The so-called ergodic hierarchy (EH) is a central part of ergodic theory. It is a hierarchy of properties that dynamical systems can possess. Its five levels are egrodicity, weak mixing, strong mixing, Kolomogorov, and Bernoulli. Although EH is a mathematical theory, its concepts have been widely used in the foundations of statistical physics, accounts of randomness, and discussions about the nature of chaos. We introduce EH and discuss its applications in these fields.

Edmund Gettier is Professor Emeritus at the University of Massachusetts, Amherst. This short piece, published in 1963, seemed to many decisively to refute an otherwise attractive analysis of knowledge. It stimulated a renewed effort, still ongoing, to clarify exactly what knowledge comprises.

The aim of this paper is to sketch a theory of justified belief. What I have in mind is an explanatory theory, one that explains in a general way why certain beliefs are counted as justified and others as unjustified. Unlike some traditional approaches, I do not try to prescribe standards for justification that differ from, or improve upon, our ordinary standards. I merely try to explicate the ordinary standards, which are, I believe, quite different from those of many classical, (...) e. g., ‘Cartesian’, accounts. (shrink)

The principal goal of this entry is to present Frege's Theorem (i.e., the proof that the Dedekind-Peano axioms for number theory can be derived in second-order logic supplemented only by Hume's Principle) in the most logically perspicuous manner. We strive to present Frege's Theorem by representing the ideas and claims involved in the proof in clear and well-established modern logical notation. This prepares one to better prepared to understand Frege's own notation and derivations, and read Frege's original work (whether in (...) German or in translation). Moreover, this should prepare the reader to understand a number of scholarly books and articles in the secondary literature on Frege's work. (shrink)

In this paper we propose to argue for two claims. The first is that a sizeable group of epistemological projects – a group which includes much of what has been done in epistemology in the analytic tradition – would be seriously undermined if one or more of a cluster of empirical hypotheses about epistemic intuitions turns out to be true. The basis for this claim will be set out in Section 2. The second claim is that, while the jury is (...) still out, there is now a substantial body of evidence suggesting that some of those empirical hypotheses are true. Much of this evidence derives from an ongoing series of experimental studies of epistemic intuitions that we have been conducting. A preliminary report on these studies will be presented in Section 3. In light of these studies, we think it is incumbent on those who pursue the epistemological projects in question to either explain why the truth of the hypotheses does not undermine their projects, or to say why, in light of the evidence we will present, they nonetheless assume that the hypotheses are false. In Section 4, which is devoted to Objections and Replies, we’ll consider some of the ways in which defenders of the projects we are criticizing might reply to our challenge. Our goal, in all of this, is not to offer a conclusive argument demonstrating that the epistemological projects we will be criticizing are untenable. Rather, our aim is to shift the burden of argument. (shrink)

Timothy Williamson has argued that a person S ’s total evidence is constituted solely by propositions that S knows. This theory of evidence entails that a false belief can not be a part of S ’s evidence base for a conclusion. I argue by counterexample that this thesis (E = K for now) forces an implausible separation between what it means for a belief to be justified and rational from one’s perspective and what it means to base one’s beliefs on (...) the evidence. Furthermore, I argue that E = K entails the implausible result that there are cases in which a well-evidenced belief necessarily can not serve as evidence for a further proposition. (shrink)

This paper explores the prospects for safety-based theories of knowledge in the light of some recent objections.

One of the key supposed 'platitudes' of contemporary epistemology is the claim that knowledge excludes luck. One can see the attraction of such a claim, in that knowledge is something that one can take credit for - it is an achievement of sorts - and yet luck undermines genuine achievement. The problem, however, is that luck seems to be an all-pervasive feature of our epistemic enterprises, which tempts us to think that either scepticism is true and that we don't know (...) very much, or else that luck is compatible with knowledge after all. In this book, Duncan Pritchard argues that we do not need to choose between these two austere alternatives, since a closer examination of what is involved in the notion of epistemic luck reveals varieties of luck that are compatible with knowledge possession and varieties that aren't. Moreover, Pritchard shows that a more nuanced understanding of the relationship between luck and knowledge can cast light on many of the most central topics in contemporary epistemology. These topics include: the externalism/internalism distinction; virtue epistemology; the problem of scepticism; metaepistemological scepticism; modal epistemology; and the problem of moral luck. All epistemologists will need to come to terms with Pritchard's original and incisive contribution. (shrink)

In this paper, I do three things. First, I offer an overview of an anti- luck epistemology, as set out in my book, Epistemic Luck. Second, I attempt to meet some of the main criticisms that one might level against the key theses that I propose in this work. And finally, third, I sketch some of the ways in which the strategy of anti- luck epistemology can be developed in new directions.

