Alan Hajek

They say that anyone of average talent can become a strong chess player by learning and internalizing certain chess heuristics—“castle early,” “avoid isolated pawns,” and so on. Analogously, philosophy has a wealth of heuristics—philosophical heuristics—although they have not been nearly so well documented and studied. Sometimes these take the form of useful heuristics for generating counterexamples, such as “check extreme cases.” Sometimes they suggest ways of generating new arguments out of old ones, as in “arguments involving possibility can often be recast as arguments involving time or space.” Sometimes they provide templates for positive arguments (e.g., ways of showing that something is possible). This chapter is partly an introduction to a larger project of mine of identifying and evaluating philosophical heuristics, illustrating them with numerous examples from the philosophical literature. I also intend it to be a contribution to the philosophy of creativity: I argue that such heuristics can enhance one’s ability to make creative contributions in philosophy.

This chapter assimilates the Sorites Paradox and the Preface Paradox, drawing parallels between reasoning with uncertainty and reasoning with vague concepts (a theme that Dorothy Edgington has explored). It discusses experiments in which subjects are taken along soritical series of coloured patches, displaying so-called _reverse hysteresis_ in their responses. The chapter offers an explanation of why reverse hysteresis is rational there. It presents a variant of the Preface Paradox—_the Progressive Preface Paradox_—that is analogous to the Sorites Paradox and its associated experiments, and it offers an analogous explanation of why reverse hysteresis is again rational. The explanation’s central idea is that ‘belief’ is context-dependent.

Pascal presents at least three ‘wagers’ for believing in God. Hacking provides three reconstructions of them using the apparatus of Bayesian decision theory — dominance reasoning and calculations of expected utilities — contending that each is valid. The argument of this chapter is that each is _invalid_. The chapter canvases McLennen’s reconstruction of the first wager as an argument from ‘superdominance’, it then shows how it can be strengthened in two respects that are faithful to Pascal’s original text; yet the resulting argument is still invalid. The chapter then turns to the second and especially the third wagers, offering two new ways of reformulating them with valid arguments. In the process new decision rules for comparing acts of infinite and indeterminate expected utility are suggested and defended.

Various philosophers are skeptical about modalities such as laws of nature, counterfactuals, dispositions, and so on. Chance is in a way the black sheep of the modal family: not only is it a modality, but it is one that comes in degrees. One might as a result be skeptical about it twice over: as modal witchcraft with spurious numbers attached! In this chapter, it is argued that the numbers provide no extra reason for skepticism. A whole range of chance values, of arbitrary precision, can be derived from very basic and intuitive comparative assumptions. In fact, for a wide range of systems these assumptions seem hard to deny. The centrepiece of the argument is a ‘magic trick’. Give me any object, any number between 0 and 1 inclusive, and a specified accuracy, and I will use the object to generate an event whose chance is the number given to the accuracy specified.

ABSTRACT‘Moore’s Paradox’ concerns sentences like ‘it’s raining and I don’t believe that it’s raining’. Philosophers have attempted to explain why asserting them, or believing their contents, would be absurd. Why are they paradoxical? The common story is that unlike contradictory sentences, Moore-paradoxical sentences could be true or might be true, so it is puzzling why they cannot be appropriately asserted or their contents rationally believed. However, it is not at all paradoxical that a particular sentence that’s merely possibly true – perhaps true only at distant worlds – cannot be so asserted or believed. Nor is this paradoxical even for a sentence that is merely epistemically possible, or merely true at nearby worlds. It is paradoxical that there are (actually) true sentences like this, which one cannot learn, and one can do much to pinpoint them oneself. I show how ubiquitous and easily identifiable are true Moore-paradoxical sentences, thus sharpening the paradox.

According to orthodox (Kolmogorovian) probability theory, conditional probabilities are by definition certain ratios of unconditional probabilities. As a result, orthodox conditional probabilities are regarded as undefined whenever their antecedents have zero unconditional probability. This has important ramifications for the notion of probabilistic independence. Traditionally, independence is defined in terms of unconditional probabilities (the factorization of the relevant joint unconditional probabilities). Various “equivalent” formulations of independence can be given using conditional probabilities. But these “equivalences” break down if conditional probabilities are permitted to have conditions with zero unconditional probability. We reconsider probabilistic independence in this more general setting. We argue that a less orthodox but more general (Popperian) theory of conditional probability should be used, and that much of the conventional wisdom about probabilistic independence needs to be rethought.

