their neighbors who have already adopted it meets the threshold. Under this adoption policy, threshold models develop dynamically with a guaranteed fixed point. We construct (...) a minimal dynamic propositional logic to describe the threshold dynamics and show that the logic is sound and complete. We then extend this framework with an epistemic dimension and investigate how information about more distant neighbors’ behaviors allows agents to anticipate changes in behavior of their closer neighbors. It is shown that this epistemic prediction dynamics is equivalent to the non-epistemic threshold model dynamics if and only if agents know exactly their neighbors’ behavior. We further show results regarding fixed points and convergence speed,and provide a partial set of reduction laws, venues for further research, and graphical representations of the dynamics. (shrink)

A sound and complete axiomatization of two philosophy tabloids is given, Leiter Logic (KB) and Deontic Leiter Logic (KDB), in single agent format, the latter of which can be extended to Shame Game Logic for multiple players. The (B) schema captures the mechanism of tabloid inference, illustrating the perils of interpreting a provability operator as an epistemic modal. To mark this hazard, and to preserve Brouwer's good name, the (B) schema interpreted to govern epistemic modals should be called The Blog (...) Schema. (shrink)

The study of graph structure has advanced in recent years with great strides: finite graphs can be described algebraically, enabling them to be constructed out of more basic elements. Separately the properties of graphs can be studied in a logical language called monadic second-order logic. In this book, these two features of graph structure are brought together for the first time in a presentation that unifies and synthesizes research over the last 25 years. The author not only provides (...) a thorough description of the theory, but also details its applications, on the one hand to the construction of graph algorithms, and, on the other to the extension of formal language theory to finite graphs. Consequently the book will be of interest to graduate students and researchers in graph theory, finite model theory, formal language theory, and complexity theory. (shrink)

A theory of two-sided containers, denoted ZF2, is introduced. This theory is then shown to be synonymous to ZF in the sense of Visser (2006), via an interpretation involving Quine pairs. Several subtheories of ZF2, and their relationships with ZF, are also examined. We include a short discussion of permutation models (in the sense of Rieger–Bernays) over ZF2. We close with highlighting some areas for future research, mostly motivated by the need to understand non-wellfounded games.