The article looks at the structure of impossible worlds, and their deployment in the analysis of some intentional notions. In particular, it is argued that one can, in fact, conceive anything, whether or not it is impossible. Thus a semantics of conceivability requires impossible worlds.

Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set (...) theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science. (shrink)

This paper presents an account of the understanding of statements involving metaphysical modality, together with dovetailing theories of their truth conditions and epistemology. The account makes modal truth an objective matter, whilst avoiding both Lewisian modal realism and mind-dependent or expressivist treatments of the truth conditions of modal sentences. The theory proceeds by formulating constraints a world-description must meet if it is to represent a genuine possibility. Modal truth is fixed by the totality of the constraints. To understand modal discourse (...) is to have tacit knowledge of the body of information stated in these constraints. Modal knowledge is attained by evaluating modal statements in accordance with the constraints. The question of the general relations between modal truth and knowability is also addressed. The paper includes a discussion of which modal logic is supported by the presented theory of truth conditions for modal statements. (shrink)

This paper criticises necessitarianism, the thesis that there is at least one necessary truth; and defends possibilism, the thesis that all propositions are contingent, or that anything is possible. The second section maintains that no good conventionalist account of necessity is available, while the third section criticises model theoretic necessitarianism. The fourth section sketches some recent technical work on nonclassical logic, with the aim of weakening necessitarian intuitions and strengthening possibilist intuitions. The fifth section considers several a prioristic attempts at (...) demonstrating that there is at least one necessary proposition and finds them inadequate. The final section emphasises the epistemic aspect of possibilism. (shrink)

There is currently intense interest in the question of the source of our presumed knowledge of truths concerning what is, or is not, metaphysically possible or necessary. Some philosophers locate this source in our capacities to conceive or imagine various actual or non-actual states of affairs, but this approach is open to certain familiar and seemingly powerful objections. A different and ostensibly more promising approach has been developed by Timothy Williamson, according to which our capacity for modal knowledge is just (...) an extension, or by-product, of our general capacity to acquire knowledge of true counterfactual conditionals — a capacity that we deploy ubiquitously in everyday life. Williamson’s account crucially involves a thesis to the effect that modal truths can be explained in terms of counterfactual truths. In this paper, I query Williamson’s account on a number of points, including this thesis. My positive proposal, which owes a debt to the work of Kit Fine on modality and essence, appeals instead to our capacity to grasp essences, understood in a neo-Aristotelian fashion, according to which essences are expressed by ‘real definitions’. (shrink)

In this paper, we develop an alternative strategy, Platonized Naturalism, for reconciling naturalism and Platonism and to account for our knowledge of mathematical objects and properties. A systematic (Principled) Platonism based on a comprehension principle that asserts the existence of a plenitude of abstract objects is not just consistent with, but required (on transcendental grounds) for naturalism. Such a comprehension principle is synthetic, and it is known a priori. Its synthetic a priori character is grounded in the fact that it (...) is an essential part of the logic in which any scientific theory will be formulated and so underlies (our understanding of) the meaningfulness of any such theory (this is why it is required for naturalism). Moreover, the comprehension principle satisfies naturalist standards of reference, knowledge, and ontological parsimony! As part of our argument, we identify mathematical objects as abstract individuals in the domain governed by the comprehension principle, and we show that our knowledge of mathematical truths is linked to our knowledge of that principle. (shrink)

I identify the special sort of stability (invariance, resilience, etc.) that distinguishes laws from accidental truths. Although an accident can have a certain invariance under counterfactual suppositions, there is no continuum between laws and accidents here; a law's invariance is different in kind, not in degree, from an accident's. (In particular, a law's range of invariance is not "broader"--at least in the most straightforward sense.) The stability distinctive of the laws is used to explicate what it would mean for there (...) to be multiple grades (or degrees) of physical necessity. Whether there are is for science to discover. (shrink)

The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous range of set-theoretic possibilities, a phenomenon that challenges the universe (...) view. In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for. (shrink)

The dream solution of the continuum hypothesis would be a solution by which we settle the continuum hypothesis on the basis of a newly discovered fundamental principle of set theory, a missing axiom, widely regarded as true. Such a dream solution would indeed be a solution, since we would all accept the new axiom along with its consequences. In this article, however, I argue that such a dream solution to $\mathrm {CH}$ is unattainable.

