We take the well-known intuitionistic modal logic of Fischer Servi with semantics in bi-relational Kripke frames, and give the natural extension to topological Kripke frames. Fischer Servi’s two interaction conditions relating the intuitionistic pre-order with the modal accessibility relation generalize to the requirement that the relation and its inverse be lower semi-continuous with respect to the topology. We then investigate the notion of topological bisimulation relations between topological Kripke frames, as introduced by Aiello and van Benthem, and show (...) that their topology-preserving conditions are equivalent to the properties that the inverse relation and the relation are lower semi-continuous with respect to the topologies on the two models. The first main result is that this notion of topological bisimulation yields semantic preservation w.r.t. topological Kripke models for both intuitionistic tense logics, and for their classical companion multi-modal logics in the setting of the Gödel translation. After giving canonical topological Kripke models for the Hilbert-style axiomatizations of the Fischer Servi logic and its classical companion logic, we use the canonical model in a second main result to characterize a Hennessy–Milner class of topological models between any pair of which there is a maximal topological bisimulation that preserve the intuitionistic semantics. (shrink)

This book is intended to serve as an advanced text and reference work on modal logic, a subject of growing importance which has applications to philosophy and linguistics. Although it is based mainly on research which I carried out during the years 1969-1973, it also includes some related results obtained by other workers in the field. Parts 0, 1 and 2, can be used as the basis of a one year graduate course in modal logic. The material which (...) they contain has been taught in such courses at Stanford since 1970. The remaining parts of the book contain more than enough material for a second course in modal logic. The exercises supplement the text and are usually difficult. I wish to thank Stanford University and Bar-Han University for making it possible for me to continue and finish this work, and A. Ungar for correcting the typescript. Bar-Ilan University, Israel Dov M. GABBA Y PART 0 AN INTRODUCTION TO GENERAL INTENSIONAL LOGICS CHAPTER 0 CONSEQUENCE RELATIONS Motivation We introduce the notions of a consequence relation and of a semantics. We show that every consequence relation is complete for a canonical semantics. We define the notion of one semantics being Dian in another and study the basic properties of this notion. The concepts of this chapter are generalizations of the various notions of logical system and possible world semantics found in the literature. (shrink)

In celebrating Arthur Prior we celebrate what he gave to the world. Much of this is measured by what others have made of his ideas after his death. The focus of this paper is a little different. It looks at what Prior himself thought he was accomplishing. In particular it considers Prior’s attitude to the semantic metatheory of the logics that he was interested in. The paper sets out some characteristics of the metalogical study of intensional languages in terms (...) of an indexical theory of truth conditions stated in the language of set theory. In examining Prior’s work using these characteristics, it emerges that Prior had serious reservations about this way of studying modal and tense logics. (shrink)

LetL be any modal or tense logic with the finite model property. For eachm, definer L (m) to be the smallest numberr such that for any formulaA withm modal operators,A is provable inL if and only ifA is valid in everyL-model with at mostr worlds. Thus, the functionr L determines the size of refutation Kripke models forL. In this paper, we will give an estimation ofr L (m) for some linear modal and tense logicsL.

ABSTRACT We critically review two extant paradigms for understanding the systematic interaction between modality and tense, as well as their respective modifications designed to do justice to the contingency of time’s structure and composition. We show that on either type of theory, as well as their respective modifications, some principles prove logically valid whose truth might sensibly be questioned on metaphysical grounds. These considerations lead us to devise a more general logical framework that allows accommodation of those metaphysical views (...) that its predecessors rule out by fiat. (shrink)

Kit Fine has since the 1970s been one of the leading contributors to work at the intersection of logic and metaphysics. This is his eagerly-awaited first book in the area. It draws together a series of essays, three of them previously unpublished, on possibility, necessity, and tense. These puzzling aspects of the way the world is have been the focus of considerable philosophical attention in recent decades. A helpful introduction orients the reader and offers a way into some of (...) the most original work in contemporary philosophy. (shrink)

