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)

This article surveys some of the key issues that arise when one tries to use tense logic as a metaphysical theory of the nature of time. Topics discussed include basic tense logic, tense logic and verb tense, the structure of the time series, instants of time, quantified tense logic, and the expressive resources of tense logic.

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)

The author's motivation for constructing the calculi of this paper\nis so that time and tense can be "discussed together in the same\nlanguage" (p. 282). Two types of enriched propositional caluli for\ntense logic are considered, both containing ordinary propositional\nvariables for which any proposition may be substituted. One type\nalso contains "clock-propositional" variables, a,b,c, etc., for\nwhich only clock-propositional variables may be substituted and that\ncorrespond to instants or moments in the semantics. The other type\nalso contains "history-propositional" variables, u,v,w, etc., for\nwhich only history-propositional (...) variables may be substituted and\nthat correspond to maximally linearly ordered subsets of instants.\nThis second type is for tense logics in which time is not linear\nand where at any moment there may exist alternative future courses\nof events. Quantifiers are included in both types but only for clock-\nand history-propositional variables. However, these quantifiers are\ngiven an essentially substitutional interpretation. In addition,\nbesides each clock-propositional variable being true at exactly one\ninstant, the semantics requires that for each instant there be at\nleast one clock-propositional variable true at that instant, which\nin effect amounts to having each instant of the model "denoted" in\nthe formalism by some clock-propositional variable. Thus it is not\nsurprising that quantification over clock-propositional variables\nturns out to have a "well behaved first-order semantics" (p. 284).\n{One axiom relating history- with clock-propositions seems to need\nrevision: CKTauTbuAUabUba. Since the clock-propositional variables\nmight be true at the same instant the consequent should include an\nalternative, namely, LCab.} Besides the standard truth-functional\nand tense operators G,H,F,P, the author includes L for "it is\nalways the case that". The model-theoretic earlier-than relation\nordering instants and the semantic truth-at (an instant) relation\nare formalized by Uab and Taα (for a wff α) which\nare defined respectively as LCbPa (whenever the clock-proposition\nb is the case the clock-proposition a was the case) and LCaα\n(α is the case whenever the clock-proposition a is the case).\nBecause the future tense operators are definable in terms of U\nand T and the latter are definable in terms of L and P, the\ndual of H, the author notes that his completeness proofs can be\nrestricted to L and H as primitive and with the future tense\noperators used for other interpretations, especially Prior, A.'s\nindeterminist "it will be the case that", where time is not linear.\nThe author concludes with some rather technical observations showing\nthat "in ordinary tense logics the instants are nonstandard elements\nof the models. This provides", according to the author, "a semantic\npressure to build the instants into tense logic, so that they will\nappear as standard elements in the models" (p. 283). (shrink)

This paper outlines a formal account of tensed sentences that is consistent with Ockhamism, a view according to which future contingents are either true or false. The account outlined substantively differs from the attempts that have been made so far to provide a formal apparatus for such a view in terms of some expressly modified version of branching time semantics. The system on which it is based is the simplest quantified modal logic.

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'. (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)

Temporal logic as a modern discipline is separate from classical logic; it is seen as an addition or expansion of the more basic propositional and predicate logics. This approach is in contrast with logic in the Middle Ages, which was primarily intended as a tool for the analysis of natural language. Because all natural language sentences have tensed verbs, medieval logic is inherently a temporal logic. This fact is most clearly exemplified in medieval theories of (...) supposition. As a case study, we look at the supposition theory of Lambert of Lagny, extracting from it a temporal logic and providing a formalization of that logic. (shrink)

We consider a formal language whose logical syntax involves both modal and tense propositional operators, as well as sortal quantifiers, sortal identities and (second order) quantifiers over sortals. We construct an intensional semantics for the language and characterize a formal logical system which we prove to be sound and complete with respect to the semantics. Conceptualism is the philosophical background of the semantic system.

We propose a new intensional semantics for modal-tense second-order languages with sortal predicates. The semantics provides a variable-domain interpretation of the second-order quantifiers. A formal logical system is characterized and proved to be sound and complete with respect to the semantics. A contemporary variant of conceptualism as a theory of universals is the philosophical background of the semantics. Justification for the variable-domain interpretation of the second-order quantifiers presupposes such a conceptualist framework.

