This chapter contains sections titled: Modal Concepts The Semantics of Modalities and Systems of Modal Logic Modality and Quantification Deontic, Epistemic, and Temporal Modalities Epistemic Logic Deontic Logic Temporal Frames Conditional Obligations and Rules of Detachment.

Nicholas Rescher. Schools.” An English translation of these sections of the text is given in Appendix A below. No matter how difficult or boring this material proved for the Muslim schoolmaster, it is of the greatest relevance for our interests.

Temporal justification logic is a new family of temporal logics of knowledge in which the knowledge of agents is modelled using a justification logic. In this paper, we present various temporal justification logics involving both past and future time modalities. We combine Artemov’s logic of proofs with linear temporal logic with past, and we also investigate several principles describing the interaction of justification and time. We present two kinds of semantics for our (...) temporal justification logics, one based on interpreted systems and Fitting models and the other based on Mkrtychev models, and further, we establish soundness and completeness. We show that the internalization property holds in some of the temporal justification logics. We further investigate the two well-known epistemic-temporal notions of no forgetting and no learning in the framework of justification logics. Finally, we present temporal justification logics that avoid the logical omniscience problem. (shrink)

In previous works, we presented a modification of the usual possible world semantics by introducing an independent temporal structure in each world and using accessibility functions to represent the relation among them. Different properties ofthe accessibility functions have been considered and axiomatic systems which define these properties have been given. Only a few ofthese systems have been proved tobe complete. The aim ofthis paper is to make a progress in the study ofcompleteness for functional systems. For this end, we (...) use indexes as names for temporal flows and give new proofs of completeness. Specifically, we focus our attention on the system which defines injectivity, because the system which defines this property without using indexes was proved to be incomplete in previous works. The only system considered which remains incomplete is the one which defines surjectivity, even ifwe consider a sequence ofnatural extensions ofthe previous one. (shrink)

The Moral Law is fulfilled iff everything that ought to be the case is the case, and The Good is realised in a possible world w at a time t iff w is deontically accessible from w at t. In this paper, I will introduce a set of temporal modal deontic systems with propositional quantifiers that can be used to prove some interesting theorems about The Moral Law and The Good. First, I will describe a set of systems (...) without any propositional quantifiers. Then, I will show how these systems can be extended by a couple of propositional quantifiers. I will use a kind of TxW semantics to describe the systems semantically and semantic tableaux to describe them syntactically. Every system will include a constant · that stands for The Good. ‘·’ is read as ‘The Good is realised’. All systems that contain the propositional quantifiers will also include a constant '*' that stands for The Moral Law. '*' is read as ‘The Moral Law is fulfilled’. I will prove that all systems (without the propositional quantifiers) are sound and complete with respect to their semantics and that all systems (including the extended systems) are sound with respect to their semantics. It is left as an open question whether or not the extended systems are complete. (shrink)

The most popular method of incorporating time into a formal logic is based on the work of Arthur Prior. It treats tenses as operators on sentences. In this essay I show a serious problem with that approach, a confusion of scheme versus proposition, which makes any system built in that way incoherent. I will compare how other formal logics deal with the scheme versus proposition distinction and find that only for formal modal logics does the same problem arise. (...) I then compare Prior’s approach to other ways of taking time into account in formal logics. (shrink)

Modal logic is the logic of necessity and possibility, of ‘must be’ and ‘may be’. These may be interpreted in various ways. If necessity is necessary truth, there is alethic modal logic; if it is moral or normative necessity, there is deontic logic [see chapter 8]. It may refer to what is known or believed to be true, in which case, there is an epistemic logic [chapter 9], or to what always has been (...) or to what henceforth always will be true, which gives an aspect of temporal logic [chapter 10]. Another interpretation is to read ‘Necessarily p′ as ‘it is provable that p′ This chapter will present the general framework of modal logic applicable to all of these, though with emphasis on alethic modal logics. (shrink)

