Branden Fitelson

We revisit a longstanding question about the shortest single axioms for positive implicational logic. Meredith discovered several 17-symbol single axioms and asked whether shorter ones exist. Later work reduced the problem to four candidate formulas of length 15. Using the automated theorem prover Vampire, we compute saturated clause sets that yield countermodels showing that three of these candidates are not single axioms. This demonstrates the effectiveness of saturation-based reasoning for such problems, which have traditionally been studied via finite model searches.

Paradoxes of individual coherence (e.g., the preface paradox for individual judgment) and group coherence (e.g., the doctrinal paradox for judgment aggregation) typically presuppose that deductive consistency is a coherence requirement for both individual and group judgment. This chapter introduces a new coherence requirement for (individual) full belief, and it explains how this new approach to individual coherence leads to an amelioration of the traditional paradoxes. In particular, it explains why this new coherence requirement gets around the standard doctrinal paradox. However, it also proves a new impossibility result, which reveals that (more complex) varieties of the doctrinal paradox can arise even for this new notion of coherence.

Lewis' (1976) triviality argument against The Equation (also known as Adams' thesis) rests on a strong assumption about the nature of (epistemic) rational requirements. Interestingly, Lewis (1980) later rejected this assumption. In his discussion of the Principal Principle, Lewis makes a weaker assumption about the nature of rational requirements. In this paper, I explain how to apply the insights of Lewis (1980) to reply to Lewis (1976). This leads to a more reasonable rendition of the equation -- one that is (a) immune to triviality and (b) a better candidate for a (bona fide) rational requirement. Further issues and problems with Adams' thesis (raised by more recent work on indicative conditionals) are also discussed.

The chapter begins with some general remarks about closure and counter-closure, and is followed with a discussion of the following: I (a) review some (alleged) counterexamples to counter-closure, I then continue by (b) discussing a popular strategy for responding to such counterexamples to counter-closure, and finally I (c) pose a dilemma for this popular strategy. Once I have discussed these three points I conclude the chapter by proposing that we reject counter-closure, but at the same time that we accept an epistemological package that includes closure and another intuitively plausible principle with regards to the psychology and epistemology of deductive inference.

In Chapter 12 of Warrant and Proper Function, Alvin Plantinga constructs two arguments against evolutionary naturalism, which he construes as a conjunction E&N. The hypothesis E says that “human cognitive faculties arose by way of the mechanisms to which contemporary evolutionary thought directs our attention” (p. 220). With respect to proposition N, Plantinga (p. 270) says “it isn't easy to say precisely what naturalism is,” but then adds that “crucial to metaphysical naturalism, of course, is the view that there is no such person as the God of traditional theism.” Plantinga tries to cast doubt on the conjunction E&N in two ways. His “preliminary argument” aims to show that the conjunction is probably false, given the fact (R) that our psychological mechanisms for forming beliefs about the world are generally reliable. His “main argument” aims to show that the conjunction E&N is self‐defeating – if you believe E&N, then you should stop believing that conjunction. Plantinga further develops the main argument in his unpublished paper “Naturalism Defeated” (Plantinga 1994). We will try to show that both arguments contain serious errors.

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.

This volume features more than fifteen essays written in honor of Peter D. Klein. It explores the work and legacy of this prominent philosopher, who has had and continues to have a tremendous influence in the development of epistemology. The essays reflect the breadth and depth of Klein's work. They engage directly with his views and with the views of his interlocutors. In addition, a comprehensive introduction discusses the overall impact of Klein's philosophical work. It also explains how each of the essays in the book fits within that legacy. Coverage includes such topics as a knowledge-first account of defeasible reasoning, felicitous falsehoods, the possibility of foundationalist justification, the many formal faces of defeat, radical scepticism, and more. Overall, the book provides readers with an overview of Klein's contributions to epistemology, his importance to twentieth and twenty-first-century philosophy, and a survey of his philosophical ideas and accomplishments. It's not only a celebration of the work of an important philosopher. It also offers readers an insightful journey into the nature of knowledge, scepticism, and justification.

The accuracy-first program attempts to ground epistemology in the norm that one’s beliefs should be as accurate as possible, where accuracy is measured using a scoring rule. We argue that considerations of scientific progress suggest that such a monism about epistemic value is untenable. In particular, we argue that counterexamples to the standard scoring rules are ubiquitous in the history of science, and hence that these scoring rules cannot be regarded as a precisification of our intuitive concept of epistemic value.

The chapter begins with some general remarks about closure and counter-closure, and is followed with a discussion of the following: I (a) review some (alleged) counterexamples to counter-closure, I then continue by (b) discussing a popular strategy for responding to such counterexamples to counter-closure, and finally I (c) pose a dilemma for this popular strategy. Once I have discussed these three points I conclude the chapter by proposing that we reject counter-closure, but at the same time that we accept an epistemological package that includes closure and another intuitively plausible principle with regards to the psychology and epistemology of deductive inference.

