This paper is a survey of results concerning embeddings of intuitionistic propositional logic and its extensions into various classical modal systems.

We study modal companions of \(K4^+\), the strictly positive fragment of _K_4. We partially find the boundary between all normal extensions of _K_4 and modal companions of \(K4^+\) among them. We also show that there is no greatest modal companion of \(K4^+\).

We study modal companions of $$K4^+$$, the strictly positive fragment of K4. We partially find the boundary between all normal extensions of K4 and modal companions of $$K4^+$$ among them. We also show that there is no greatest modal companion of $$K4^+$$.

Normal propositional modal systems This first chapter has two main aims. One is to give a general account of the propositional modal systems that we shall ...

This paper investigates partitions of lattices of modal logics based on superintuitionistic logics which are defined by forming, for each superintuitionistic logic L and classical modal logic , the set L[] of L-companions of . Here L[] consists of those modal logics whose non-modal fragments coincide with L and which axiomatize if the law of excluded middle p V p is added. Questions addressed are, for instance, whether there exist logics with the disjunction property in L[], (...) whether L[] contains a smallest element, and whether L[] contains lower covers of . Positive solutions as concerns the last question show that there are (uncountably many) superclean modal logics based on intuitionistic logic in the sense of Vakarelov [28]. Thus a number of problems stated in [28] are solved. As a technical tool the paper develops the splitting technique for lattices of modal logics based on superintuitionistic logics and ap plies duality theory from [34]. (shrink)

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)

The logic BN4 was defined by R.T. Brady as a four-valued extension of Routley and Meyer’s basic logic B. The system EF4 is defined as a companion to BN4 to represent the four-valued system of implication. The system Ł was defined by J. Łukasiewicz and it is a four-valued modal logic that validates what is known as strong Łukasiewicz-type modal paradoxes. The systems EF4-M and EF4-Ł are defined as alternatives to Ł without modal paradoxes. This paper (...) aims to define a Belnap-Dunn semantics for EF4, EF4-M and EF4-Ł. It is shown that EF4, EF4-M and EF4-Ł are strongly sound and complete w.r.t. their respective semantics and that EF4-M and EF4-Ł are free from strong Łukasiewicz-type modal paradoxes. (shrink)

In Bayesian belief revision a Bayesian agent revises his prior belief by conditionalizing the prior on some evidence using Bayes’ rule. We define a hierarchy of modal logics that capture the logical features of Bayesian belief revision. Elements in the hierarchy are distinguished by the cardinality of the set of elementary propositions on which the agent’s prior is defined. Inclusions among the modal logics in the hierarchy are determined. By linking the modal logics in the hierarchy to (...) the strongest modal companion of Medvedev’s logic of finite problems it is shown that the modal logic of belief revision determined by probabilities on a finite set of elementary propositions is not finitely axiomatizable. (shrink)

In Bayesian belief revision a Bayesian agent revises his prior belief by conditionalizing the prior on some evidence using Bayes’ rule. We define a hierarchy of modal logics that capture the logical features of Bayesian belief revision. Elements in the hierarchy are distinguished by the cardinality of the set of elementary propositions on which the agent’s prior is defined. Inclusions among the modal logics in the hierarchy are determined. By linking the modal logics in the hierarchy to (...) the strongest modal companion of Medvedev’s logic of finite problems it is shown that the modal logic of belief revision determined by probabilities on a finite set of elementary propositions is not finitely axiomatizable. (shrink)

This long-awaited book replaces Hughes and Cresswell's two classic studies of modal logic: _An Introduction to Modal Logic_ and _A Companion to Modal Logic_. _A New Introduction to Modal Logic_ is an entirely new work, completely re-written by the authors. They have incorporated all the new developments that have taken place since 1968 in both modal propositional logic and modal predicate logic, without sacrificing tha clarity of exposition and approachability that were essential features (...) of their earlier works. The book takes readers from the most basic systems of modal propositional logic right up to systems of modal predicate with identity. It covers both technical developments such as completeness and incompleteness, and finite and infinite models, and their philosophical applications, especially in the area of modal predicate logic. (shrink)

We study a modal language for negative operators—an intuitionistic-like negation and its paraconsistent dual—added to (bounded) distributive lattices. For each non-classical negation an extra operator is hereby adjoined in order to allow for standard logical inferences to be opportunely restored. We present abstract characterizations and exhibit the main properties of each kind of negative modality, as well as of the associated connectives that express consistency and determinedness at the object-language level. Appropriate sequent-style proof systems and adequate kripke semantics are (...) also introduced, characterizing the minimal normal logic and a few other basic logics containing such negative modalities and their companions. (shrink)