Modal logics, originally conceived in philosophy, have recently found many applications in computer science, artificial intelligence, the foundations of mathematics, linguistics and other disciplines. Celebrated for their good computational behaviour, modal logics are used as effective formalisms for talking about time, space, knowledge, beliefs, actions, obligations, provability, etc. However, the nice computational properties can drastically change if we combine some of these formalisms into a many-dimensional system, say, to reason about knowledge bases developing in time or moving objects. (...) To study the computational behaviour of many-dimensional modal logics is the main aim of this book. On the one hand, it is concerned with providing a solid mathematical foundation for this discipline, while on the other hand, it shows that many seemingly different applied many-dimensional systems (e.g., multi-agent systems, description logics with epistemic, temporal and dynamic operators, spatio-temporal logics, etc.) fit in perfectly with this theoretical framework, and so their computational behaviour can be analyzed using the developed machinery. We start with concrete examples of applied one- and many-dimensional modal logics such as temporal, epistemic, dynamic, description, spatial logics, and various combinations of these. Then we develop a mathematical theory for handling a spectrum of 'abstract' combinations of modal logics - fusions and products of modal logics, fragments of first-order modal and temporal logics - focusing on three major problems: decidability, axiomatizability, and computational complexity. Besides the standard methods of modal logic, the technical toolkit includes the method of quasimodels, mosaics, tilings, reductions to monadic second-order logic, algebraic logic techniques. Finally, we apply the developed machinery and obtained results to three case studies from the field of knowledge representation and reasoning: temporal epistemic logics for reasoning about multi-agent systems, modalized description logics for dynamic ontologies, and spatio-temporal logics. The genre of the book can be defined as a research monograph. It brings the reader to the front line of current research in the field by showing both recent achievements and directions of future investigations (in particular, multiple open problems). On the other hand, well-known results from modal and first-order logic are formulated without proofs and supplied with references to accessible sources. The intended audience of this book is logicians as well as those researchers who use logic in computer science and artificial intelligence. More specific application areas are, e.g., knowledge representation and reasoning, in particular, terminological, temporal and spatial reasoning, or reasoning about agents. And we also believe that researchers from certain other disciplines, say, temporal and spatial databases or geographical information systems, will benefit from this book as well. Key Features: Integrated approach to modern modal and temporal logics and their applications in artificial intelligence and computer science Written by internationally leading researchers in the field of pure and applied logic Combines mathematical theory of modal logic and applications in artificial intelligence and computer science Numerous open problems for further research Well illustrated with pictures and tables. (shrink)

In this paper we improve the results of [2] by proving the product f.m.p. for the product of minimal n-modal and minimal n-temporal logic. For this case we modify the finite depth method introduced in [1]. The main result is applied to identify new fragments of classical first-order logic and of the equational theory of relation algebras, that are decidable and have the finite model property.

ABSTRACT: The modal systems of the Stoic logician Chrysippus and the two Hellenistic logicians Philo and Diodorus Cronus have survived in a fragmentary state in several sources. From these it is clear that Chrysippus was acquainted with Philo’s and Diodorus’ modal notions, and also that he developed his own in contrast of Diodorus’ and in some way incorporated Philo’s. The goal of this paper is to reconstruct the three modal systems, including their modal definitions and (...) class='Hi'>modal theorems, and to make clear the exact relations between them; moreover, to elucidate the philosophical reasons that may have led Chrysippus to modify his predessors’ modal concept in the way he did. It becomes apparent that Chrysippus skillfully combined Philo’s and Diodorus’ modal notions, with making only a minimal change to Diodorus’ concept of possibility; and that he thus obtained a modal system of modalities (logical and physical) which fit perfectly fit into Stoic philosophy. (shrink)

Let ${{\mathcal L}^{\square\circ}}$ be a propositional language with standard Boolean connectives plus two modalities: an S4-ish topological modality □ and a temporal modality ◦, understood as ‘next’. We extend the topological semantic for S4 to a semantics for the language ${{\mathcal L}^{\square\circ}}$ by interpreting ${{\mathcal L}^{\square\circ}}$ in dynamic topological systems, i.e., ordered pairs 〈X, f〉, where X is a topological space and f is a continuous function on X. Artemov, Davoren and Nerode have axiomatized a logic S4C, and (...) have shown that S4C is sound and complete for this semantics. S4C is also complete for continuous functions on Cantor space (Mints and Zhang, Kremer), and on the real plane (Fernández Duque); but incomplete for continuous functions on the real line (Kremer and Mints, Slavnov). Here we show that S4C is complete for continuous functions on the rational numbers. (shrink)

