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Modal Logic

Cambridge University Press (2002)

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  1. Arthur Prior and Hybrid Logic.Patrick Blackburn - 2006 - Synthese 150 (3):329-372.
    Contemporary hybrid logic is based on the idea of using formulas as terms, an idea invented and explored by Arthur Prior in the mid-1960s. But Prior’s own work on hybrid logic remains largely undiscussed. This is unfortunate, since hybridisation played a role that was both central to and problematic for his philosophical views on tense. In this paper I introduce hybrid logic from a contemporary perspective, and then examine the role it played in Prior’s work.
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  • Determining the Environment: A Modal Logic for Closed Interaction.Jan Broersen, Rosja Mastop, John-Jules Meyer & Paolo Turrini - 2009 - Synthese 169 (2):351-369.
    The aim of the work is to provide a language to reason about Closed Interactions, i.e. all those situations in which the outcomes of an interaction can be determined by the agents themselves and in which the environment cannot interfere with they are able to determine. We will see that two different interpretations can be given of this restriction, both stemming from Pauly Representation Theorem. We will identify such restrictions and axiomatize their logic. We will apply the formal tools to (...)
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  • Multi-Modal CTL: Completeness, Complexity, and an Application. [REVIEW]Thomas Ågotnes, Wiebe Van der Hoek, Juan A. Rodríguez-Aguilar, Carles Sierra & Michael Wooldridge - 2009 - Studia Logica 92 (1):1-26.
    We define a multi-modal version of Computation Tree Logic (ctl) by extending the language with path quantifiers E δ and A δ where δ denotes one of finitely many dimensions, interpreted over Kripke structures with one total relation for each dimension. As expected, the logic is axiomatised by taking a copy of a ctl axiomatisation for each dimension. Completeness is proved by employing the completeness result for ctl to obtain a model along each dimension in turn. We also show that (...)
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  • Beyond Regularity for Presburger Modal Logic.Facundo Carreiro & Stéphane Demri - 2012 - In Thomas Bolander, Torben Braüner, Silvio Ghilardi & Lawrence Moss (eds.), Advances in Modal Logic, Volume 9. CSLI Publications. pp. 161-182.
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  • What One May Come to Know.van Benthem Johan - 2004 - Analysis 64 (2):95–105.
    The general verificationist thesis says that What is true can be known or formally: φ → ◊Kφ VT Fitch's argument trivializes this principle. It uses a weak modal epistemic logic to show that VT collapses truth and knowledge, by taking a clever substitution instance for φ: P ∧ ¬KP → ◊ K(P ∧ ¬KP) Then we have the following chain of three conditionals (a) ◊ K(P ∧ ¬KP) → ◊ (KP ∧ K¬KP) in the minimal modal logic for the knowledge (...)
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  • On Definability in Multimodal Logic.Joseph Y. Halpern, Dov Samet & Ella Segev - 2009 - Review of Symbolic Logic 2 (3):451-468.
    Three notions of definability in multimodal logic are considered. Two are analogous to the notions of explicit definability and implicit definability introduced by Beth in the context of first-order logic. However, while by Beth’s theorem the two types of definability are equivalent for first-order logic, such an equivalence does not hold for multimodal logics. A third notion of definability, reducibility, is introduced; it is shown that in multimodal logics, explicit definability is equivalent to the combination of implicit definability and reducibility. (...)
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  • Admissible Bases Via Stable Canonical Rules.Nick Bezhanishvili, David Gabelaia, Silvio Ghilardi & Mamuka Jibladze - 2016 - Studia Logica 104 (2):317-341.
    We establish the dichotomy property for stable canonical multi-conclusion rules for IPC, K4, and S4. This yields an alternative proof of existence of explicit bases of admissible rules for these logics.
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  • Dynamic Logics of Knowledge and Access.Tomohiro Hoshi & Eric Pacuit - 2010 - Synthese 177 (1):29 - 49.
    A recurring issue in any formal model representing agents' (changing) informational attitudes is how to account for the fact that the agents are limited in their access to the available inference steps, possible observations and available messages. This may be because the agents are not logically omniscient and so do not have unlimited reasoning ability. But it can also be because the agents are following a predefined protocol that explicitly limits statements available for observation and/or communication. Within the broad literature (...)