Standard unary modal logic and binary modal logic, i.e. modal logic with one binary operator, are shown to be definitional extensions of one another when an additional axiom |$U$| is added to the basic axiomatization of the binary side. This is a strengthening of our previous results. It follows that all unary modal logics extending Classical Modal Logic, in other words all unary modal logics with a neighborhood semantics, can equivalently be seen as binary (...) modal logics. This in particular applies to standard modal logics, which can be given simple natural axiomatizations in binary form. We illustrate this in the logic K. We call such logics binary expansions of the unary modal logics. There are many more such binary expansions than the ones given by the axiom |$U$|⁠. We initiate an investigation of the properties of these expansions and in particular of the maximal binary expansions of a logic. Our results directly imply that all sub- and superintuitionistic logics with a standard modal companion also have binary modal companions. The latter also applies to the weak subintuitionistic logic WF of our previous papers. This logic doesn’t seem to have a unary modal companion. (shrink)

For each natural number n we study the modal logic determined by the class of transitive Kripke frames in which there are no cycles of length greater than n and no strictly ascending chains. The case n=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=0$$\end{document} is the Gödel-Löb provability logic. Each logic is axiomatised by adding a single axiom to K4, and is shown to have the finite model property and be decidable. We then consider a number of (...) extensions of these logics, including restricting to reflexive frames to obtain a corresponding sequence of extensions of S4. When n=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=1$$\end{document}, this gives the famous logic of Grzegorczyk, known as S4Grz, which is the strongest modal companion to intuitionistic propositional logic. A topological semantic analysis shows that the n-th member of the sequence of extensions of S4 is the logic of hereditarily n+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n+1$$\end{document}-irresolvable spaces when the modality ◊\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Diamond $$\end{document} is interpreted as the topological closure operation. We also study the definability of this class of spaces under the interpretation of ◊\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Diamond $$\end{document} as the derived set operation. The variety of modal algebras validating the n-th logic is shown to be generated by the powerset algebras of the finite frames with cycle length bounded by n. Moreover each algebra in the variety is a model of the universal theory of the finite ones, and so is embeddable into an ultraproduct of them. (shrink)

This Companion provides a comprehensive guide to ancient logic. The first part charts its chronological development, focussing especially on the Greek tradition, and discusses its two main systems: Aristotle's logic of terms and the Stoic logic of propositions. The second part explores the key concepts at the heart of the ancient logical systems: truth, definition, terms, propositions, syllogisms, demonstrations, modality and fallacy. The systematic discussion of these concepts allows the reader to engage with some specific logical and exegetical issues (...) and to appreciate their transformations across different philosophical traditions. The intersections between logic, mathematics and rhetoric are also explored. The third part of the volume discusses the reception and influence of ancient logic in the history of philosophy and its significance for philosophy in our own times. Comprehensive coverage, chapters by leading international scholars and a critical overview of the recent literature in the field will make this volume essential for students and scholars of ancient logic. (shrink)

This chapter contains sections titled: Introduction Modal propositional Logics (MPLs) Modal Quantificational Logics(QMLs).

This chapter contains sections titled: Enriching Extensional Logic with Intensional Notions Changing Views of Modal Logic A Précis of Basic Modal Logic The Major Applications Fine‐Structure of Expressive Power System Combination: Action and Information Back to the Heartland Conclusion.

Dummett's formulation presupposes a realistic attitude towards modality. And as also remarked, realism in this sense has not gone unchallenged. This chapter discusses a different form the realism it opposes may assume which entails, but goes appreciably beyond, the comparatively modest variety just sketched, and which has been the focus of much recent discussion: realism about possible worlds. It begins by drawing some distinctions among different notions of necessity and possibility. Probably the single most important distinction to be drawn is (...) between absolute and relative kinds or senses. In an early defense of realism, David Lewis gives some prominence to an argument which represents the central thesis of his realism. Lewis is chary of saying outright whether worlds themselves are concrete entities, pending clarification of the abstract/concrete distinction. Lewis's own discussion of the charge that his position is epistemologically bankrupt largely in presupposing that it will be based upon a causal constraint on knowledge. (shrink)

Nominalizations of modal predicates have received little, if any, attention in the semantic or philosophical literature. This paper will argue that nominalizations of modal predicates require recognizing a novel ontological category of modal objects and it will outline a new semantics of modals based on modal objects.