If evidence is propositional, is one’s evidence limited to true propositions or might false propositions constitute evidence? In this paper, I consider three recent attempts to show that there can be ‘false evidence,’ and argue that each of these attempts fails. The evidence for the thesis that evidence consists of truths is much stronger than the evidence offered in support of the theoretical assumptions that people have relied on to argue against this thesis. While I shall not defend the view (...) that evidence is propositional, I shall defend the view that any propositional evidence must be true. (shrink)

In this impressive second edition of Theory of Knowledge, Keith Lehrer introduces students to the major traditional and contemporary accounts of knowing. Beginning with the traditional definition of knowledge as justified true belief, Lehrer explores the truth, belief, and justification conditions on the way to a thorough examination of foundation theories of knowledge,the work of Platinga, externalism and naturalized epistemologies, internalism and modern coherence theories, contextualism, and recent reliabilist and causal theories. Lehrer gives all views careful examination and concludes that (...) external factors must be matched by appropriate internal factors to yield knowledge. This match of internal and external factors follows from Lehrer’s new coherence theory of undefeated justification. In addition to doing justice to the living epistemological traditions, the text smoothly integrates several new lines that will interest scholars. Also, a feature of special interest is Lehrer’s concept of a justification game.This second edition of Theory of Knowledge is a thoroughly revised and updated version that contains several completely new chapters. Written by a well-known scholar and contributor to modern epistemology, this text is distinguished by clarity of structure, accessible writing, and an elegant mix of traditional material, contemporary ideas, and well-motivated innovation. (shrink)

Suppose we want to know whether a person justifiably believes a certain claim. Further, suppose that our interest in this question is because we take such justification to be necessary for knowledge. To justifiably believe a claim requires more than there being a justification for that claim. Presumably, there is a justification for accepting all sorts of scientific theories of which I have no awareness; because of my lack of awareness, I do not justifiably believe those theories. Further, even if (...) I, by chance, did believe one of those theories, I might not justifiably believe it; for I may be completely unaware of any justification for it. I then would not believe it.. (shrink)

Knowledge and Lotteries is organized around an epistemological puzzle: in many cases, we seem consistently inclined to deny that we know a certain class of propositions, while crediting ourselves with knowledge of propositions that imply them. In its starkest form, the puzzle is this: we do not think we know that a given lottery ticket will be a loser, yet we normally count ourselves as knowing all sorts of ordinary things that entail that its holder will not suddenly acquire a (...) large fortune. After providing a number of specific and general characterizations of the puzzle, Hawthorne carefully examines the competing merits of candidate solutions. In so doing, he explores a number of central questions concerning the nature and importance of knowledge, including the relationship of knowledge to assertion and practical reasoning, the status of epistemic closure principles, the merits of various brands of scepticism, the prospects for a contextualist account of knowledge, and the potential for other sorts of salience-sensitive accounts. Along the way, he offers a careful treatment of pertinent issues at the foundations of semantics. His book will be of interest to anyone working in the field of epistemology, as well as to philosophers of language. (shrink)

An attempt to analyse what it is for belief to be based on reasons becomes involved with questions about the goodness of reasons and the gettier examples. intuitions about knowledge and the "gettier effect" can be used to decide when reasoning has occurred and what reasoning there has been. explanation by reasons is not deterministic.

Since Edmund L. Gettier reminded us recently of a certain important inadequacy of the traditional analysis of "S knows that p," several attempts have been made to correct that analysis. In this paper I shall offer still another analysis (or a sketch of an analysis) of "S knows that p," one which will avert Gettier's problem. My concern will be with knowledge of empirical propositions only, since I think that the traditional analysis is adequate for knowledge of nonempirical truths.

Russian translation of Gettier E. L. Is Justified True Belief Knowledge? // Analysis, vol. 23, 1963. Translated by Lev Lamberov with kind permission of the author.