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  • Positive Logic with Adjoint Modalities: Proof Theory, Semantics, and Reasoning About Information: Positive Logic with Adjoint Modalities.Mehrnoosh Sadrzadeh - 2010 - Review of Symbolic Logic 3 (3):351-373.
    We consider a simple modal logic whose nonmodal part has conjunction and disjunction as connectives and whose modalities come in adjoint pairs, but are not in general closure operators. Despite absence of negation and implication, and of axioms corresponding to the characteristic axioms of _T_, _S4_, and _S5_, such logics are useful, as shown in previous work by Baltag, Coecke, and the first author, for encoding and reasoning about information and misinformation in multiagent systems. For the propositional-only fragment of such (...)
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  • Social Laws in Alternating Time: Effectiveness, Feasibility, and Synthesis.Wiebe van Der Hoek, Mark Roberts & Michael Wooldridge - 2007 - Synthese 156 (1):1-19.
    Since it was first proposed by Moses, Shoham, and Tennenholtz, the social laws paradigm has proved to be one of the most compelling approaches to the offline coordination of multiagent systems. In this paper, we make four key contributions to the theory and practice of social laws in multiagent systems. First, we show that the "Alternating-time Temporal Logic" of Alur, Henzinger, and Kupferman provides an elegant and powerful framework within which to express and understand social laws for multiagent systems. Second, (...)
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  • The Quasi-Lattice of Indiscernible Elements.Mauri Cunha do Nascimento, Décio Krause & Hércules Araújo Feitosa - 2011 - Studia Logica 97 (1):101-126.
    The literature on quantum logic emphasizes that the algebraic structures involved with orthodox quantum mechanics are non distributive. In this paper we develop a particular algebraic structure, the quasi-lattice (J-lattice), which can be modeled by an algebraic structure built in quasi-set theory Q. This structure is non distributive and involve indiscernible elements. Thus we show that in taking into account indiscernibility as a primitive concept, the quasi-lattice that 'naturally' arises is non distributive.
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  • Belief Fusion and Revision: An Overview Based on Epistemic Logic Semantics.Churn-Jung Liau - 2004 - Journal of Applied Non-Classical Logics 14 (3):247-274.
    In this paper, we formulate some approaches to belief fusion and revision using epistemic logic semantics. Fusion operators considered in this paper are majority merging, arbitration, and general merging. Some modalities corresponding to belief fusion and revision operators are incorporated into epistemic logics. The Kripke semantics of these extended logics are presented. While most existing approaches treat belief fusion and revision operators as meta-level constructs, we directly incorporate these operators into our object logic language. By doing so, we both extend (...)
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  • Dynamic Logics of the Region-Based Theory of Discrete Spaces.Philippe Balbiani, Tinko Tinchev & Dimiter Vakarelov - 2007 - Journal of Applied Non-Classical Logics 17 (1):39-61.
    The aim of this paper is to give new kinds of modal logics suitable for reasoning about regions in discrete spaces. We call them dynamic logics of the region-based theory of discrete spaces. These modal logics are linguistic restrictions of propositional dynamic logic with the global diamond E. Their formulas are equivalent to Boolean combinations of modal formulas like E where A and B are Boolean terms and α is a relational term. Examining what we can say about dynamic models (...)
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  • On the Strength and Scope of DLS.Willem Conradie - 2006 - Journal of Applied Non-Classical Logics 16 (3-4):279-296.
    We provide syntactic necessary and sufficient conditions on the formulae reducible by the second-order quantifier elimination algorithm DLS. It is shown that DLS is compete for all modal Sahlqvist and Inductive formulae, and that all modal formulae in a single propositional variable on which DLS succeeds are canonical.
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  • BDD-Based Decision Procedures for the Modal Logic K ★.Guoqiang Pan, Ulrike Sattler & Moshe Y. Vardi - 2006 - Journal of Applied Non-Classical Logics 16 (1-2):169-207.