Time and modality show remarkable similarities. Each of the most discussed theories in philosophy of time finds an analogous counterpart in modal metaphysics, suggesting that the parallel between the two notions is metaphysically deep. This chapter offers a brief overview of their analogies. Section 1 addresses the analogy between presentism and actualism. Section 2 explores the analogy between non-presentist theories and possibilism. Section 3 discusses the analogy between temporal and modal persistence.

This chapter contains sections titled: Abstract Epistemic Modals Contextualism Contextualism about Epistemic Modals Relativist Proposals Relativists' Arguments Against Contextualism Conclusion References.

The Continuum Companion to Philosophy of Language offers the definitive guide to contemporary philosophy of language. The book covers all the fundamental questions asked by the philosophy of language - areas that have continued to attract interest historically as well as topics that have emerged more recently as active areas of research. Ten specially commissioned essays from an international team of experts reveal where important work continues to be done in the area and, most valuably, the exciting new directions (...) the field is taking. The Companion explores issues pertaining to the nature of language, form semantics, theories of meaning, reference, intensional contexts, context-dependence, pragmatics, the normativity of language, analyticity, a priority and modality. Featuring a series of indispensable research tools, including an A to Z of key terms and concepts, a detailed list of resources and a fully annotated bibliography, this is the essential reference tool for anyone working in the philosophy of language. (shrink)

For Spinoza substances are of certain kinds. Spinoza captures this with the help of attributes. For example, thought and extension are infinite attributes of God‐substance. For Spinoza the essence of substance is its existence and existence is power. An important albeit difficult thing in Spinoza's conception of essence is that an expression of an essence, such as the physical reality or the mental reality, is always a full expression of the essence or power. Spinoza shared with Descartes a well‐known tenet (...) according to which thinking of the finite is posterior to thinking of the infinite. The chapter deals with a citation where Spinoza denies that God is subject to fate, and provides a rough sketch of how Spinoza tries to shape the universe in causal form. Spinoza never considers how such free acts of thought or adequate ideas, are consistent with his necessitarianism. (shrink)

This chapter contains section titled: Acknowledgments References and Further Reading.

[1] In twelve quite demanding chapters, outstanding scholars provide an overall view of the key issues of Scotus’s philosophical thought. To this a very concise introduction is added, concerning the life and works of John Duns (very good, especially the survey of works and the information on critical editions etc.). Throughout the book, I find the information clear and the difficult topics well explained. Moreover, the volume gives a quick entrance to the vast literature. Among the topics discussed are: ‘Metaphysics’ (...) (Peter King), ‘Universals and Individuation’ (Timothy Noone), ‘Modal Theory’ (Calvin Normore), ‘Natural Theology’ (James Ross & Todd Bates), ‘Philosophy of Mind’ (Richard Cross), ‘Cognition’ (Robert Pasnau), ‘Moral Dispositions’ (Bonnie Kent). What strikes the eye is the absence of important theological subjects: Trinity, Christology, sin and grace, to name a few. Since the cover text promises that ‘the essays in this volume systematically survey the full range of Scotus’ thought’, this omission is remarkable. It stems, I guess, from the strict philosophical scope of the series of the Cambridge Companions, but such a limitation should have been recognised explicitly: this companion provides, in fact, an introduction to John Duns’s philosophy—i.e., philosophy in our modern sense. Of course, this separation of philosophical from theological thought is not from Scotus. Most of his innovative ‘philosophical’ ideas are developed in a profoundly theological context! (shrink)

In many logics dealing with information one needs to make statements not only about cognitive states, but also about transitions between them. In this paper we analyze a dynamic modal logic that has been designed with this purpose in mind. On top of an abstract information ordering on states it has instructions to move forward or backward along this ordering, to states where a certain assertion holds or fails, while it also allows combinations of such instructions by means of (...) operations from relation algebra. In addition, the logic has devices for expressing whether in a given state a certain instruction can be carried out, and whether that state can be arrived at by carrying out a certain instruction. This paper deals mainly with technical aspects of our dynamic modal logic. It gives an exact description of the expressive power of this language; it also contains results on decidability for the language with 'arbitrary' structures and for the special case with a restricted class of admissible structures. In addition, a complete axiomatization is given. The paper concludes with a remark about the modal algebras appropriate for our dynamic modal logic, and some questions for further work. The paper only contains some sketchy examples showing how the logic can be used to capture situations of dynamic interest, far more detailed applications are given in a companion to this paper (De Rijke [33]). (shrink)