Various four- and three-valued modal propositional logics are studied. The basic systems are modal extensions BK and BS4 of Belnap and Dunn's four-valued logic of firstdegree entailment. Three-valued extensions of BK and BS4 are considered as well. These logics are introduced semantically by means of relational models with two distinct evaluation relations, one for verification and the other for falsification. Axiom systems are defined and shown to be sound and complete with respect to the relational semantics and (...) with respect to twist structures over modal algebras. Sound and complete tableau calculi are presented as well. Moreover, a number of constructive non-modal logics with strong negation are faithfully embedded into BS4, into its three-valued extension B3S4, or into temporal BS4, BtS4. These logics include David Nelson's three-valued logic N3, the four-valued logic N4 bottom, the connexive logic C, and several extensions of bi-intuitionistic logic by strong negation. (shrink)

We study the modal logic M L r of the countable random frame, which is contained in and `approximates' the modal logic of almost sure frame validity, i.e. the logic of those modal principles which are valid with asymptotic probability 1 in a randomly chosen finite frame. We give a sound and complete axiomatization of M L r and show that it is not finitely axiomatizable. Then we describe the finite frames of that (...) class='Hi'>logic and show that it has the finite frame property and its satisfiability problem is in EXPTIME. All these results easily extend to temporal and other multi-modal logics. Finally, we show that there are modal formulas which are almost surely valid in the finite, yet fail in the countable random frame, and hence do not follow from the extension axioms. Therefore the analog of Fagin's transfer theorem for almost sure validity in first-order logic fails for modal logic. (shrink)

We present the propositional logic LEC for the two epistemic modalities of current and stable knowledge used by an agent who system-atically enriches his language. A change in the linguistic resources of an agent as a result of certain cognitive processes is something that commonly happens. Our system is based on the logic LC intended to formalize the idea that the occurrence of changes induces the passage of time. Here, the primitive operator C read as: it changes that, (...) defines the temporal succession of states of the world. The notion of current knowledge concerns variable components of the world and it may change over time. We represent it by the primitive operator k read as: the agent currently knows that, and assume that it has S5 properties. The second type of knowledge, symbolized by the primitive operator K read as: the agent stably knows that, relates to constant components of the world and it does not change. As a result of the axiomatic entanglement of C, K and k we show that stable knowledge satisfies axioms of S4.3. K and k modalities are not mutually definable, stable knowledge implies the current one and if the latter never changes, then it comes to be stable. The combination of K and k with the idea of an expanding language allows questioning of the so-called perfect recall principle. It cannot be maintained for both types of knowledge just because of changes in the vocabulary of the agent and possibly the growing spectrum of possible states of the world. We interpret LEC in the semantics of histories of epistemic changes and show that it is complete. Finally, we compare our logic with selected epistemic logics based on the concept of linear discrete time. (shrink)

This paper concerns modal logics appearing from the temporal ordering of domains in two-dimensional Minkowski spacetime. As R. Goldblatt has proved recently, the logic of the whole plane isS4.2. We consider closed or open convex polygons and closed or open domains bounded by simple differentiable curves; this leads to the logics:S4,S4.1,S4.2 orS4.1.2.

The first monograph on Stoic modal logic. Part 1 discusses the Stoic notion of propositions (assertibles, axiomata): their definition; their truth-criteria; the relation between sentence and proposition; propositions that perish; propositions that change their truth-value; the temporal dependency of propositions; the temporal dependency of the Stoic notion of truth; pseudo-dates in propositions. Part 2 discusses Stoic modal logic: the Stoic definitions of their modal notions (possibility, impossibility, necessity, non-necessity); the logical relations between the (...) modalities; modalities as properties of propositions; contingent propositions; the relation between the Stoic modal notions and those of Diodorus Cronus and Philo of Megara; the role of ‘external hindrances’ for the modalities; the temporal dependency of the modalities; propositions that change their modalities; the principle that something possible can follow from something impossible; the interpretations of the Stoic modal system by B. Mates, M. Kneale, M. Frede, J. Vuillemin and M. Mignucci are evaluated. -/- For a shorter, updated, English version of Part 1 of the book see S. Bobzien, ‘Stoic Logic’, in K. Algra et al. (eds), The Cambridge History of Hellenistic Philosophy, Cambridge 1999, 92-157. For a shorter, updated, English version of Part 2 of the book see S. Bobzien 'Chrysippus' Modal Logic and its Relation to Philo and Diodorus', in K. Doering / Th. Ebert (eds) Dialektiker und Stoiker (Stuttgart 1993) 63-84. (shrink)

