"This book is valuable as expounding in full a theory of meaning that has its roots in the work of Frege and has been of the widest influence.... The chief virtue of the book is its systematic character. From Frege to Quine most philosophical logicians have restricted themselves by piecemeal and local assaults on the problems involved. The book is marked by a genial tolerance. Carnap sees himself as proposing conventions rather than asserting truths. However he provides plenty of matter (...) for argument."-Anthony Quinton, Hibbert Journal. (shrink)

In this paper, I investigate a system of quantified modal logic, due in many respects to Bressan (see [2]), from several perspectives -- both semantic and proof-theoretic. As Anderson and Belnap note in [1]: "It seems to be generally conceded that formal systems are natural or substantial if they can be looked at from several points of view. We tend to think of systems as artificial or ad hoc if most of their formal properties arise from some one notational system (...) in terms of which they are described." My efforts in this paper will be in part to lend substantiality to the system in question. Several formulation of the system are given and proved equivalent in appropriate senses. Also, some comments are made concerning possible alternative formulations. (shrink)

Modern modal logic originated as a branch of philosophical logic in which the concepts of necessity and possibility were investigated by means of a pair of dual operators that are added to a propositional or first-order language. The field owes much of its flavor and success to the introduction in the 1950s of the “possible-worlds” semantics in which the modal operators are interpreted via some “accessibility relation” connecting possible worlds. In subsequent years, modal logic has received attention as an attractive (...) approach towards formalizing such diverse notions as time, knowledge, or action. Nowadays, modal logics are applied in various disciplines, ranging from economics to linguistics and computer science. Consequently, there is by now a large variety of modal languages, with an even greater wealth of interpretations. For instance, many applications require a poly-modal framework consisting of a language with a family of modal operators and a semantics in which the corresponding accessibility relations are connected somehow. (shrink)

In Modal Logic as Metaphysics, Timothy Williamson claims that the possibilism-actualism (P-A) distinction is badly muddled. In its place, he introduces a necessitism-contingentism (N-C) distinction that he claims is free of the confusions that purportedly plague the P-A distinction. In this paper I argue first that the P-A distinction, properly understood, is historically well-grounded and entirely coherent. I then look at the two arguments Williamson levels at the P-A distinction and find them wanting and show, moreover, that, when the N-C (...) distinction is broadened (as per Williamson himself) so as to enable necessitists to fend off contingentist objections, the P-A distinction can be faithfully reconstructed in terms of the N-C distinction. However, Williamson’s critique does point to a genuine shortcoming in the common formulation of the P-A distinction. I propose a new definition of the distinction in terms of essential properties that avoids this shortcoming. (shrink)

This is a thorough treatment of first-order modal logic. The book covers such issues as quantification, equality (including a treatment of Frege's morning star/evening star puzzle), the notion of existence, non-rigid constants and function symbols, predicate abstraction, the distinction between nonexistence and nondesignation, and definite descriptions, borrowing from both Fregean and Russellian paradigms.

When it comes to Kripke-style semantics for quantified modal logic, there’s a choice to be made concerning the interpretation of the quantifiers. The simple approach is to let quantifiers range over all possible objects, not just objects existing in the world of evaluation, and use a special predicate to make claims about existence. This is the constant domain approach. The more complicated approach is to assign a domain of objects to each world. This is the varying domain approach. Assuming that (...) all terms denote, the semantics of predication on the constant domain approach is obvious: either the denoted object has the denoted property in the world of evaluation, or it hasn’t. On the varying domain approach, there’s a third possibility: the object in question doesn’t exist. Terms may denote objects not included in the domain of the world of evaluation. The question is whether an atomic formula then should be evaluated as true or false, or if its truth value should be undefined. This question, however, cannot be answered in isolation. The consequences of one’s choice depends on the interpretation of molecular formulas. Should the negation of a formula whose truth value is undefined also be undefined? What about conjunction, universal quantification and necessitation? The main contribution of this paper is to identify two partial semantics for logical operators, a weak and a strong one, which uniquely satisfy a list of reasonable constraints. I also show that, provided that the point of using varying domains is to be able to make certain true claims about existence without using any existence predicate, this result yields two possible partial semantics for quantified modal logic with varying domains. (shrink)