This paper discusses some serious difficulties for what we shall call the standard account of various kinds of relative necessity, according to which any given kind of relative necessity may be defined by a strict conditional - necessarily, if C then p - where C is a suitable constant proposition, such as a conjunction of physical laws. We argue, with the help of Humberstone, that the standard account has several unpalatable consequences. We argue that Humberstone’s alternative account has certain disadvantages, (...) and offer another - considerably simpler - solution. (shrink)

Evidentialism is a view about the conditions under which a person is epistemically justified in having a particular doxastic attitude toward a proposition. Evidentialism holds that the justified attitudes are determined entirely by the person's evidence. This is the traditional view of justification. It is now widely opposed. The essays included in this volume develop and defend the tradition.Evidentialism has many assets. In addition to providing an intuitively plausible account of epistemic justification, it helps to resolve the problem of the (...) criterion, helps to disentangle epistemic and ethical evaluations, and illuminates the relationship between epistemic evaluations of beliefs and the evaluation of the methods used to form beliefs. These issues are all addressed in the essays presented here. External world skepticism poses the classic problem for an epistemological theory. The final essay in this volume argues that evidentialism is uniquely well qualified to make sense of skepticism and to respond to its challenge.Evidentialism is a version of epistemic internalism. Recent epistemology has included many attacks on internalism and has seen the development of numerous externalist theories. The essays included here respond to those attacks and raise objections to externalist theories, especially the principal rival, reliabilism. Internalism generally has been criticized for having unacceptable deontological implications, for failing to connect epistemic justification to truth, and for failing to provide an adequate account of what makes basic beliefs justified. Each of these charges is answered in these essays. The collection includes two previously unpublished essays and new afterwords to five of the reprinted essays; it will be the definitive resource on evidentialism for all epistemologists. (shrink)

I defend a version of Kripke's claim that the metaphysically necessary and the knowable a priori are independent. On my version, there are two independent families of modal notions, metaphysical and epistemic, neither stronger than the other. Metaphysical possibility is constrained by the laws of nature. Logical validity, I suggest, is best understood in terms of epistemic necessity.

I defend a version of Kripke's claim that the metaphysically necessary and the knowable a priori are independent. On my version, there are two independent families of modal notions, metaphysical and epistemic, neither stronger than the other. Metaphysical possibility is constrained by the laws of nature. Logical validity, I suggest, is best understood in terms of epistemic necessity.

Michael Dummett is a leading contemporary philosopher whose work on the logic and metaphysics of language has had a lasting influence on how these subjects are conceived and discussed. This volume contains some of the most provocative and widely discussed essays published in the last fifteen years, together with a number of unpublished or inaccessible writings. Essays included are: "What is a Theory of Meaning?," "What do I Know When I Know a Language?," "What Does the Appeal to Use Do (...) for the Theory of Meaning?," "Language and Truth," "Truth and Meaning," "Language and Communication," "The Source of the Concept of Truth," "Mood, Force, and Convention," "Frege and Husserl on Reference," "Realism," "Existence," "Does Quantification Involve Identity?," "Could there be Unicorns?," "Causal Loops," "Common Sense and Physics," "Testimony and Memory," "What is Mathematics About?," "Wittgenstein on Necessity: Some Reflections," and "Realism and Anti-Realism." Serving well as a companion to Dummett's other collections, the essays in this volume are not forbiddingly technical or specialized, and have relevance to many areas of analytic philosophy. (shrink)

This paper considers how counterfactuals should be evaluated on the assumption that determinism is true. I argue against Lewis's influential view that the actual laws of nature would have been false if something had happened that never actually happened, and in favour of the competing view that history would have been different all the way back. I argue that we can do adequate justice to our ordinary practice of relying on a wide range of historical truths in evaluating counterfactuals by (...) saying that, in typical cases, history would have been only *very slightly* different until shortly before the relevant time. The paper also draws some connections between the puzzle about counterfactuals under determinism and the debate about whether determinism entails that no-one can ever do otherwise than they in fact do. (shrink)

Conditional sentences are among the most intriguing and puzzling features of language, and analysis of their meaning and function has important implications for, and uses in, many areas of philosophy. Jonathan Bennett, one of the world's leading experts, distils many years' work and teaching into this book, making it the fullest and most authoritative treatment of the subject.