    We describe BDD-based decision procedures for the modal logic K. Our approach is inspired by the automata-theoretic approach, but we avoid explicit automata construction. Instead, we compute certain fixpoints of a set of types — which can be viewed as an on-the-fly emptiness of the automaton. We use BDDs to represent and manipulate such type sets, and investigate different kinds of representations as well as a “level-based” representation scheme. The latter turns out to speed up construction and reduce memory consumption (...)
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  • Establishing Connections Between Aristotle's Natural Deduction and First-Order Logic.Edgar Jose Andrade & Edward Samuel Becerra - 2008 - History and Philosophy of Logic 29 (4):309-325.
    This article studies the mathematical properties of two systems that model Aristotle's original syllogistic and the relationship obtaining between them. These systems are Corcoran's natural deduction syllogistic and ?ukasiewicz's axiomatization of the syllogistic. We show that by translating the former into a first-order theory, which we call T RD, we can establish a precise relationship between the two systems. We prove within the framework of first-order logic a number of logical properties about T RD that bear upon the same properties (...)
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  • On Bellissima’s Construction of the Finitely Generated Free Heyting Algebras, and Beyond.Luck Darnière & Markus Junker - 2010 - Archive for Mathematical Logic 49 (7-8):743-771.
    We study finitely generated free Heyting algebras from a topological and from a model theoretic point of view. We review Bellissima’s representation of the finitely generated free Heyting algebra; we prove that it yields an embedding in the profinite completion, which is also the completion with respect to a naturally defined metric. We give an algebraic interpretation of the Kripke model used by Bellissima as the principal ideal spectrum and show it to be first order interpretable in the Heyting algebra, (...)
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  • The Expressive Power of Memory Logics.Carlos Areces, Diego Figueira, Santiago Figueira & Sergio Mera - 2011 - Review of Symbolic Logic 4 (2):290-318.
    We investigate the expressive power of memory logics. These are modal logics extended with the possibility to store (or remove) the current node of evaluation in (or from) a memory, and to perform membership tests on the current memory. From this perspective, the hybrid logic (↓), for example, can be thought of as a particular case of a memory logic where the memory is an indexed list of elements of the domain.
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  • Canonicity Results of Substructural and Lattice-Based Logics.Tomoyuki Suzuki - 2011 - Review of Symbolic Logic 4 (1):1-42.
    In this paper, we extend the canonicity methodology in Ghilardi & Meloni (1997) to arbitrary lattice expansions, and syntactically describe canonical inequalities for lattice expansions consisting of -meet preserving operations, -multiplicative operations, adjoint pairs, and constants. This approach gives us a uniform account of canonicity for substructural and lattice-based logics. Our method not only covers existing results, but also systematically accounts for many canonical inequalities containing nonsmooth additive and multiplicative uniform operations. Furthermore, we compare our technique with the approach in (...)
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  • Decidability of Independence-Friendly Modal Logic.Merlijn Sevenster - 2010 - Review of Symbolic Logic 3 (3):415-441.
    In this paper we consider an independence-friendly modal logic, IFML. It follows from results in the literature that qua expressive power, IFML is a fragment of second-order existential logic, , that cannot be translated into first-order logic. It is also known that IFML lacks the tree structure property. We show that IFML has the , a weaker version of the tree structure property, and that its satisfiability problem is solvable in 2NEXP. This implies that this paper reveals a new decidable (...)
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  • Arrow Update Logic.Barteld Kooi & Bryan Renne - 2011 - Review of Symbolic Logic 4 (4):536-559.
    We present Arrow Update Logic, a theory of epistemic access elimination that can be used to reason about multi-agent belief change. While the belief-changing of Arrow Update Logic can be transformed into equivalent belief-changing from the popular Dynamic Epistemic Logic approach, we prove that arrow updates are sometimes exponentially more succinct than action models. Further, since many examples of belief change are naturally thought of from Arrow Update Logicrelativized” common knowledge familiar from the Dynamic Epistemic Logic literature.
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  • Logic in a Social Setting.Johan van Benthem - 2011 - Episteme 8 (3):227-247.