The main project of this dissertation is to analyze various temporal conceptions of modality for discrete n-dimensional spacetime. The first chapter contains an introduction to the problem and known results. Chapter 2 consists of a study of logics which are analogues of the so-called 'logic of today and tomorrow' and 'logic of tomorrow' investigated by Segerberg and others. We consider the analogues of these successor logics for 2-dimensional integral spacetime. We provide axiomatizations in monomodal and multimodal languages (...) and prove completeness theorems. We also establish that the irreflexive successor logic in the "standard" modal language is not finitely axiomatizable. ;Chapter 3 investigates the axiomatization problem for 2-dimensional integral spacetime frames with Robb's irreflexive 'after' relation. We provide an axiomatization in a multimodal language and prove that this axiomatization is complete. We use this result to prove the system in the "standard" modal language is decidable. ;In Chapter 4 we take up the problem of what the correct Diodorean modal logic is for 2-dimensional integral spacetime. We show that the corresponding Diodorean modal logic is not S4.2, nor indeed any known modal system. We then present several candidate axioms and prove their independence in the context of S4.2. We also study some of the sublogics of the Diodorean modal logic for 2-dimensional discrete spacetime . ;The final chapter contains some general results. First we prove that for every n ≥ 2 the n-dimensional frame determines a unique Diodorean modal logic. We also consider temporal logics in the language with F and P for two kinds of irreflexive spacetime frames. We prove that for every n ≥ 2 the n-dimensional frame with Robb's 'after' relation determines a unique temporal logic. We also prove a generalized completeness theorem for n-dimensional irreflexive successor logics. Finally, in an appendix, we provide a list of selected open problems and indicate directions for future research in this area. (shrink)

We present a modal logic for the class of subset spaces based on discretely descending chains of sets. Apart from the usual modalities for knowledge and effort the standard temporal connectives are included in the underlying language. Our main objective is to prove completeness of a corresponding axiomatization. Furthermore, we show that the system satisfies a certain finite model property and is decidable thus.

Let $\mathcal{L}$ be a propositional language with standard Boolean connectives plus two modalities: an S4-ish topological modality $\square$ and a temporal modality $\bigcirc$ , understood as ‘next’. We extend the topological semantic for S4 to a semantics for the language $\mathcal{L}$ by interpreting $\mathcal{L}$ in dynamic topological systems, i.e. ordered pairs $\langle X, f\rangle$ , where X is a topological space and f is a continuous function on X. Artemov, Davoren and Nerode have axiomatized a logic S4C, and (...) have shown that S4C is sound and complete for this semantics. Zhang and Mints have shown that S4C is complete relative to a particular topological space, Cantor space. The current paper produces an alternate proof of the Zhang-Mints result. (shrink)

In this paper, we study the introduction of modal past temporal operators in Temporal Equilibrium Logic, an hybrid formalism that mixes linear-time modalities and logic programs interpreted under stable models and their characterisation in terms of Equilibrium Logic. We show that Kamp’s translation can also be used to translate the new extension of TEL with past operators into Quantified Equilibrium Logic. Additionally, we provide a method for removing past operators that consists in replacing (...) past-time subformulas by fresh auxiliary atoms, obtaining an equivalent formula, modulo the original alphabet. (shrink)

In this paper we suggest adding to predicate modal and temporal logic a locality predicate W which gives names to worlds (or time points). We also study an equal time predicate D(x, y)which states that two time points are at the same distance from the root. We provide the systems studied with complete axiomatizations and illustrate the expressive power gained for modal logic by simulating other logics. The completeness proofs rely on the fairly intuitive notion (...) of a configuration in order to use a proof technique similar to a Henkin completion mixed with a tableau construction. The main elements of the completeness proofs are given for each case, while purely technical results are grouped in the appendix. (shrink)

This paper introduces a framework for direct surface composition by online update. The surface string is interpreted as is, with each morpheme in turn updating the input state of information and attention. A formal representation language, Logic of Centering, is defined and some crosslinguistic constraints on lexical meanings and compositional operations are formulated.