The focus of this paper is the prima facie plausible view, expressed by the principle of Counter-Closure, that knowledge-yielding competent deductive inference must issue from known premises. I construct a case that arguably falsifies this principle and consider five available lines of response that might help retain Counter-Closure. I argue that three are problematic. Of the two remaining lines of response, the first relies on non-universal intuitions and forces one to view the case I construct as exhibiting a justified, true (...) belief to which none of the usual diagnoses of knowledge failure in Gettier cases apply. The second line involves claiming that Fake Barns and its ilk are misdiagnosed by epistemological orthodoxy as Gettier cases. We are thus confronted by a trilemma: either the case I discuss undermines the first-blush plausible principle of Counter-Closure; or the case I discuss instantiates a novel kind of Gettier case; or a popular conception of a key range of alleged Gettier cases must be rejected. No matter which horn we choose, the case points to a philosophically curious conclusion. (shrink)

This paper advances the debate over the question whether false beliefs may nevertheless have warrant, the property that yields knowledge when conjoined with true belief. The paper’s first main part—which spans Sections 2–4—assesses the best argument for Warrant Infallibilism, the view that only true beliefs can have warrant. I show that this argument’s key premise conflicts with an extremely plausible claim about warrant. Sections 5–6 constitute the paper’s second main part. Section 5 presents an overlooked puzzle about warrant, and uses (...) that puzzle to generate a new argument for Warrant Fallibilism, the view that false beliefs can have warrant. Section 6 evaluates this pro-Fallibilism argument, finding ultimately that it defeats itself in a surprising way. I conclude that neither Infallibilism nor Fallibilism should now constrain theorizing about warrant. (shrink)

According to some philosophers who accept a propositional conception of evidence, someone's evidence includes a proposition only if it is true. I argue against this thesis by appealing to the possibility of knowledge from falsehood.

In this important new text, Keith Lehrer introduces students to the major traditional and contemporary accounts of knowing. Beginning with the accepted definition of knowledge as justified true belief, Lehrer explores the truth, belief and justification conditions on the way to a thorough examination of foundation theories of knowledge, externalism and naturalized epistemologies, internalism and modern coherence theories as well as recent reliabilist and causal theories. Lehrer gives all views careful examination and concludes that external factors must be matched by (...) appropriate internal ones to yield knowledge. Readers of Professor Lehrer's earlier book _Knowledge_ will want to know that this text adopts the framework of that classic text. But _Theory of Knowledge_ is a completely rewritten and updated version of that book that has been simplified throughout for student use. (shrink)

In this important new text, Keith Lehrer introduces students to the major traditional and contemporary accounts of knowing. Beginning with the accepted definition of knowledge as justified true belief, Lehrer explores the truth, belief and justification conditions on the way to a thorough examination of foundation theories of knowledge, externalism and naturalized epistemologies, internalism and modern coherence theories as well as recent reliabilist and causal theories. Lehrer gives all views careful examination and concludes that external factors must be matched by (...) appropriate internal ones to yield knowledge. Readers of Professor Lehrer's earlier book _Knowledge_ will want to know that this text adopts the framework of that classic text. But _Theory of Knowledge_ is a completely rewritten and updated version of that book that has been simplified throughout for student use. (shrink)

The standard view of philosophical methodology is that philosophers rely on intuitions as evidence. Herman Cappelen argues that this claim is false: it is not true that philosophers rely extensively on intuitions as evidence. At worst, analytic philosophers are guilty of engaging in somewhat irresponsible use of 'intuition'-vocabulary. While this irresponsibility has had little effect on first order philosophy, it has fundamentally misled meta-philosophers: it has encouraged meta-philosophical pseudo-problems and misleading pictures of what philosophy is.

Alvin Plantinga responds to the essays in a concluding chapter.

Presented throughout in an accessible style, this book will prove particularly useful for students, researchers and general readers of philosophy who are ...

This entry addresses the nature and epistemological role of intuition by considering the following questions: (1) What are intuitions?, (2) What roles do they serve in philosophical (and other “armchair”) inquiry?, (3) Ought they serve such roles?, (4) What are the implications of the empirical investigation of intuitions for their proper roles?, and (5) What is the content of intuitions prompted by the consideration of hypothetical cases?