    Taking Backward Induction as its running example, this paper explores avenues for a logic of information-driven social action. We use recent results on limit phenomena in knowledge updating and belief revision, procedural rationality, and a ‘Theory of Play’ analyzing how games are played by different agents.
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  • Technical Modal Logic.Marcus Kracht - 2011 - Philosophy Compass 6 (5):350-359.
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  • Axiomatizing Relativistic Dynamics Using Formal Thought Experiments.Attila Molnár & Gergely Székely - 2015 - Synthese 192 (7):2183-2222.
    Thought experiments are widely used in the informal explanation of Relativity Theories; however, they are not present explicitly in formalized versions of Relativity Theory. In this paper, we present an axiom system of Special Relativity which is able to grasp thought experiments formally and explicitly. Moreover, using these thought experiments, we can provide an explicit definition of relativistic mass based only on kinematical concepts and we can geometrically prove the Mass Increase Formula in a natural way, without postulates of conservation (...)
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  • 'Knowable' as 'Known After an Announcement'.Philippe Balbiani, Alexandru Baltag, Hans van Ditmarsch, Andreas Herzig, Tomohiro Hoshi & Tiago de Lima - 2008 - Review of Symbolic Logic 1 (3):305-334.
    Public announcement logic is an extension of multiagent epistemic logic with dynamic operators to model the informational consequences of announcements to the entire group of agents. We propose an extension of public announcement logic with a dynamic modal operator that expresses what is true after any announcement: after which , does it hold that Kφ? We give various semantic results and show completeness for a Hilbert-style axiomatization of this logic. There is a natural generalization to a logic for arbitrary events.
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  • Modal and Guarded Characterisation Theorems Over Finite Transition Systems.Martin Otto - 2004 - Annals of Pure and Applied Logic 130 (1-3):173-205.
    We explore the finite model theory of the characterisation theorems for modal and guarded fragments of first-order logic over transition systems and relational structures of width two. A new construction of locally acyclic bisimilar covers provides a useful analogue of the well known tree-like unravellings that can be used for the purposes of finite model theory. Together with various other finitary bisimulation respecting model transformations, and Ehrenfeucht–Fraïssé game arguments, these covers allow us to upgrade finite approximations for full bisimulation equivalence (...)
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  • A Logical Characterisation of Qualitative Coalitional Games.Paul E. Dunne, Wiebe van der Hoek & Michael Wooldridge - 2007 - Journal of Applied Non-Classical Logics 17 (4):477-509.
    Qualitative coalitional games were introduced as abstract formal models of goal-oriented cooperative systems. A QCG is a game in which each agent is assumed to have some goal to achieve, and in which agents must typically cooperate with others in order to satisfy their goals. In this paper, we show how it is possible to reason about QCGs using Coalition Logic, a formalism intended to facilitate reasoning about coalitional powers in game-like multiagent systems. We introduce a correspondence relation between QCGs (...)
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  • Multi-Modal CTL: Completeness, Complexity, and an Application.Thomas Ågotnes, Wiebe Van der Hoek, Juan A. Rodríguez-Aguilar, Carles Sierra & Michael Wooldridge - 2009 - Studia Logica 92 (1):1 - 26.
    We define a multi-modal version of Computation Tree Logic (CTL) by extending the language with path quantifiers $E^\delta $ and $E^\delta $ where δ denotes one of finitely many dimensions, interpreted over Kripke structures with one total relation for each dimension. As expected, the logic is axiomatised by taking a copy of a CTL axiomatisation for each dimension. Completeness is proved by employing the completeness result for CTL to obtain a model along each dimension in turn. We also show that (...)
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  • Dynamics We Can Believe In: A View From the Amsterdam School on the Centenary of Evert Willem Beth.Cédric Dégremont & Jonathan Zvesper - 2011 - Synthese 179 (2):223 - 238.
    Logic is breaking out of the confines of the single-agent static paradigm that has been implicit in all formal systems until recent times. We sketch some recent developments that take logic as an account of information-driven interaction. These two features, the dynamic and the social, throw fresh light on many issues within logic and its connections with other areas, such as epistemology and game theory.
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  • Coalgebraic Logic for Stochastic Right Coalgebras.Ernst-Erich Doberkat & Christoph Schubert - 2009 - Annals of Pure and Applied Logic 159 (3):268-284.