The history of contemporary modal logic dates back to the writings of C. S. Lewis in the early part of this century. Since then, a growing body of literature has attested to professional interest in the area, and in a number of related issues in philosophical logic which have received wide attention. The recent development of powerful formal techniques for modal system building, together with an increasing interest in modal logic as a tool for (...) philosophical analysis, have created a need for an up-to-date text to introduce students to this material. Snyder's book attempts to answer this need. Assuming an understanding of elementary propositional logic, Snyder introduces a reductive technique called cancellation for detecting the theorems of a system of logic. This technique, analogous to the construction of Smullyan trees but more compact, is extended to modal and quantified contexts. Cancellation versions of the systems M and Mn of G. H. von Wright, and S4 and S5 of Lewis are developed. Snyder uses a Hintikka type of semantics, showing how the various formal systems are differentiated at the semantic level by differing conditions which must be imposed on the model systems used as interpretations for them. These model systems, and the conditions imposed upon them, are in turn described by means of a metalanguage which is essentially first-order quantificational logic. The power of this technique stems from the fact that for every object language formula of a system, there is a corresponding formula in the metalanguage which describes the conditions an interpretation must meet in order to satisfy the original formula. The conditions that the interpretations of a given system of logic must meet are reflected in this metalanguage as "antecedent assumptions," and these in turn correspond to cancellation rules for the object language. This correspondence is constructed in such a way that the metalinguistic counterpart of each theorem of the system will be a theorem of first-order logic. Snyder's primary interest is in showing how modal logic can be used, with the help of these techniques, to build a large variety of formal systems and to "tailor" such systems to meet a variety of analytical tasks. Simple modal systems are developed for the articulation of a number of modal concepts, in addition to the classic alethic ones. Examples are taken from temporal, deontic, and epistemic logic to show how specific interpretations of the meanings of the relevant operators lead to the incorporation of appropriate conditions in the formal systems used to articulate them. An entire chapter is devoted to a discussion of the paradoxes of material and strict implication, and to an attempt to articulate the notion of entailment through the development of formal systems which include a dyadic modal operator that is free of these paradoxes. The final chapter discusses such classical issues as proper names, reference, fictional entities, definite descriptions, and existence presuppositions. Appendices deal with such matters as the equivalence of the cancellation systems to more classical axiomatic-deduction systems, and the sketch of a proof of soundness and completeness for all the modal systems presented. While designed as a text, this volume should be of interest to both logicians and those working in metaphysics and language analysis, since the primary concern of the author is to develop techniques that will facilitate the usefulness of modal logic as a tool for philosophical analysis. The instructor's manual contains many suggestions derived from the author's experience in teaching this non-standard treatment of the subject.--K. T. (shrink)

This paper establishes a connection between structure sensitive categorial inference and classical modal logic. The embedding theorems for non-associative Lambek Calculus and the whole class of its weak Sahlqvist extensions demonstrate that various resource sensitive regimes can be modelled within the framework of unimodal temporal logic. On the semantic side, this requires decomposition of the ternary accessibility relation to provide its correlation with standard binary Kripke frames and models.

In this article, the author studies some central concepts in Avicenna's and sī's modal logics as presented in Avicenna's Al-Ish r t wa'l Tan īh t ( Pointers and Reminders ) and in sī's commentary. In this work, Avicenna introduces some remarkable distinctions in order to interpret Aristotle's modal syllogistic in the Prior Analytics . The author outlines a new interpretation of absolute sentences as temporally indefinite sentences and argues on the basis of this that Avicenna seems to (...) subscribe to the Principle of Plenitude. He also shows that he has no valid proof of the modal conversion rules and that he uses some rather ad hoc distinctions to show that Aristotle's modal syllogistic is correct. The author also notes some interesting differences between Avicenna's and sī's approaches to modal logic. (shrink)