Two of the main debates in philosophy of language concerning time and tense are the debate about the semantics of the tenses in the English language and the debate over whether propositions can be transiently true or false as opposed to always being eternally true or false. The latter quarrel is also known as the "temporalism–eternalism debate." Given standard semantics, the two debates are not logically independent, as we will see. Those who believe propositions are eternally true or false (...) needn’t treat the tenses as operators. Their opponents, on the other hand, appear to be committed to an operator theory of the tenses, given a standard semantic framework. In this chapter I will focus primarily on these two debates. (shrink)

We show that monadic intuitionistic quantifiers admit the following temporal interpretation: “always in the future” (for$\forall $) and “sometime in the past” (for$\exists $). It is well known that Prior’s intuitionistic modal logic${\sf MIPC}$axiomatizes the monadic fragment of the intuitionistic predicate logic, and that${\sf MIPC}$is translated fully and faithfully into the monadic fragment${\sf MS4}$of the predicate${\sf S4}$via the Gödel translation. To realize the temporal interpretation mentioned above, we introduce a new tense extension${\sf TS4}$of${\sf S4}$and provide a full (...) and faithful translation of${\sf MIPC}$into${\sf TS4}$. We compare this new translation of${\sf MIPC}$with the Gödel translation by showing that both${\sf TS4}$and${\sf MS4}$can be translated fully and faithfully into a tense extension of${\sf MS4}$, which we denote by${\sf MS4.t}$. This is done by utilizing the relational semantics for these logics. As a result, we arrive at the diagram of full and faithful translations shown in Figure 1 which is commutative up to logical equivalence. We prove the finite model property (fmp) for${\sf MS4.t}$using algebraic semantics, and show that the fmp for the other logics involved can be derived as a consequence of the fullness and faithfulness of the translations considered. (shrink)

One of Arthur Prior’s constructions of the relational calculus for times within tense logic plus propositional quantifiers is considered using Malcolm Rennie’s multimodal semantics and found wantin...

Intensional logic is the technical study of such intensional phenomena in human reasoning as modality, knowledge, or flow of time. These all require a richer semantic picture than standard truth values in one static environment. Such a picture is provided by so-called possible worlds semantics, a paradigm which is surveyed in this book, both as to its external sources of motivation and as to the internal dynamics of the resulting program. In particular, Manual of Intensional Logic presents the (...) major classical topics, including modal logic, tense logic, and conditional logic, all of which illustrate motivations coming from philosophy and linguistics. The Book also discusses recent computational applications in computer science and AI. Finally, Manual of Intensional Logic takes up recent developments in the study of language and information making themselves felt in the area. The Book examines the role of partial information--with illustrations drawn from different branches of Intensional Logic--and various influences stemming from current theories of the semantics of natural language, involving generalized quantifiers and theories of types. (shrink)

'Wolterstorff (2009) provides an important explanation to the question: What caused the surprising resurgence in philosophical theology that has occurred over the last 50 years—a resurgence that rivals its zenith in the Middle Ages? This article supplements that with a more fine-grained answer to the question. Recent discoveries in Arthur Norman Prior’s correspondence with J.J.C Smart and Mary Prior, between November 1953 and August 1954 on the possibility of necessary existence, demonstrates the importance of Prior’s discussion of the Barcan formulae (...) in Time and Modality (1957) for the resurgence of analytic theology. The correspondence establishes that Prior’s discovery of tense-logic, and his discussion of quantified tense-logic constituted the perfect opportunity for him to challenge key anti-metaphysical assumptions in analytic philosophy, from which four important consequences can be drawn for the resurgence of philosophical theology. First, Prior’s discussion of time and existence challenged the idea of Russell (1945) and Findlay (1948) on the logical status of a necessary existing being. Second, the discussion challenged the Analytic school’s view of analysis and gave Prior the opportunity to introduce a different perspective on the relationship between logic and metaphysics. Third, it gave Prior a good opportunity to demonstrate that the then-prevailing attitude towards medieval logic was wrong. Fourth, it made it possible for Prior to demonstrate that the highly surprising metaphysical conclusions of quantified tense-logic brings modern logicians into a discussion with the theologically minded medieval logicians. (shrink)