    We generalize stochastic Kripke models and Markov transition systems to stochastic right coalgebras. These are coalgebras for a functor with as an endofunctor on the category of analytic spaces, and is the subprobability functor. The modal operators are generalized through predicate liftings which are set-valued natural transformations involving the functor. Two states are equivalent iff they cannot be separated by a formula. This equivalence relation is used to construct a cospan for logical equivalent coalgebras under a separation condition for the (...)
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  • Syntactic Awareness in Logical Dynamics.Davide Grossi & Fernando R. Velázquez-Quesada - 2015 - Synthese 192 (12):4071-4105.
    The paper develops an interface between syntax-based logical models of awareness and dynamic epistemic logic. The framework is shown to be able to accommodate a variety of notions of awareness and knowledge, as well as their dynamics. This, it is argued, offers a natural formal environment for the analysis of epistemic phenomena typical of multi-agent information exchange, such as how agents become aware of relevant details, how they perform inferences and how they share their information within a group. Technically, the (...)
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  • Probabilization of Logics: Completeness and Decidability. [REVIEW]Pedro Baltazar - 2013 - Logica Universalis 7 (4):403-440.
    The probabilization of a logic system consists of enriching the language (the formulas) and the semantics (the models) with probabilistic features. Such an operation is said to be exogenous if the enrichment is done on top, without internal changes to the structure, and is called endogenous otherwise. These two different enrichments can be applied simultaneously to the language and semantics of a same logic. We address the problem of studying the transference of metaproperties, such as completeness and decidability, to the (...)
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  • Dynamic Epistemic Logic I: Modeling Knowledge and Belief.Eric Pacuit - 2013 - Philosophy Compass 8 (9):798-814.
    Dynamic epistemic logic, broadly conceived, is the study of logics of information change. This is the first paper in a two-part series introducing this research area. In this paper, I introduce the basic logical systems for reasoning about the knowledge and beliefs of a group of agents.
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  • Verso una ontologia formale "tommasiana": la contro-implicazione stretta causale.Francesco Panizzoli - 2015 - Epistemologia 38 (1):47-67.
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  • Vagueness: Subvaluationism.Pablo Cobreros - 2013 - Philosophy Compass 8 (5):472-485.
    Supervaluationism is a well known theory of vagueness. Subvaluationism is a less well known theory of vagueness. But these theories cannot be taken apart, for they are in a relation of duality that can be made precise. This paper provides an introduction to the subvaluationist theory of vagueness in connection to its dual, supervaluationism. A survey on the supervaluationist theory can be found in the Compass paper of Keefe (2008); our presentation of the theory in this paper will be short (...)
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  • Constrained Consequence.Katarina Britz, Johannes Heidema & Ivan Varzinczak - 2011 - Logica Universalis 5 (2):327-350.
    There are various contexts in which it is not pertinent to generate and attend to all the classical consequences of a given premiss—or to trace all the premisses which classically entail a given consequence. Such contexts may involve limited resources of an agent or inferential engine, contextual relevance or irrelevance of certain consequences or premisses, modelling everyday human reasoning, the search for plausible abduced hypotheses or potential causes, etc. In this paper we propose and explicate one formal framework for a (...)
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  • Distributed Knowledge.Floris Roelofsen - 2007 - Journal of Applied Non-Classical Logics 17 (2):255-273.
    This paper provides a complete characterization of epistemic models in which distributed knowledge complies with the principle of full communication (van der Hoek et al., 1999; Gerbrandy, 1999). It also introduces an extended notion of bisimulation and corresponding model comparison games that match the expressive power of distributed knowledge operators.
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  • Bi-Simulating in Bi-Intuitionistic Logic.Guillermo Badia - 2016 - Studia Logica 104 (5):1037-1050.
    Bi-intuitionistic logic is the result of adding the dual of intuitionistic implication to intuitionistic logic. In this note, we characterize the expressive power of this logic by showing that the first order formulas equivalent to translations of bi-intuitionistic propositional formulas are exactly those preserved under bi-intuitionistic directed bisimulations. The proof technique is originally due to Lindstrom and, in contrast to the most common proofs of this kind of result, it does not use the machinery of neither saturated models nor elementary (...)