Viewing the language of modal logic as a language for describing directed graphs, a natural type of directed graph to study modally is one where the nodes are sets and the edge relation is the subset or superset relation. A well-known example from the literature on intuitionistic logic is the class of Medvedev frames $\langle W,R\rangle$ where $W$ is the set of nonempty subsets of some nonempty finite set $S$, and $xRy$ iff $x\supseteq y$, or more liberally, where $\langle (...) W,R\rangle$ is isomorphic as a directed graph to $\langle \wp(S)\setminus\{\emptyset\},\supseteq\rangle$. Prucnal [32] proved that the modal logic of Medvedev frames is not finitely axiomatizable. Here we continue the study of Medvedev frames with extended modal languages. Our results concern definability. We show that the class of Medvedev frames is definable by a formula in the language of tense logic, i.e., with a converse modality for quantifying over supersets in Medvedev frames, extended with any one of the following standard devices: nominals (for naming nodes), a difference modality (for quantifying over those $y$ such that $x\not= y$), or a complement modality (for quantifying over those $y$ such that $x\not\supseteq y$). It follows that either the logic of Medvedev frames in one of these tense languages is finitely axiomatizable---which would answer the open question of whether Medvedev's [31] "logic of finite problems'' is decidable---or else the minimal logics in these languages extended with our defining formulas are the beginnings of infinite sequences of frame-incomplete logics. (shrink)

In an earlier paper by the author, Åqvist , I presented an approach to the logic of historical necessity, or inevitability, in the sense of a “two-dimensional” combination of tense and modal logic for worlds, or histories, with the same time order, known as T × W logic. Distinctive features of that approach were, apart from its two-dimensionality, its being based on discrete and finite time, and its use of so-called systematic frame constants in order to enable us (...) to indicate longitudes and latitudes of any points in the co-ordinate systems under consideration. This led us to study and axiomatize an infinite hierarchy HTWxy of two-dimensional modal tense logics with the characteristic operators for historical necessity and possibility added to the original basic vocabulary. The main purpose of the present paper is then twofold: to extend the logics HTWxy to the interesting branch of philosophical logic constituted by deontic logic as combined with tense logic; and to deal with a curious puzzle known as the so-called epistemic obligation paradox – a well known stumbling-block in this area of research in philosophical logic. We argue for a solution to both these problems, which appeals to a new infinite hierarchy DHTWxym of extensions of the HTWxy in the sense of logics combining dyadic deontic modalities with temporal ones such as those for historical necessity and other two-dimensional modalities. (shrink)

We provide a direct method for proving Craig interpolation for a range of modal and intuitionistic logics, including those containing a "converse" modality. We demonstrate this method for classical tense logic, its extensions with path axioms, and for bi-intuitionistic logic. These logics do not have straightforward formalisations in the traditional Gentzen-style sequent calculus, but have all been shown to have cut-free nested sequent calculi. The proof of the interpolation theorem uses these calculi and is purely syntactic, (...) without resorting to embeddings, semantic arguments, or interpreted connectives external to the underlying logical language. A novel feature of our proof includes an orthogonality condition for defining duality between interpolants. (shrink)

In recent years combinations of tense and modality have moved intothe focus of logical research. From a philosophical point of view, logical systems combining tense and modality are of interest because these logics have a wide field of application in original philosophical issues, for example in the theory of causation, of action, etc. But until now only methods yielding completeness results for propositional languages have been developed. In view of philosophical applications, analogous results with respect to languages (...) of predicate logic are desirable, and in this paper I present two such results. The main developments in this area can be split into two directions, differing in the question whether the ordering of time is world-independent or not. Semantically, this difference appears in the discussion whether T x W-frames or Kamp-frames (resp. Ockham-frames) provide a suitable semantics for combinations of tense and modality. Here, two calculi are presented, the first adequate with respect to Kamp-semantics, the second to T x Wsemantics. (Both calculi contain an appropriate version of Gabbay's irreflexivity rule.) Furthermore, the proposed constructions of canonical frames simplify some of those which have hitherto been discussed. (shrink)

According to Hans Kamp and Frank Vlach, the two-dimensional tense operators “now” and “then” are ineliminable in quantified tense logic. This is often adduced as an argument against tense logic, and in favor of an extensional account that makes use of explicit quantification over times. The aim of this paper is to defend tense logic against this attack. It shows that “now” and “then” are eliminable in quantified tense logic, provided we endow it with enough (...) quantificational structure. The operators might not be redundant in some other systems of tense logic, but this merely indicates a lack of quantificational resources and does not show any deep-seated inability of tense logic to express claims about time. The paper closes with a brief discussion of the modal analogue of this issue, which concerns the role of the actuality operator in quantified modal logic. (shrink)

We shall describe the set of strongly meet irreducible logics in the lattice ϵLin.t of normal tense logics of weak orderings. Based on this description it is shown that all logics in ϵLin.t are independently axiomatizable. Then the description is used in order to investigate tense logics with respect to decidability, finite axiomatizability, axiomatization problems and completeness with respect to Kripke semantics. The main tool for the investigation is a translation of bimodal formulas into (...) a language talking about partitions of general frames into intervals so that relative to both Kripke frames and descriptive frames the expressive power of both languages coincides. (shrink)