Future Logic is an original, and wide-ranging treatise of formal logic. It deals with deduction and induction, of categorical and conditional propositions, involving the natural, temporal, extensional, and logical modalities. Traditional and Modern logic have covered in detail only formal deduction from actual categoricals, or from logical conditionals (conjunctives, hypotheticals, and disjunctives). Deduction from modal categoricals has also been considered, though very vaguely and roughly; whereas deduction from natural, temporal and extensional forms of conditioning (...) has been all but totally ignored. As for induction, apart from the elucidation of adductive processes (the scientific method), almost no formal work has been done. This is the first work ever to strictly formalize the inductive processes of generalization and particularization, through the novel methods of factorial analysis, factor selection and formula revision. This is the first work ever to develop a formal logic of the natural, temporal and extensional types of conditioning (as distinct from logical conditioning), including their production from modal categorical premises. Future Logic contains a great many other new discoveries, organized into a unified, consistent and empirical system, with precise definitions of the various categories and types of modality (including logical modality), and full awareness of the epistemological and ontological issues involved. Though strictly formal, it uses ordinary language, wherever symbols can be avoided. Among its other contributions: a full list of the valid modal syllogisms (which is more restrictive than previous lists); the main formalities of the logic of change (which introduces a dynamic instead of merely static approach to classification); the first formal definitions of the modal types of causality; a new theory of class logic, free of the Russell Paradox; as well as a critical review of modern metalogic. But it is impossible to list briefly all the innovations in logical science — and therefore, epistemology and ontology — this book presents; it has to be read for its scope to be appreciated. (shrink)

Branching-time temporal logics have proved to be an extraordinarily successful tool in the formal specification and verification of distributed systems. Much of their success stems from the tractability of the model checking problem for the branching time logic CTL, which has made it possible to implement tools that allow designers to automatically verify that systems satisfy requirements expressed in CTL. Recently, CTL was generalised by Alur, Henzinger, and Kupferman in a logic known as Alternating-time Temporal (...) class='Hi'>Logic (ATL). The key insight in ATL is that the path quantifiers of CTL could be replaced by cooperation modalities, of the form , where is a set of agents. The intended interpretation of an ATL formula is that the agents can cooperate to ensure that holds (equivalently, that have a winning strategy for ). In this paper, we extend ATL with knowledge modalities, of the kind made popular in the work of Fagin, Halpern, Moses, Vardi and colleagues. Combining these knowledge modalities with ATL, it becomes possible to express such properties as group can cooperate to bring about iff it is common knowledge in that . The resulting logic — Alternating-time Temporal Epistemic Logic (ATEL) — shares the tractability of model checking with its ATL parent, and is a succinct and expressive language for reasoning about game-like multiagent systems. (shrink)

In classical modal semantics, a binary accessibility relation connects worlds. In this paper, we present a uniform and systematic treatment of modal semantics with a continuous accessibility relation alongside the continuous accessibility modal logics that they model. We develop several such logics for a variety of philosophical applications. Our main conclusions are as follows. Modal logics with a continuous accessibility relation are sound and complete in their natural classes of models. The class of Kripke frames where (...) a continuous accessibility relation has a magnitude characterizing its degree of accessibility is not modally definable, and this has unappreciated significance to completeness proofs for such logics, revealing a methodological advantage of using classical multimodal semantics over fuzzy modal semantics. There is a pseudometric space modal logic that is complete in the class of pseudometric spaces, a natural semantic setting for quantitative modal reasoning about similarity. There is a metric space modal logic that is complete in the class of metric spaces, a natural semantic setting for quantitative modal reasoning about neighborhoods and counterfactual stability. There is a real line continuous temporal logic that is canonical for real lines, a natural semantic setting for quantitative modal reasoning about time. (shrink)

Shows that a certain temporal logic has infinitely many non-equivalent formulae in a single variable.

Branching-time temporal logics have proved to be an extraordinarily successful tool in the formal specification and verification of distributed systems. Much of their success stems from the tractability of the model checking problem for the branching time logic CTL, which has made it possible to implement tools that allow designers to automatically verify that systems satisfy requirements expressed in CTL. Recently, CTL was generalised by Alur, Henzinger, and Kupferman in a logic known as "Alternating-time Temporal (...) class='Hi'>Logic". The key insight in ATL is that the path quantifiers of CTL could be replaced by "cooperation modalities", of the form $\langle \langle \Gamma \rangle \rangle $, where Γ is a set of agents. The intended interpretation of an ATL formula $\langle \langle \Gamma \rangle \rangle \varphi $ is that the agents Γ can cooperate to ensure that φ holds. In this paper, we extend ATL with knowledge modalities, of the kind made popular in the work of Fagin, Halpern, Moses, Vardi and colleagues. Combining these knowledge modalities with ATL, it becomes possible to express such properties as "group Γ can cooperate to bring about φ iff it is common knowledge in Γ that ψ". The resulting logic -- Alternating-time Temporal Epistemic Logic -- shares the tractability of model checking with its ATL parent, and is a succinct and expressive language for reasoning about game-like multiagent systems. (shrink)