This paper has two parts. In part I, I review two older accounts of the logical forms of modifiers, and suggest that they may be combined with each other so as to yield a theory that is better than either of its parts taken singly. Part of this theory involves the idea that certain sentences refer to events, states, or processes; Part II of this paper shows how to use this idea to account for tenses and temporal adverbials, and offers (...) a new account of ordinary language quantification. (shrink)

This paper has two parts. In part I, I review two older accounts of the logical forms of modifiers, and suggest that they may be combined with each other so as to yield a theory that is better than either of its parts taken singly. Part of this theory involves the idea that certain sentences refer to events, states, or processes; Part II of this paper shows how to use this idea to account for tenses and temporal adverbials, and offers (...) a new account of ordinary language quantification. (shrink)

ABSTRACT Backwards-looking operators Saarinen, E. [1979. “Backwards-Looking Operators in Tense Logic and in Natural Language.” In Essays on Mathematical and Philosophical Logic, edited by J. Hintikka, I. Niiniluoto, and E. Saarinen, 341–367. Dordrecht: Reidel] that have the material in their scope depend on higher intensional operators, are known to increase the expressivity of some intensional languages and have thus played a central role in debates about approaches to intensionality in terms of implicit parameters vs. variables explicitly quantifying (...) over them. The current contribution takes a look at these operators from a type-logical perspective. It is shown that extending Gallin's translation from intensional type logic to two-sorted type theory so as to include a version of Yanovich's backwards-looking operators, does not increase the expressive power of formulae with exclusively intensional parameters. The result, which makes use of a theorem by Zimmermann : 65–77]), is illustrated by pertinent examples from the literature. The paper closes by indicating alternative strategies of incorporating backwards-looking operators. (shrink)

The paper contains a survey of existence problems in the folowing calculi of formal logic: calculi of classes, syllogistics, classic predicate calculus, Lesniewski's ontology, quantificational modal calculi, quantificational tense logics, sentential calculi. The particular attention is given to metaphysical (ontological) presuppositions and commitments of logical calculi especially to the problems of empty set, existential axiom (axiom of the non-emptiness of universe of discourse), existential operators (quantifiers and functors), possible worlds, temporal predicates and operators, objects of sentences.

Could one give expression to a doctrine of essentialism without running afoul of semantical problems that are alleged to beggar systems of quantified modal logic? An affirmative answer is, I believe, called for at least in the case of individual essentialism. Individual essentialism is an ontological thesis concerning a kind of necessary connection between objects and their (essential) properties. It is not or anyhow not primarily a semantic thesis, a thesis about meanings, for example. And thus we are (...) implicitly counselled not to seek for the logical reconstruction of such necessities in theories in which necessity is a de ditto matter. A more natural-seeming target is an understanding of the structure of idioms of de re necessity. Of course, de re necessities may turn out to be special cases of necessities de ditto, with what consequences it is interesting to speculate upon, but it would be preferable if the theory of individual essences were to be prosecuted initially at least without prejudice to this other matter. A theory of individual essences therefore may find it a convenience to recognize a special category of open sentence by means of which to represent de re attributions of necessity. (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 Craft of Formal Logic” is Arthur Prior’s unpublished textbook, written in 1950–51, in which he developed a theory of modality as quantification over possible worlds-like objects. This theory predates most of the prominent pioneering texts in possible worlds semantics and anticipates the significance of its basic concept in modal logic. Prior explicitly defines modal operators as quantifiers of ‘entities’ with modal character. Although he talks about these ‘entities’ only informally, and hesitates how to name them, using alternately (...) the phrases ‘possible states of affairs’, ‘chances’, ‘cases’ or ‘peculiar objects’, he is nevertheless very clear that they should be the fundamental concept of any theory of modality as a form of quantity. Without the assumption that modal operators quantify over such modal objects, the modal system will be incapable of distinguishing an actually true proposition from a necessarily true one. Due to the fact that Prior never made any direct reference to this theory in his subsequently published papers, it remained largely unknown. The comparison of “The Craft” with some of his papers on tense logic suggests that this early theory of modality underlies his later work on temporality. (shrink)

This work aims to make tense logic a more robust tool for ontologists, philosophers, knowledge engineers and programmers by outlining a fusion of tense logic and ontology of time. In order to make tense logic better understandable, the central formal primitives of standard tense logic are derived as theorems from an informal and intuitive ontology of time. In order to make formulation of temporal propositions easier, temporal operators that were introduced by Georg (...) Henrik von Wright are developed, and mapped to the ontology of time. (shrink)

With the past and future tense propositional operators in its syntax, a formal logical system for sortal quantifiers, sortal identity and (second order) quantification over sortal concepts is formulated. A completeness proof for the system is constructed and its absolute consistency proved. The completeness proof is given relative to a notion of logical validity provided by an intensional semantic system, which assumes an approach to sortals from a modern form of conceptualism.