It has been argued by Bernard Linsky and Edward Zalta, and independently by Timothy Williamson, that the best quantified modal logic is one that validates both the Barcan Formula and its converse. This requires that domains be fixed across all possible worlds. All objects exist necessarily; some – those we would usually consider contingent – are concrete at some worlds and non-concrete (but still existent) at others. Linsky and Zalta refer to such objects as ‘contingently non-concrete’. I defend the standard (...) usage of the word ‘exists’, and the view that many objects exist only contingently. I argue that the Linsky/Zalta analysis, and to a lesser extent Williamson’s, suffers not only from a peculiar ontology but also from two related formal difficulties. Their analysis gives either counter-intuitive or ad hoc results about essences, and it fails to accommodate contingently existing abstracta. (shrink)

The new edition of this widely used and respected textbook includes three new chapters on conditional logic. Other chapters have been revised and updated, making the second edition a fully comprehensive introduction to modal logics and their application. Unlike most modal logic textbooks, which are both forbidding mathematically and short on philosophical discussion, Modal Logics and Philosophy focuses on showing how useful modal logic can be as a tool for formal philosophical analysis. In Part 1, the reader is introduced to (...) some standard systems of modal logic and provided with a series of exercises that encourage proficiency in manipulating these logics. Girle emphasizes possible world semantics for modal logics and its formal method, Jeffrey-style truth-trees, in which standard truth-trees are extended in a simple and transparent way to take possible worlds into account. Part 2 explores the applications of modal logic to philosophical issues such as truth, time, processes, knowledge and belief, and obligation and permission. (shrink)

Quantified modal logic has reputation for complexity. Completeness results for the various systems appear piecemeal. Different tactics are used for different systems, and success of a given method seems sensitive to many factors, including the specific combination of choices made for the quantifiers, terms, identity, and the strength of the underlying propositional modal logic. The lack of a unified framework in which to view QMLs and their completeness properties puts pressure on those who develop, apply, and teach QML to work (...) with the simplest systems, namely those that adopt the Barcan Formulas and predicate logic rules for the quantifiers. In these systems, the quantifier ranges over a fixed domain of possible individuals, so advocates of these logics are sometimes called possibilists. A literature has grown up rationalizing the choice of possibilist logics despite ordinary intuitions that the resulting theorems are too strong.Williamson even takes the view that the complications to be faced within the weaker logics “are a warning sign of philosophical error”. It is the purpose of this paper to show that abandonment of the weaker QMLs is excessively fainthearted, since most QMLs can be given relatively simple formulations within one general framework. Given the straightforward nature of the systems and their completeness results, the purported complications evaporate, along with any philosophical warnings one might have associated with them. (shrink)

This text aims to convey some of the interest and charm of modal logic, and to put a reader new to the subject in a position to have an informed opinion as to its applicability to each of several areas of philosophical concern in which the merits of a modal approach' have been controversial. he main focus, for these purposes, is on normal modal logics, though some attention is given to the non-normal side of the picture.

A (normal) system of propositional modal logic is said to be complete iff it is characterized by a class of (Kripke) frames. When we move to modal predicate logic the question of completeness can again be raised. It is not hard to prove that if a predicate modal logic is complete then it is characterized by the class of all frames for the propositional logic on which it is based. Nor is it hard to prove that if a propositional modal (...) logic is incomplete then so is the predicate logic based on it. But the interesting question is whether a complete propositional modal logic can have an incomplete extension. In 1967 Kripke announced the incompleteness of a predicate extension of S4. The purpose of the present article is to present several such systems. In the first group it is the systems with the Barcan Formula which are incomplete, while those without are complete. In the second group it is those without the Barcan formula which are incomplete, while those with the Barcan Formula are complete. But all these are based on propositional systems which are characterized by frames satisfying in each case a single first-order sentence. (shrink)

This is part I of a two-part essay introducing case-intensional first order logic, an easy-to-use, uniform, powerful, and useful combination of first-order logic with modal logic resulting from philosophical and technical modifications of Bressan’s General interpreted modal calculus. CIFOL starts with a set of cases; each expression has an extension in each case and an intension, which is the function from the cases to the respective case-relative extensions. Predication is intensional; identity is extensional. Definite descriptions are context-independent terms, and lambda-predicates (...) and -operators can be introduced without constraints. These logical resources allow one to define, within CIFOL, important properties of properties, viz., extensionality and absoluteness, Bressan’s chief innovation that allows tracing an individual across cases without recourse to any notion of “rigid designation” or “trans-world identity.” Thereby CIFOL abstains from incorporating any metaphysical principles into the quantificational machinery, unlike extant frameworks of quantified modal logic. We claim that this neutrality makes CIFOL a useful tool for discussing both metaphysical and scientific arguments involving modality and quantification, and we illustrate by discussing in diagrammatic detail a number of such arguments involving the extensional identification of individuals via absolute properties, essential properties, de re vs. de dicto, and the results of possible tests. (shrink)