Developed by Arthur Prior, Temporal Logic allows to represent temporal information on a logical system using modal operators such as P, F, H or G, whose intuitive meaning is “it was sometime in the Past...”, “it will be sometime in the Future...”, “it Has always been in the past...” and “it will always Going to be in the future...” respectively. Valuation of formulae built from these operators are carried out on Kripke semantics, so Modal (...) class='Hi'>Logic and Temporal Logic are consequently related. In fact, Temporal Logic is an extension of Modal one. Even when both logics mechanisms are able to formalize modal-temporal information with some accuracy, they suffer from a lack of expressiveness which Hybrid Logic can solve. Indeed, one of the problems of Modal Logic consists in its incapacity of naming specific points inside a model. As Temporal Logic is based on it, it cannot make such a thing neither. But First-Order Logic does can by means of constants and equality relation. Hybrid Logic, which results from combining Modal Logic and First-Order Logic, may solve this shortcoming. The main aim of this paper is to explain how Hybrid Logic emanates from Modal and Temporal ones in order to show what it adds to both logics with regard to information representation, why it is more expressive than them and what relation it maintains with the First-Order Correspondence Language. (shrink)

Developed by Arthur Prior, Temporal Logic allows to represent temporal information on a logical system using modal operators such as P, F, H or G, whose intuitive meaning is “it was sometime in the Past...”, “it will be sometime in the Future...”, “it Has always been in the past...” and “it will always Going to be in the future...” respectively. Valuation of formulae built from these operators are carried out on Kripke semantics, so Modal (...) class='Hi'>Logic and Temporal Logic are consequently related. In fact, Temporal Logic is an extension of Modal one. Even when both logics mechanisms are able to formalize modal-temporal information with some accuracy, they suffer from a lack of expressiveness which Hybrid Logic can solve. Indeed, one of the problems of Modal Logic consists in its incapacity of naming specific points inside a model. As Temporal Logic is based on it, it cannot make such a thing neither. But First-Order Logic does can by means of constants and equality relation. Hybrid Logic, which results from combining Modal Logic and First-Order Logic, may solve this shortcoming. The main aim of this paper is to explain how Hybrid Logic emanates from Modal and Temporal ones in order to show what it adds to both logics with regard to information representation, why it is more expressive than them and what relation it maintains with the First-Order Correspondence Language. (shrink)

It is shown how the temporal-modal system of events TM (axiomatized in Appendix) allows for the avoidance of the logical determinism without the rejection of the principle of bivalence. The point is that the temporal and the modal parts of TM are so inter-related that modalities are in-the-real-world-inherent modalities independently of whether they concern actual or only possible events. Though formulated in a tenseless language, whose interpretation does not require the assumption of tense facts at the (...) basic level of reality, TM implies an objective, observer-independent difference between tenses based only on the way in which modalities are distributed along the time continuum. The conclusion is that the arrow of time is an intra-model characteristic of any model of TM that describes the non-deterministic real world up to a certain point of its history, while the flow of time is an inter-model characteristic of the continuous transition between these models. (shrink)

We contrast Bonanno’s ‘Belief Revision in a Temporal Framework’ [15] with preference change and belief revision from the perspective of dynamic epistemic logic (DEL). For that, we extend the logic of communication and change of [11] with relational substitutions [8] for preference change, and show that this does not alter its properties. Next we move to a more constrained context where belief and knowledge can be defined from preferences [29; 14; 5; 7], prove completeness of a very (...) expressive logic of belief revision, and define a mechanism for updating belief revision models using a combination of action priority update [7] and preference substitution [8]. (shrink)

We introduce and study hierarchies of extensions of the propositional modal and temporal languages with pairs of new syntactic devices: point of reference-reference pointer which enable semantic references to be made within a formula. We propose three different but equivalent semantics for the extended languages, discuss and compare their expressiveness. The languages with reference pointers are shown to have great expressive power (especially when their frugal syntax is taken into account), perspicuous semantics, and simple deductive systems. For instance, (...) Kamp's and Stavi's temporal operators, as well as nominals (names, clock variables), are definable in them. Universal validity in these languages is proved undecidable. The basic modal and temporal logics with reference pointers are uniformly axiomatized and a strong completeness theorem is proved for them and extended to some classes of their extensions. (shrink)