The forward convergence constraint is important to time analysis. Without it, given two future moments to the same point, the time branches. This is unacceptable if one assumes that time is linear. Nevertheless, one may wish to consider time-branching in order to discuss future possibilities. One can have both a linear order for the time and branching through the combination of the tense logic semantics with those of an alethic logic which allows the evaluation of the timelines of other (...) possible worlds. In this paper, I give the semantics and the tableaux systems for some alethic-tense logics. I review one in which worlds differ on their time orders, another in which they agree on the time order, a pair of conditional tense logics ), and the first-order version of MT. A brief philosophical discussion arises from every system. (shrink)

In this paper we examine Prior’s reconstruction of Master Argument [4] in some modal-tense logic. This logic consists of a purely tense part and Diodorean definitions of modal alethic operators. Next we study this tense logic in the pure tense language. It is the logic K t 4 plus a new axiom ( P ): ‘ p Λ G p ⊃ P G p ’. This formula was used by Prior in his original analysis (...) of Master Argument. ( P ) is usually added as an extra axiom to an axiomatization of the logic of linear time. In that case the set of moments is a total order and must be left-discrete without the least moment. However, the logic of Master Argument does not require linear time. We show what properties of the set of moments are exactly forced by ( P ) in the reconstruction of Prior. We make also some philosophical remarks on the analyzed reconstruction. (shrink)

This monograph is designed to provide an introduction to the principal areas of tense logic. Many of the developments in this ever-growing field have been intentionally excluded to fulfill this aim. Length also dictated a choice between the alternative notations of A. N. Prior and Nicholas Rescher - two pioneers of the subject. I choose Prior's because of the syntactical parallels with the language it symbolizes and its close ties with other branches of logi cal theory, especially modal (...) logic. The first chapter presents a wider view of the material than later chapters. Several lines of development are consequently not followed through the remainder of the book, most notably metric systems. Although it is import ant to recognize that the unadorned Prior-symbolism can be enriched in vari ous ways it is an advanced subject as to how to actually carry off these enrichments. Readers desiring more information are referred to the appropri ate literature. Specialists will notice that only the first of several quantifi cational versions of tense logic is proven complete in the final chapter. Again constraints of space are partly to blame. The proof for the 'star' systems is wildly complex and at the time of this writing is not yet ready for publi cation. (shrink)

An interpretation in modal and tense logic is proposed for Boethius's reconciliation of God's foreknowledge with human freedom from The consolation of philosophy, Book V. The interpretation incorporates a suggestion by Paul Spade that God's special status in time be explained as a restriction of God's knowledge to eternal sentences. The argument proves valid, and the seeming restriction on omnipotence is mitigated by the very strong expressive power of eternal sentences.

Tense logics formulated in the bimodal propositional language are investigated with respect to Kripke-completeness (completeness) and decidability. It is proved that all minimal tense extensions of modal logics of finite width (in the sense of K. Kine) as well as all minimal tense extensions of cofinal subframe logics (in the sense of M. Zakharyaschev) are complete. The decidability of all finitely axiomatizable minimal tense extensions of cofinal subframe logics is shown. A (...) number of variations and extensions of these results are also presented. (shrink)

Tense logics formulated in the bimodal propositional language are investigated with respect to Kripke-completeness and decidability. It is proved that all minimal tense extensions of modal logics of finite width as well as all minimal tense extensions of cofinal subframe logics are complete. The decidability of all finitely axiomatizable minimal tense extensions of cofinal subframe logics is shown. A number of variations and extensions of these results are also presented.

The paper introduces a first-order theory in the language of predicate tense logic which contains a single simple axiom. It is shewn that this theory enables times to be referred to and sentences involving ‘now’ and ‘then’ to be formalised. The paper then compares this way of increasing the expressive capacity of predicate tense logic with other mechanisms, and indicates how to generalise the results to other modal and tense systems.