This paper follows Part I of our essay on case-intensional first-order logic ). We introduce a framework of branching histories to take account of indeterminism. Our system BH-CIFOL adds structure to the cases, which in Part I formed just a set: a case in BH-CIFOL is a moment/history pair, specifying both an element of a partial ordering of moments and one of the total courses of events that that moment is part of. This framework allows us to define the familiar (...) Ockhamist temporal/modal connectives, most notably for past, future, and settledness. The novelty of our framework becomes visible in our discussion of substances in branching histories, i.e., in its first-order part. That discussion shows how the basic idea of tracing an individual thing from case to case via an absolute property is applicable in a branching histories framework. We stress the importance of keeping apart extensionality and moment-definiteness, and give a formal account of how the specification of natural sortals and natural qualities turns out to be a coordination task in BH-CIFOL. We also provide a detailed answer to Lewis’s well-known argument against branching histories, exposing the fallacy in that argument. (shrink)

The article deals with the interpretation of propositional attitudes in the framework of modal predicate logic. The first part discusses the classical puzzles arising from the interplay between propositional attitudes, quantifiers and the notion of identity. After comparing different reactions to these puzzles it argues in favor of an analysis in which evaluations of de re attitudes may vary relative to the ways of identifying objects used in the context of use. The second part of the article gives this analysis (...) a precise formalization from a model-and proof-theoretic perspective. (shrink)

Since the first chapter of this book presents an intro duction to the present state of game-theoretical semantics (GTS), there is no point in giving a briefer survey here. Instead, it may be helpful to indicate what this volume attempts to do. The first chapter gives a short intro duction to GTS and a survey of what is has accomplished. Chapter 2 puts the enterprise of GTS into new philo sophical perspective by relating its basic ideas to Kant's phi losophy (...) of mathematics, space, and time. Chapters 3-6 are samples of GTS's accomplishments in understanding different kinds of semantical phenomena, mostly in natural languages. Beyond presenting results, some of these chapters also have other aims. Chapter 3 relates GTS to an interesting line of logical and foundational studies - the so-called functional interpretations - while chapter 4 leads to certain important methodological theses. Chapter 7 marks an application of GTS in a more philo sophical direction by criticizing the Frege-Russell thesis that words like "is" are multiply ambiguous. This leads in turn to a criticism of recent logical languages (logical notation), which since Frege have been based on the ambi guity thesis, and also to certain methodological sug gestions. In chapter 8, GTS is shown to have important implications for our understanding of Aristotle's doctrine of categories, while chapter 9 continues my earlier criticism of Chomsky's generative approach to linguistic theorizing. (shrink)

A collection of papers by leading researchers in modal logic and theoretical computer science.

Modal logic is the study of modalities - expressions that qualify assertions about the truth of statements - like the ordinary language phrases necessarily, possibly, it is known/believed/ought to be, etc., and computationally or mathematically motivated expressions like provably, at the next state, or after the computation terminates. The study of modalities dates from antiquity, but has been most actively pursued in the last three decades, since the introduction of the methods of Kripke semantics, and now impacts on a wide (...) range of disciplines, including the philosophy of language and linguistics ('possible words' semantics for natural language), constructive mathematics (intuitionistic logic), theoretical computer science (dynamic logic, temporal and other logics for concurrency), and category theory (sheaf semantics). This volume collects together a number of the author's papers on modal logic, beginning with his work on the duality between algebraic and set-theoretic modals, and including two new articles, one on infinitary rules of inference, and the other about recent results on the relationship between modal logic and first-order logic. Another paper on the 'Henkin method' in completeness proofs has been substantially extended to give new applications. Additional articles are concerned with quantum logic, provability logic, the temporal logic of relativistic spacetime, modalities in topos theory, and the logic of programs. (shrink)

This much-needed book provides a thorough account of temporal logic, one of the most important areas of logic in computer science today. The book begins with a solid introduction to semantical and axiomatic approaches to temporal logic. It goes on to cover predicate temporal logic, meta-languages, general theories of axiomatization, many dimensional systems, propositional quantifiers, expressive power, Henkin dimension, temporalization of other logics, and decidability results. With its inclusion of cutting-edge results and unifying methodologies, this book is an indispensable reference (...) for both the pure logician and the theoretical computer scientist. (shrink)

This modern, advanced textbook reviews modal logic, a field which caught the attention of computer scientists in the late 1970's.