We present an embedding of quantified multimodal logics into simple type theory and prove its soundness and completeness. A correspondence between QKπ models for quantified multimodal logics and Henkin models is established and exploited. Our embedding supports the application of off-the-shelf higher-order theorem provers for reasoning within and about quantified multimodal logics. Moreover, it provides a starting point for further logic embeddings and their combinations in simple type theory.

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)

Modal sentences of the form "every F might be G" and "some F must be G" have a threefold ambiguity. in addition to the familiar readings "de dicto" and "de re", there is a third reading on which they are examples of the "plural de re": they attribute a modal property to the F's plurally in a way that cannot in general be reduced to an attribution of modal properties to the individual F's. The plural "de re" readings of modal (...) sentences cannot be captured within standard quantified modal logic. I consider various strategies for extending standard quantified modal logic so as to provide analyses of the readings in question. I argue that the ambiguity in question is associated with the scope of the general term 'F'; and that plural quantifiers can be introduced for purposes of representing the scope of a general term. Moreover, plural quantifiers provide the only fully adequate solution that keeps within the framework of quantified modal logic. (shrink)

The paper presents a semantics for quantified modal logic which has a weaker axiomatization than the usual Kripke semantics. In particular, the Barcan Formula and its converse are not valid with the proposed semantics. Subclasses of models which validate BF and other interesting formulas are presented. A completeness theorem is proved, and the relation between this result and completeness with respect to Kripke models is investigated.

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)

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)

The first book devoted to a systematic investigation of the logic of the high Middle Ages, this work demonstrates the magnitude of the achievement of medieval logicians. Broadie focuses on the work of some of the great figures of the 14th century, including Walter Burley, William Ockham, John Buridan, Albert of Saxony, and Paul of Venice, and analyzes their theories of truth conditions and valid conditions. Among the topics considered are the medieval exposition of the quantifier shift fallacy, and (...) the rules of valid inference devised to deal with arguments whose premises are not all present-tensed. Revealing how much of what seems characteristically 20th-century logic was actually familiar long ago, Broadie here demonstrates how medieval logic may yet contribute to the solution of 20th-century problems. (shrink)

We add a limited but useful form of quantification to Coalition Logic, a popular formalism for reasoning about cooperation in game-like multi-agent systems. The basic constructs of Quantified Coalition Logic (QCL) allow us to express such properties as "every coalition satisfying property P can achieve φ" and "there exists a coalition C satisfying property P such that C can achieve φ". We give an axiomatisation of QCL, and show that while it is no more expressive than Coalition (...) Logic, it is nevertheless exponentially more succinct. The complexity of QCL model checking for symbolic and explicit state representations is shown to be no worse than that of Coalition Logic, and satisfiability for QCL is shown to be no worse than satisfiability for Coalition Logic. We illustrate the formalism by showing how to succinctly specify such social choice mechanisms as majority voting, which in Coalition Logic require specifications that are exponentially long in the number of agents. (shrink)

This volume constitutes the Proceedings of a workshop on formal seman tics of natural languages which was held in Tiibingen from the 1st to the 3rd of December 1977. Its main body consists of revised versions of most of the papers presented on that occasion. Three supplementary papers are included because they seem to be of particular interest in their respective fields. The area covered by the work of scholars engaged in philosophical logic and the formal analysis of natural (...) languages testifies to the live liness in those disciplines. It would have been impossible to aim at a complete documentation of relevant research within the limits imposed by a short conference whereas concentration on a single topic would have conveyed the false impression of uniformity foreign to a young and active field. It is hoped that the essays collected in this volume strike a reasonable balance between the two extremes. The topics discussed here certainly belong to the most important ones enjoying the attention of linguists and philosophers alike: the analysis of tense in formal and natural languages, the quickly expanding domain of generalized quantifiers, the problem of vagueness, the connected areas of pronominal reference and presupposition and, last but not least, modal logic as a sort of all-embracing theoretical framework. The workshop which led to this collection formed part of the activities celebrating the 500th anniversary of Tiibingen University. (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)