We consider a version of so called T x W logic for historical necessity in the sense of R.H. Thomason (1984), which is somewhat special in three respects: (i) it is explicitly based on two-dimensional modal logic in the sense of Segerberg (1973); (ii) for reasons of applicability to interesting fields of philosophical logic, it conceives of time as being discrete and finite in the sense of having a beginning and an end; and (iii) it utilizes the technique of (...) systematic frame constants in order to handle the problem of irreflexivity in tense logics, well known since Gabbay (1981). Axiomatizations are given for two infinite hierarchies of two-dimensional modal tense logics, one without and one with the characteristic operators for historical necessity and possibility. Strong and weak completeness results are obtained for both hierarchies as well as a result to the effect that two approaches to their semantics are equivalent, much in the spirit of Di Maio and Zanardo (1996) and von Kutschera (1997). (shrink)

This paper explains how to obtain quantification over times in a tense logic in which all temporal distinctions are ultimately spelled out in terms of the two simple tense operators “it was the case that” and “it will be the case that.” The account of times defended here is similar to what is known as “linguistic ersatzism” about possible worlds, but there are noteworthy differences between these two cases. For example, while linguistic ersatzism would support actualism, the view (...) of times defended here does not support presentism. (shrink)

In this paper we examine Prior's reconstruction of Master Argument [4] in some modal-tense logic. This logic consists of a purely tense part and Diodorean definitions of modal alethic operators. Next we study this tense logic in the pure tense language. It is the logic $K_t 4$ plus a new axiom $:p \wedge Gp \supset PGp'$. This formula was used by Prior in his original analysis of Master Argument. is usually added as an extra (...) axiom to an axiomatization of the logic of linear time. In that case the set of moments is a total order and must be left-discrete without the least moment. However, the logic of Master Argument does not require linear time. We show what properties of the set of moments are exactly forced by in the reconstruction of Prior. We make also some philosophical remarks on the analyzed reconstruction. (shrink)

We introduce translations between display calculus proofs and labeled calculus proofs in the context of tense logics. First, we show that every derivation in the display calculus for the minimal tense logic Kt extended with general path axioms can be effectively transformed into a derivation in the corresponding labeled calculus. Concerning the converse translation, we show that for Kt extended with path axioms, every derivation in the corresponding labeled calculus can be put into a special form that (...) is translatable to a derivation in the associated display calculus. A key insight in this converse translation is a canonical representation of display sequents as labeled polytrees. Labeled polytrees, which represent equivalence classes of display sequents modulo display postulates, also shed light on related correspondence results for tense logics. (shrink)

Based on the results of [11] this paper delivers uniform algorithms for deciding whether a finitely axiomatizable tense logic has the finite model property, is complete with respect to Kripke semantics, is strongly complete with respect to Kripke semantics, is d-persistent, is r-persistent.It is also proved that a tense logic is strongly complete iff the corresponding variety of bimodal algebras is complex, and that a tense logic is d-persistent iff it is complete and its Kripke frames form (...) a first order definable class. From this we obtain many natural non-d-persistent tense logics whose corresponding varieties of bimodal algebras are complex. (shrink)

This is a revised and expanded edition of a seminal work in the logic and philosophy of time, originally published in 1968. Arthur N. Prior (1914-1969) was the founding father of temporal logic, and his book offers an excellent introduction to the fundamental questions in the field. Several important papers have been added to the original selection, as well as a comprehensive bibliography of Prior's work and an illuminating interview with his widow, Mary Prior. In addition, the Polish logic which (...) made Prior's writings difficult for many readers has been replaced by standard logical notation. This new edition will secure the classic status of the book. (shrink)

Neither question (1) nor question (2) posed on page 446 have been adequately answered in this paper. Regarding (1) we have merely given functor maps onto the object languages of physical theories and regarding (2) we have merely described the algebraic structure of observables. A more satisfactory treatment will most likely involve (1) a generalization to algebraic categories, universal algebra and model theory in such a way as to capture the full inference structure of (perhaps van Fraassen's modal) quantum (...) logic, and (2) the introduction of the category of differentiable manifolds in order to capture the full symplectic structure of (perhaps Chernoff-Marsden's) Hamiltonian dynamics. (See van Fraassen [4] and Chernoff-Marsden [5] in Other References.)Finally, the intriguing suggestion of Finkelstein that “category theory is to metaphysics what group theory is to physics” deserves serious consideration. Indeed, although the theme has not been touched on explicitly in this paper it was central in motivating the project initially. However, it is more properly to be developed in a context external to a symposium on quantum logic. (shrink)

We explain two ways of revising a tense logic like kripke's (1963) modal logic by adding predicate modifiers. first we show that modifiers allow us to render valid some mixing formulas--conditionals reversing the order of a quantifier and an operator--within a complete bivalent system. then we show how modifiers enable a tense logic to give analyses close to the surface form for sentences with temporal qualifications of singular terms, e.g., 'toby was fatter then than william is today'.