Modality is standardly thought to come in two varieties: de dicto and de re. De re modality concerns the attribution of modal features to things or individuals, and enshrines a commitment to Aristotelian essentialism. This chapter considers how David Lewis's conception of de re modality fits into his overall metaphysics. The hypothesis is that the driving force behind his metaphysics in general, and his adherence to counterpart theory in particular, is the distinctly Humean thought that necessary connections between distinct (...) existences are literally unintelligible. Lewis's appeal to counterpart theory in his account of truthmakers is explicitly aimed at delivering a truthmaker principle that eschews necessary connections. Lewis's attitude towards several well‐rehearsed debates in contemporary ontology and the reasons underlying that attitude closely mirror Hume's incendiary verdict on "divinity and school metaphysics." Finally, de re modality is connected with his adherence to the doctrine of "Humean supervenience". (shrink)

This chapter contains sections titled: Validity and Necessity The Validity Paradox Gödel Arithmetizing the Validity Paradox The Validity Paradox in S5 Validity, Necessity, and Deductive Inference.

These lecture notes were composed while teaching a class at Stanford and studying the work of Brian Chellas (Modal Logic: An Introduction, Cambridge: Cambridge University Press, 1980), Robert Goldblatt (Logics of Time and Computation, Stanford: CSLI, 1987), George Hughes and Max Cresswell (An Introduction to Modal Logic, London: Methuen, 1968; A Companion to Modal Logic, London: Methuen, 1984), and E. J. Lemmon (An Introduction to Modal Logic, Oxford: Blackwell, 1977). The Chellas text influenced me the (...) most, though the order of presentation is inspired more by Goldblatt.2 My goal was to write a text for dedicated undergraduates with no previous experience in modal logic. The text had to meet the following desiderata: (1) the level of difficulty should depend on how much the student tries to prove on his or her own—it should be an easy text for those who look up all the proofs in the appendix, yet more difficult for those who try to prove everything themselves; (2) philosophers (i.e., colleagues) with a basic training in logic should be able to work through the text on their own; (3) graduate students should find it useful in preparing for a graduate course in modal logic; (4) the text should prepare people for reading advanced texts in modal logic, such as Goldblatt, Chellas, Hughes and Cresswell, and van Benthem, and in particular, it should help the student to see what motivated the choices in these texts; (5) it should link the two conceptions of logic, namely, the conception of a logic as an axiom system (in which the set of theorems is constructed from the bottom up through proof sequences) and the conception of a logic as a set containing initial ‘axioms’ and closed under ‘rules of inference’ (in which the set of theorems is constructed from the top down, by carving out the logic from the set of all formulas as the smallest set closed under the rules); finally, (6) the pace for the presentation of the completeness theorems should be moderate—the text should be intermediate between Goldblatt and Chellas in this regard (in Goldblatt, the completeness proofs come too quickly for the undergraduate, whereas in Chellas, too many unrelated.... (shrink)

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.

This chapter focuses on relativism, and outlines debate about relativism about epistemic modals. The debate will be helpful to say a bit more about the structure of contextualist theories, since contextualism is the main competitor to relativism, and probably is the default starting point view. Accordingly, much of the motivation for relativism comes from the purported inadequacy of the contextualist options. The chapter looks at some of the important features of contextualist views in general. It discusses the internal workings of (...) two different sorts of relativist theories and some standard arguments relativists have deployed to motivate relativism over contextualism. The chapter traces a few of the first steps of the ensuing dialectic of reply and objection and looks at a number of the standard replies to relativist arguments. It looks at some of the anti‐relativists' responses to the relativists' positive arguments. (shrink)

Many contemporary philosophers accept Hume's Dictum, according to which there are no metaphysically necessary connections between distinct, intrinsically typed entities. Tacit in Lewis 's work is a potential motivation for HD, according to which one should accept HD as presupposed by the best account of the range of metaphysical possibilities---namely, a combinatorial account, applied to spatiotemporal fundamentalia. Here I elucidate and assess this Ludovician motivation for HD. After refining HD and surveying its key, recurrent role in Lewis ’s work, I (...) present Lewis ’s appeal to HD as providing a broadly axiomatic generating basis for the space of metaphysical modality, and canvas the prima facie advantages of the resulting combinatorial principle---HD ---as being principled, extensionally adequate and modally reductive. Most criticisms of Lewis 's combinatorialism have targeted seeming ways in which the theory overgenerates the desired space; I rather argue that HD seriously undergenerates the desired space in three different ways. For each way I argue that available means of overcoming the undergeneration either fail to close the gap, undermine the claim that HD is a principled generator of metaphysical modal space, undermine the reductive status of Lewis 's combinatorialism, or call into question the truth of HD. (shrink)

This chapter contains sections titled: Introduction Intensions Models About Quantification Truth in Models Equality Rigidity De Re/De Dicto Partial Designation Designation and Existence Definite Descriptions What Next?