Prior has conjectured that the tense-logical system Gli obtained by adding to a complete basis for the classical propositional calculus the primitive symbolG, the definitionsDf. F:Fα=NGNαDf. L:Lα=KαGα,and the postulatesis complete for the logic of linear, infinite, transitive, discrete future time. In this paper it is demonstrated that that conjecture is correct and it is shown that Gli has the finite model property: see [4]. The techniques used are in part suggested by those used in Bull [2] and [3]:Gli (...) can be shown to be complete for the logic of linear, infinite, transitive, discrete future time in the sense that every formula of Gli which is true of such time can be proved as a theorem of Gli. For this purpose the notion of truth needs to be formalized. This formalization is effected by the construction of a model for linear, infinite, transitive, discrete future time. (shrink)

One way to obtain a comprehensive semantics for various systems of modal logic is to use a general notion of non-normal world. In the present article, a general notion of modal system is considered together with a semantic framework provided by such a general notion of non-normal world. Methodologically, the main purpose of this paper is to provide a logical framework for the study of various modalities, notably prepositional attitudes. Some specific systems are studied together with semantics using non-normal (...) worlds of different kinds. (shrink)

We present a general approach to quantified modal logics that can simulate most other approaches. The language is based on operators indexed by terms which allow to express de re modalities and to control the interaction of modalities with the first-order machinery and with non-rigid designators. The semantics is based on a primitive counterpart relation holding between n-tuples of objects inhabiting possible worlds. This allows an object to be represented by one, many, or no object in an accessible world. (...) Moreover by taking as primitive a relation between n-tuples we avoid some shortcoming of standard individual counterparts. Finally, we use cut-free labelled sequent calculi to give a proof-theoretic characterisation of the quantified extensions of each first-order definable propositional modal logic. In this way we show how to complete many axiomatically incomplete quantified modal logics. (shrink)

Exhumation and study of the 1945 paradox of confirmation brings out the defect of its formulation. In the context of quantifier conditional-probability logic it is shown that a repair can be accomplished if the truth-functional conditional used in the statement of the paradox is replaced with a connective that is appropriate to the probabilistic context. Description of the quantifier probability logic involved in the resolution of the paradox is presented in stages. Careful distinction is maintained between a formal (...) logic language and its semantics, as the same language may be outfitted with different semantics. An acquaintance with sections 1? 5 of Hailperin (2006) covering the sentential aspects of probability logic is assumed as background information for quantifier probability logic. (shrink)

This paper argues that the Einstein-Minkowski space-time of special relativity provides an adequate model for classical tense logic, including rigorous definitions of tensed becoming and of the logical priority of proper time. In addition, the extension of classical tense logic with an operator for predicate-term negation provides us with a framework for interpreting and defending the significance of future contingency in special relativity. The framework for future contingents developed here involves the dual falsehood of non-logical contraries, (...) only one of which becomes true. This has several methodological, metaphysical and physical advantages over the alternative traditional frameworks for handling future contingents. (shrink)

This paper deals with the historical and philosophical background of the introduction of the notion of branching time in philosophical logic as it is revealed in the hitherto unpublished mail-correspondence between Saul Kripke and A.N. Prior in the late 1950s. The paper reveals that the idea was first suggested by Saul Kripke in a letter to A.N. Prior, dated September 3, 1958, and it is shown how the elaboration of the idea in the course of the correspondence was intimately (...) intervowen with considerations of how to represent indeterminism and of the adequacy of tensed logic in light of special relativity. The correspondence underpins the point that Prior's later development of branching time may be understood as a crucial part of his attempt at the formulating a conceptual framework integrating basic human notions of time and free choice. (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)

We may distinguish between two ways of understanding tense-logical formulae, depending on whether we construe tense operators as operators on sentences or on predicates. Bearing this distinction in mind helps us formalise the premisses of Diodorus Cronus' Master Argument correctly, and give a formal reconstruction of the Argument itself.

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)