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  • A STIT Logic for Reasoning About Social Influence.Emiliano Lorini & Giovanni Sartor - 2016 - Studia Logica 104 (4):773-812.
    In this paper we propose a method for modeling social influence within the STIT approach to action. Our proposal consists in extending the STIT language with special operators that allow us to represent the consequences of an agent’s choices over the rational choices of another agent.
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  • On a New Tentative Solution to Fitch’s Paradox.Alessandro Giordani - 2016 - Erkenntnis 81 (3):597-611.
    In a recent paper, Alexander argues that relaxing the requirement that sound knowers know their own soundness might provide a solution to Fitch’s paradox and introduces a suitable axiomatic system where the paradox is avoided. In this paper an analysis of this solution is proposed according to which the effective move for solving the paradox depends on the axiomatic treatment of the ontic modality rather than the limitations imposed on the epistemic one. It is then shown that, once the ontic (...)
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  • Logics of Temporal-Epistemic Actions.Bryan Renne, Joshua Sack & Audrey Yap - 2016 - Synthese 193 (3):813-849.
    We present Dynamic Epistemic Temporal Logic, a framework for reasoning about operations on multi-agent Kripke models that contain a designated temporal relation. These operations are natural extensions of the well-known “action models” from Dynamic Epistemic Logic. Our “temporal action models” may be used to define a number of informational actions that can modify the “objective” temporal structure of a model along with the agents’ basic and higher-order knowledge and beliefs about this structure, including their beliefs about the time. In essence, (...)
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  • My Beliefs About Your Beliefs: A Case Study in Theory of Mind and Epistemic Logic.Hans Van Ditmarsch & Willem Labuschagne - 2007 - Synthese 155 (2):191 - 209.
    We model three examples of beliefs that agents may have about other agents' beliefs, and provide motivation for this conceptualization from the theory of mind literature. We assume a modal logical framework for modelling degrees of belief by partially ordered preference relations. In this setting, we describe that agents believe that other agents do not distinguish among their beliefs ('no preferences'), that agents believe that the beliefs of other agents are in part as their own ('my preferences'), and the special (...)
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  • Paradoxes of Interaction?Johannes Stern & Martin Fischer - 2015 - Journal of Philosophical Logic 44 (3):287-308.
    Since Montague’s work it is well known that treating a single modality as a predicate may lead to paradox. In their paper “No Future”, Horsten and Leitgeb show that if the two temporal modalities are treated as predicates paradox might arise as well. In our paper we investigate whether paradoxes of multiple modalities, such as the No Future paradox, are genuinely new paradoxes or whether they “reduce” to the paradoxes of single modalities. In order to address this question we develop (...)
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  • Ontological Symmetry in Language: A Brief Manifesto.Philippe Schlenker - 2006 - Mind Language 21 (4):504-539.
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  • A Generalization of the Routley-Meyer Semantic Framework.Morgan Thomas - 2015 - Journal of Philosophical Logic 44 (4):411-427.
    We develop an axiomatic theory of “generalized Routley-Meyer logics.” These are first-order logics which are can be characterized by model theories in a certain generalization of Routley-Meyer semantics. We show that all GRM logics are subclassical, have recursively enumerable consequence relations, satisfy the compactness theorem, and satisfy the standard structural rules and conjunction and disjunction introduction/elimination rules. We also show that the GRM logics include classical logic, intuitionistic logic, LP/K3/FDE, and the relevant logics.
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  • Modelling Simultaneous Games in Dynamic Logic.Johan Van Benthem, Sujata Ghosh & Fenrong Liu - 2008 - Synthese 165 (2):247 - 268.
    We make a proposal for formalizing simultaneous games at the abstraction level of player's powers, combining ideas from dynamic logic of sequential games and concurrent dynamic logic. We prove completeness for a new system of 'concurrent game logic' CDGL with respect to finite non-determined games. We also show how this system raises new mathematical issues, and throws light on branching quantifiers and independence-friendly evaluation games for first-order logic.
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