In this work we present some new contributions towards two different directions in the study of modal logic. First we employ tense logics to provide a temporal interpretation of intuitionistic quantifiers as “always in the future” and “sometime in the past.” This is achieved by modifying the Gödel translation and resolves an asymmetry between the standard interpretation of intuitionistic quantifiers.Then we generalize the classic Gelfand–Naimark–Stone duality between compact Hausdorff spaces and uniformly complete bounded archimedean $\ell $ -algebras (...) to a duality encompassing compact Hausdorff spaces with continuous relations. This leads to the notion of modal operators on bounded archimedean $\ell $ -algebras and in particular on rings of continuous real-valued functions on compact Hausdorff spaces. This new duality is also a generalization of the classic Jónsson-Tarski duality in modal logic.Abstract taken directly from the thesis.E-mail: [email protected]: https://www.proquest.com/openview/5d284dbfb954383da9364149fa312b6f/1?pq-origsite=gscholar&cbl=18750& diss=y. (shrink)

We prove completeness and decidability results for a family of combinations of propositional dynamic logic and unimodal doxastic logics in which the modalities may interact. The kind of interactions we consider include three forms of commuting axioms, namely, axioms similar to the axiom of perfect recall and the axiom of no learning from temporal logic, and a Church–Rosser axiom. We investigate the influence of the substitution rule on the properties of these logics and propose a new semantics (...) for the test operator to avoid unwanted side effects caused by the interaction of the classic test operator with the extra interaction axioms. (shrink)

The purpose of this paper is to describe a set of quantified temporal alethic-deontic systems, i.e., systems that combine temporal alethicdeontic logic with predicate logic. We consider three basic kinds of systems: constant, variable and constant and variable domain systems. These systems can be augmented by either necessary or contingent identity, and every system that includes identity can be combined with descriptors. All logics are described both semantically and proof theoretically. We use a kind of possible (...) world semantics, inspired by the so-called T × W semantics, to characterize them semantically and semantic tableaux to characterize them proof theoretically. We also show that all systems are sound and complete with respect to their semantics. (shrink)

In the 4th century BC, the Greek philosopher Diodoros Chronos gave a temporal definition of necessity. Because it connects modality and temporality, this definition is of interest to philosophers working within branching time or branching space-time models. This definition of necessity can be formalized and treated within a logical framework. We give a survey of the several known modal and temporal logics of abstract space-time structures based on the real numbers and the integers, considering three different accessibility (...) relations between spatio-temporal points. (shrink)

Like modal logic, temporal logic, and description logic, separation logic has become a popular class of logical formalisms in computer science, conceived as assertion languages for Hoare-style proof systems with the goal to perform automatic program analysis. In a broad sense, separation logic is often understood as a programming language, an assertion language and a family of rules involving Hoare triples. In this survey, we present similarities between separation logic as an assertion (...) language and modal and temporal logics. Moreover, we propose a selection of landmark results about decidability, complexity and expressive power. (shrink)

The essay introduces a non-Diodorean, non-Kantian temporal modal semantics based on part-whole, rather than class, theory. Formalizing Edmund Husserl’s theory of inner time consciousness, §3 uses his protention and retention concepts to define a relation of self-awareness on intentional events. §4 introduces a syntax and two-valued semantics for modal first-order predicate object-languages, defines semantic assignments for variables and predicates, and truth for formulae in terms of the axiomatic version of Edmund Husserl’s dependence ontology (viz. the Calculus [CU] (...) of Urelements) introduced by The Ontology of Intentionality I & II. It then uses the §3 results to define the modalities of truth, and §5 extends the semantics to identity claims. §6 defines and contrasts synthetic a priori truths to analytic a priori truths, and §7 compares Brentano School noetic semantic and Leibnizian possible-world semantic perspectives on modality. The essay argues that the modal logics it defines semantically are two-valued, first-order versions of the type of language which Husserl viewed as the language of any ontology of experience (i.e. of any science), and conceived as the logic of intentionality. (shrink)

We generalize an observation made by Goldblatt in "Diodorean modality in Minkowski spacetime" by proving that each -dimensional integral spacetime frame equipped with Robb's irreflexive `after' relation determines a unique temporal logic. Our main result is that, unlike -dimensional spacetime where, as Goldblatt has shown, the Diodorean modal logic is the same for each frame , in the case of -dimensional integral spacetime, the frame determines a unique Diodorean modal logic.