Many contemporary philosophers accept a strong generalization of Hume's denial of necessary causal connections, in the form of Hume's dictum (HD), according to which there are no metaphysically necessary connections between distinct, intrinsically typed entities. Hume's version of his dictum occurs during his investigation into the source of the idea of causal connection. The most powerful role that HD plays in Lewis's system concerns its providing a basis for, as Lewis puts it, a "principle of plentitude" that will guarantee "that (...) the worlds are abundant, and logical space is somehow complete". Lewis's combinatorialism promises to provide a principled, extensionally adequate, and reductive generator of metaphysical modal space. Beyond combinatorialism, there are many other accounts of metaphysical modality that need to be considered, not just with an eye to extensional adequacy but also to the clear desideratum to provide an account of the space of metaphysical modality that is principled. (shrink)

The current work investigated whether differences in phonological overlap between the past- and present-tense forms of regular and irregular verbs can account for the graded neurophysiological effects of verb regularity observed in past-tense priming designs. Event-related potentials were recorded from 16 healthy participants who performed a lexical-decision task in which past-tense primes immediately preceded present-tense targets. To minimize intra-modal phonological priming effects, cross-modal presentation between auditory primes and visual targets was employed, and results were compared to a (...) class='Hi'>companion intra-modal auditory study (Justus, T., Larsen, J., de Mornay Davies, P., Swick, D. (2008). Interpreting dissociations between regular and irregular past-tense morphology: evidence from event-related potentials. Cognitive, Affective, Behavioral Neuroscience, 8, 178–194.). For both regular and irregular verbs, faster response times and reduced N400 components were observed for present-tense forms when primed by the corresponding past-tense forms. Although behavioral facilitation was observed with a pseudopast phonological control condition, neither this condition nor an orthographic-phonological control produced significant N400 priming effects. Instead, these two types of priming were associated with a post-lexical anterior negativity (PLAN). Results are discussed with regard to dual- and single-system theories of inflectional morphology, as well as intra- and cross-modal prelexical priming. (shrink)

This volume succeeds the same authors' well-known An Introduction to Modal Logic and A Companion to Modal Logic. We designate the three books and their authors NIML, IML, CML and H&C respectively. Sadly, George Hughes died partway through the writing of NIML.

Induction, conceived as the class of rational non-truth preserving inferences, has been a perennial problem in philosophy. Aside from the problem of justification of induction, a less debated issue is the problem of properly describing inductive inferences. The purpose of this book is to conceptually investigate this descriptive problem of induction from the standpoint of the nonmonotonic logical tradition raised inside the field of Artificial Intelligence in the last thirty years. As we try to show, an essential part of this (...) enterprise involves the analysis of a pragmatically oriented notion of probability, called by us inductive plausibility. This in turn is made from the viewpoint of paraconsistency ? which according to some is the unavoidable companion of nonmonotonicity ? and modal logic. We therefore put forward a nonmonotonic, paraconsistent and modal approach to induction and plausibility. The book shall be useful to philosophers, logicians and computer scientists interested in the problem of induction in general and in the logical analysis of induction and plausibility in particular. (shrink)

Peter King’s essay on Scotus’s metaphysics belongs to the first type. King introduces the reader in a clear and lively manner to some of the major themes of Scotist metaphysics. One may only regret that the Scotist’s doctrine of the univocity of being is mentioned all too briefly and that the author does not fully explore the tension it creates with the doctrine of God’s transcendence. In “Universal and Individuation” Timothy Noone offers a remarkably clear analysis of this intricate topic (...) and presents Scotus’s solution in dialogue with his predecessors and contemporaries. Discussing modal theory, Calvin Normore rightly takes his distance from the possible-world semantic model that has been imposed on Scotus, and he shows that Scotus never completely divorced time and modalities, “retaining a significant distinction between the modal status of the past and that of the future and the use of notions of priority and posteriority modeled on temporal relations in his account of the contingency of the present”. Scotus’s theology is presented in two essays by James Ross and Todd Bates on “Duns Scotus on Natural Theology” and William Mann’s lively discussion of “Duns Scotus on Natural and Supernatural Theology.”. (shrink)