This is a first course in propositional modal logic, suitable for mathematicians, computer scientists and philosophers. Emphasis is placed on semantic aspects, in the form of labelled transition structures, rather than on proof theory. The book covers all the basic material - propositional languages, semantics and correspondence results, proof systems and completeness results - as well as some topics not usually covered in a modal logic course. It is written from a mathematical standpoint. To help the reader, the material is (...) covered in short chapters, each concentrating on one topic. These are arranged into five parts, each with a common theme. An important feature of the book is the many exercises and an extensive set of solutions is provided. (shrink)

Modal logic, developed as an extension of classical propositional logic and first-order quantification theory, integrates the notions of possibility and necessity and necessary implication. Arguments whose understanding depends on some fundamental knowledge of modal logic have always been important in philosophy of religion, metaphysics, and epistemology. Moreover, modal logic has become increasingly important with the use of the concept of "possible worlds" in these areas. Introductory Modal Logic fills the need for a basic text on modal logic, accessible to students (...) of elementary symbolic logic. Kenneth Konyndyk presents a natural deduction treatment of propositional modal logic and quantified modal logic, historical information about its development, and discussions of the philosophical issues raised by modal logic. Characterized by clear and concrete explanations, appropriate examples, and varied and challenging exercises, Introductory Modal Logic makes both modal logic and the possible-worlds metaphysics readily available to the introductory level student. (shrink)

Timothy Williamson gives an original and provocative treatment of deep metaphysical questions about existence, contingency, and change, using the latest resources of quantified modal logic. Contrary to the widespread assumption that logic and metaphysics are disjoint, he argues that modal logic provides a structural core for metaphysics.

This book introduces the reader to relevant logic and provides the subject with a philosophical interpretation. The defining feature of relevant logic is that it forces the premises of an argument to be really used in deriving its conclusion. The logic is placed in the context of possible world semantics and situation semantics, which are then applied to provide an understanding of the various logical particles and natural language conditionals. The book ends by examining various applications of relevant logic and (...) presenting some interesting open problems. It will be of interest to a range of readers including advanced students of logic, philosophical and mathematical logicians, and computer scientists. (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)

A textbook on modal logic, intended for readers already acquainted with the elements of formal logic, containing nearly 500 exercises. Brian F. Chellas provides a systematic introduction to the principal ideas and results in contemporary treatments of modality, including theorems on completeness and decidability. Illustrative chapters focus on deontic logic and conditionality. Modality is a rapidly expanding branch of logic, and familiarity with the subject is now regarded as a necessary part of every philosopher's technical equipment. Chellas here offers an (...) up-to-date and reliable guide essential for the student. (shrink)

This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency. Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians. Here it receives its first scientific application since its invention. Modal logic is concerned with the notions of necessity and possibility. What George Boolos does (...) is to show how the concepts, techniques, and methods of modal logic shed brilliant light on the most important logical discovery of the twentieth century: the incompleteness theorems of Kurt Godel and the 'self-referential' sentences constructed in their proof. The book explores the effects of reinterpreting the notions of necessity and possibility to mean provability and consistency. (shrink)

The Handbook of Modal Logic contains 20 articles, which collectively introduce contemporary modal logic, survey current research, and indicate the way in which the field is developing. The articles survey the field from a wide variety of perspectives: the underling theory is explored in depth, modern computational approaches are treated, and six major applications areas of modal logic (in Mathematics, Computer Science, Artificial Intelligence, Linguistics, Game Theory, and Philosophy) are surveyed. The book contains both well-written expository articles, suitable for beginners (...) approaching the subject for the first time, and advanced articles, which will help those already familiar with the field to deepen their expertise. Please visit: http://people.uleth.ca/~woods/RedSeriesPromo_WP/PubSLPR.html - Compact modal logic reference - Computational approaches fully discussed - Contemporary applications of modal logic covered in depth. (shrink)

Why is two-dimensional semantics important? One can think of it as the most recent act in a drama involving three of the central concepts of philosophy: meaning, reason, and modality. First, Kant linked reason and modality, by suggesting that what is necessary is knowable a priori, and vice versa. Second, Frege linked reason and meaning, by proposing an aspect of meaning (sense) that is constitutively tied to cognitive signi?cance. Third, Carnap linked meaning and modality, by proposing an aspect of meaning (...) (intension) that is constitutively tied to possibility and necessity. (shrink)