The last chapter examines the tense system used by ordinarily competent speakers of English to discuss past, present, and future events, actual and possible events, and various combinations of these. I present a systematic method for translating English sentences containing certain compound verb tenses and embedded tense constructions into a logical language using tense operators. Finally I show how the usual semantics for these operators reflects the truth conditions of the original English sentences. I argue, however, that (...) a tense logical semantics reflecting the use of tense in English will contain a partially ordered tense structure rather than one that is linearly ordered or branching. ;The third chapter discusses logic of 'unless'. I show that many of the previously puzzling features of this word can be understood by considering the tenses of the verbs in the sentences joined by 'unless'. In a number of cases it turns out that 'unless' should not be interpreted as a truth-functional connective at all, but is best understood as an intensional, tense connective. ;In the second chapter I discuss Quine's doctrine of eternal sentences. I take Quine's position to be that all the information important for logic and science presented in tensed English sentences can also be presented by eternal sentences which use timeless verbs and data and clock times. I argue that in so far as logic and science are concerned with an analysis of the reasons for human action, and with the explanation and prediction of specific events, eternal sentences lose indexical information that is necessary to accomplish these tasks. ;The first chapter investigates what may be called the standard view of propositions. On this view propositions are abstract objects that are timelessly true , and serve both as the meanings of sentences and as the objects of propositional attitudes. Although this view was widely held by a previous generation of logicians and is found in many textbooks today, there are serious difficulties in the view. I show that many of these difficulties stem from the claim that propositions are timelessly true . The standard view has been revised in two general ways. The first sort of revision allows propositions to be sometimes true and sometimes false. This position is consistent with the development of tense logic as one of the intensional logics. The second sort of revision replaces the notion of timeless propositions with that of eternal sentences which always have the same truth value. On this view tense logic is not worth serious philosophic attention since its basic notion, that of truth at a time, can be eliminated in favor of eternal truth. ;The first part of the project examines two major sources of criticism of tense logic: the view that logic is concerned with timeless propositions, and Quine's doctrine of eternal sentences. The second part of the project discusses several problems in logical theory that the use of tense logic can clarify or resolve, and that alternative theories are unable to present accurately, much less to elucidate. ;In the past few years there has been increasing interest in the logic of indexicals, including the temporal indexical 'now', and in the relationship between tense and modality. Although modern tense logic has reached a high level of technical sophistication, often specifically in response to this interest, many philosophers remain skeptical about the value of tense logic as a philosophic tool. This thesis addresses that skepticism and argues that it is unjustified. (shrink)

Despite its brevity, McArthur’s monograph offers a helpful introduction to tense logic to readers who have already been introduced to the syntactical and semantical metatheory of classical logic. Familiarity with substitutional quantification, free-logic, and modal logic will help readers appreciate techniques used and the directions in which topics are developed.

Ulrich Meyer's book The Nature of Time uses tense logic to argue for a `modal' view of time, which replaces substantial times with `ersatz times' constructed using conceptually basic tense operators. He also argues against Bertrand Russell's relationist theory, in which times are classes of events, and against the idea that relativity compels the integration of time and space. I find fault with each of these negative arguments, as well as with Meyer's purported reconstruction of empty spacetime (...) from tense operators and substantial spatial points. I suggest that Meyer's positive project is best conceived as an elimination of time in the mode of Julian Barbour's The End of Time. (shrink)

Is what could have happened but never did as real as what did happen? What did happen, but isn't happening now, happened at another time. Analogously, one can say that what could have happened happens in another possible world. Whatever their views about the reality of such things as possible worlds, philosophers need to take this analogy seriously. Adriane Rini and Max Cresswell exhibit, in an easy step-by-step manner, the logical structure of temporal and modal discourse, and show that (...) every temporal construction has an exact parallel that requires a language that can refer to worlds, and vice versa. They make precise, in a way which can be articulated and tested, the claim that the parallel is at work behind even ordinary talk about time and modality. The book gives metaphysicians a sturdy framework for the investigation of time and modality - one that does not presuppose any particular metaphysical view. (shrink)

This is a revised and expanded edition of a seminal work in the logic and philosophy of time, originally published in 1968. Arthur N. Prior was the founding father of temporal logic, and his book offers an excellent introduction to the fundamental questions in the field. Several important papers have been added to the original selection, as well as a comprehensive bibliography of Prior's work and an illuminating interview with his widow, Mary Prior. In addition, the Polish logic which made (...) Prior's writings difficult for many readers has been replaced by standard logical notation. This new edition will secure the classic status of the book. (shrink)