To an unusual extent, philosophers agree that counterfactuals have truth conditions involving the most similar possible worlds where their antecedents are true, in the style of the celebrated and path‐breaking Stalnaker/Lewis accounts. Roughly, these accounts say that the counterfactual if A were the case, C would be the case is true if and only if at the most similar A‐worlds, C is true. I will argue that there are general structural problems with the appeals to both “the most” and “similar”. I will challenge any fixation on ‘the most __ worlds’, however we fill in the blank with a non‐trivial ordering of worlds: in ignoring worlds that are later in the ordering, it adjudicates various implausibly specific counterfactuals to be true. I will then raise foundational problems for appealing to ‘similarity’—from consequents that are chancy, disjunctive antecedents, and unspecific antecedents more generally. I will also raise further problems for a number of specific proposals for understanding ‘similarity’. A recurring theme will be the tension that may arise between probability and similarity considerations. I will end by arguing for a paradigm shift, replacing ‘the most similar worlds’ approach with one based on conditional chances.

So-called “traditional epistemology” and “Bayesian epistemology” share a word, but it may often seem that the enterprises hardly share a subject matter. They differ in their central concepts. They differ in their main concerns. They differ in their main theoretical moves. And they often differ in their methodology. However, in the last decade or so, there have been a number of attempts to build bridges between the two epistemologies. Indeed, many would say that there is just one branch of philosophy here—epistemology. There is a common subject matter after all. In this paper, we begin by playing the role of a “bad cop,” emphasizing many apparent points of disconnection, and even conflict, between the approaches to epistemology. We then switch role, playing a “good cop” who insists that the approaches are engaged in common projects after all. We look at various ways in which the gaps between them have been bridged, and we consider the prospects for bridging them further. We conclude that this is an exciting time for epistemology, as the two traditions can learn, and have started learning, from each other.

The philosophy of probability has been alive and well for several decades in Australia and New Zealand. Some distinctive lines of thought have emerged, resonating with broader themes that have come to be associated with Australasian philosophers: realist/objectivist accounts of various theoretical entities; an ongoing concern with logic, including the development of non­classical logics; and enthusiasm for conceptual analysis, rooted in commonsense but informed by science. In this article I concentrate on work by philosophers on the interpretation of probability, its logical foundations, and its philosophical applications.1 My nomination for the earliest major Australasian philosopher of probability may surprise some readers: Karl Popper. He counts as Australasian by dint of his employment at the University of Canterbury from 1937 until the end of World War II; he counts as a major philosopher of probability by any estimation. Two of his contributions have initiated research programs in the foundations of probability that are still thriving: his (1959a) axiomatization of primitive conditional probability functions (so­called ‘Popper functions’), and his ‘propensity’ interpretation of probability (1959b), intended to illuminate single­case attributions of objective probabilities, as are putatively found in quantum mechanics.

Philosophy has a wealth of heuristics—philosophical heuristics—although they have not been well documented or studied. Sometimes they draw attention to a problem with a philosophical position—for example, it involves a problematic definite description, or it has to make a choice that seems arbitrary. Sometimes they provide solutions to a problem—for example, there are many techniques for handling arbitrariness. Sometimes they suggest ways of replacing hard problems with easier ones, with strategies for approaching the latter—for example, replacing intensional notions with extensional surrogates, and then perhaps diagramming the latter. Sometimes they appeal to fertile modes of thinking more generally—for example, continuity reasoning. Philosophers have been becoming increasingly self-conscious of their methodology, yet I believe that the study of such heuristics has been surprisingly neglected. This chapter highlights the importance of philosophical heuristics to philosophical methodology. It considers especially a cluster involving the notion of resemblance.

As van Fraassen pointed out in his opening remarks, Henry Kyburg's lottery paradox has long been known to raise difficulties in attempts to represent full belief as a probability greater than or equal to p, where p is some number less than 1. Recently, Patrick Maher has pointed out that to identify full belief with probability equal to 1 presents similar difficulties. In his paper, van Fraassen investigates ways of representing full belief by personal probability which avoid the difficulties raised by Maher's measure-theoretic version of the lottery paradox. Van Fraassen's more subtle representation dissolves the simple identification of full belief with maximal personal probability. His investigation exploits the richer resources for representing opinion provided by taking conditional, rather than unconditional, personal probability as fundamental. It has interesting implications for equivalent alternative approaches based on non-Archimedean probability, as well as for equivalent approaches in which assumption contexts representing full belief relative to suppositions are taken as fundamental.

David Lewis was one of the most important philosophers of the 20th century working in the Anglo-American analytic tradition. His corpus is extraordinary for its breadth of subject matter and for its systematicity. For both these reasons, it is difficult to do justice to his work in a short space—there are rich interconnections among his myriad writings, and numerous possible entry points. This article approaches Lewis and his work in three passes: first, a biographical tracing of his intellectual influences; second, a summary of his metaphilosophy; third, a survey of his more specific philosophical views, mostly following their order of conceptual dependence.