Classical logic has been attacked by adherents of rival, anti-realist logical systems: Ian Rumfitt comes to its defence. He considers the nature of logic, and how to arbitrate between different logics. He argues that classical logic may dispense with the principle of bivalence, and may thus be liberated from the dead hand of classical semantics.

In The Myth of Morality, Richard Joyce argues that moral discourse is hopelessly flawed. At the heart of ordinary moral judgements is a notion of moral inescapability, or practical authority, which, upon investigation, cannot be reasonably defended. Joyce argues that natural selection is to blame, in that it has provided us with a tendency to invest the world with values that it does not contain, and demands that it does not make. Should we therefore do away with morality, as we (...) did away with other faulty notions such as witches? Possibly not. We may be able to carry on with morality as a 'useful fiction' - allowing it to have a regulative influence on our lives and decisions, perhaps even playing a central role - while not committing ourselves to believing or asserting falsehoods, and thus not being subject to accusations of 'error'. (shrink)

Boris Kment takes a new approach to the study of modality that emphasises the origin of modal notions in everyday thought. He argues that the concepts of necessity and possibility originate in counterfactual reasoning, which allows us to investigate explanatory connections. Contrary to accepted views, explanation is more fundamental than modality.

During the middle of the twentieth century, philosophers generally agreed that, by contrast with science, philosophy should offer no substantial thoughts about the general nature of concrete reality. Instead, philosophers offered conceptual truths. It is widely assumed that, since 1970, things have changed greatly.

This revised and considerably expanded 2nd edition brings together a wide range of topics, including modal, tense, conditional, intuitionist, many-valued, paraconsistent, relevant, and fuzzy logics. Part 1, on propositional logic, is the old Introduction, but contains much new material. Part 2 is entirely new, and covers quantification and identity for all the logics in Part 1. The material is unified by the underlying theme of world semantics. All of the topics are explained clearly using devices such as tableau proofs, and (...) their relation to current philosophical issues and debates are discussed. Students with a basic understanding of classical logic will find this book an invaluable introduction to an area that has become of central importance in both logic and philosophy. It will also interest people working in mathematics and computer science who wish to know about the area. (shrink)

There is a long tradition in philosophy of using a priori methods to draw conclusions about what is possible and what is necessary, and often in turn to draw conclusions about matters of substantive metaphysics. Arguments like this typically have three steps: first an epistemic claim , from there to a modal claim , and from there to a metaphysical claim.

An approach to explaining the nature and source of logic and its laws with a rich historical tradition takes the laws of logic to be laws of thought. This view seems intuitively compelling, after all, logic seems to be intimately related with how we think. But how exactly should we understand it? And what arguments can we give in favour? I will propose one line of argument for the claim that the laws of logic are laws of thought. I will (...) motivate the claim that there is a certain phenomenon, namely, that there are logical principles which are immune to rational doubt. I will then give an argument to the best explanation; I will argue that the best explanation of this phenomenon is to take the laws of logic to be constitutive-normative laws of thought. The proposal, and some responses to potential objections, will have a notably Kantian flavour. (shrink)

It is argued that there are three main forms of necessity --the metaphysical, the natural and the normative--and that none of them is reducible to the others or to any other form of necessity. In arguing for a distinctive form of natural necessity, it is necessary to refute a version of the doctrine of scientific essentialism; and in arguing for a distinctive form of normative necessity, it is necessary to refute certain traditional and contemporary versions of ethical naturalism.