There are many things—call them ‘experts’—that you should defer to in forming your opinions. The trouble is, many experts are modest: they’re less than certain that they are worthy of deference. When this happens, the standard theories of deference break down: the most popular (“Reflection”-style) principles collapse to inconsistency, while their most popular (“New-Reflection”-style) variants allow you to defer to someone while regarding them as an anti-expert. We propose a middle way: deferring to someone involves preferring to make any decision using their opinions instead of your own. In a slogan, deferring opinions is deferring decisions. Generalizing the proposal of Dorst (2020a), we first formulate a new principle that shows exactly how your opinions must relate to an expert’s for this to be so. We then build off the results of Levinstein (2019) and Campbell-Moore (2020) to show that this principle is also equivalent to the constraint that you must always expect the expert’s estimates to be more accurate than your own. Finally, we characterize the conditions an expert’s opinions must meet to be worthy of deference in this sense, showing how they sit naturally between the too-strong constraints of Reflection and the too-weak constraints of New Reflection.

Suppositions can be introduced in either the indicative or subjunctive mood. The introduction of either type of supposition initiates judgments that may be either qualitative, binary judgments about whether a given proposition is acceptable or quantitative, numerical ones about how acceptable it is. As such, accounts of qualitative/quantitative judgment under indicative/subjunctive supposition have been developed in the literature. We explore these four different types of theories by systematically explicating the relationships canonical representatives of each. Our representative qualitative accounts of indicative and subjunctive supposition are based on the belief change operations provided by AGM revision and KM update respectively; our representative quantitative ones are offered by conditionalization and imaging. This choice is motivated by the familiar approach of understanding supposition as `provisional belief revision' wherein one temporarily treats the supposition as true and forms judgments by making appropriate changes to their other opinions. To compare the numerical judgments recommended by the quantitative theories with the binary ones recommended by the qualitative accounts, we rely on a suitably adapted version of the Lockean thesis. Ultimately, we establish a number of new results that we interpret as vindicating the often-repeated claim that conditionalization is a probabilistic version of revision, while imaging is a probabilistic version of update.

In this paper, we compare and contrast two methods for the revision of qualitative beliefs. The first method is generated by a simplistic diachronic Lockean thesis requiring coherence with the agent’s posterior credences after conditionalization. The second method is the orthodox AGM approach to belief revision. Our primary aim is to determine when the two methods may disagree in their recommendations and when they must agree. We establish a number of novel results about their relative behavior. Our most notable finding is that the inverse of the golden ratio emerges as a non-arbitrary bound on the Bayesian method’s free-parameter—the Lockean threshold. This “golden threshold” surfaces in two of our results and turns out to be crucial for understanding the relation between the two methods.

Themes from Klein: Knowledge, Scepticism, and Justification is a collection of essays written to honor retiring philosopher Peter D. Klein, whose work has been and continues to be influential in the ongoing development of contemporary epistemology. Klein has done important work on skepticism, the Gettier problem, the structure of justification, defeasibility theory and defeaters, and his own hotly debated theories, including infinitism and his attempt to systematize what he calls “useful falsehoods.” Klein’s ideas have been influenced, he tells us, by twentieth-century predecessors such as Roderick Chisholm, Ludwig Wittgenstein, and A. J. Ayer. But the reader of Klein’s work will be struck not by whatever similarity it bears to what has come before, nor by the extent to which it departs from the past, but rather by its intellectual integrity. Neither a conformist nor a reactionary, Peter Klein marches to the beat of one drummer: his reasoning. This introduction provides a broad overview of Klein’s ideas on each of the book’s themes (knowledge, skepticism, and justification).

As every philosopher knows, “the design argument” concludes that God exists from premisses that cite the adaptive complexity of organisms or the lawfulness and orderliness of the whole universe. Since 1859, it has formed the intellectual heart of creationist opposition to the Darwinian hypothesis that organisms evolved their adaptive features by the mindless process of natural selection. Although the design argument developed as a defense of theism, the logic of the argument in fact encompasses a larger set of issues. William Paley saw clearly that we sometimes have an excellent reason to postulate the existence of an intelligent designer. If we find a watch on the heath, we reasonably infer that it was produced by an intelligent watchmaker. This design argument makes perfect sense. Why is it any different to claim that the eye was produced by an intelligent designer? Both critics and defenders of the design argument need to understand what the ground rules are for inferring that an intelligent designer is the unseen cause of an observed effect.

The conjunction fallacy has been a key topic in debates on the rationality of human reasoning and its limitations. Despite extensive inquiry, however, the attempt of providing a satisfactory account of the phenomenon has proven challenging. Here, we elaborate the suggestion (first discussed by Sides et al., 2001) that in standard conjunction problems the fallacious probability judgments experimentally observed are typically guided by sound assessments of confirmation relations, meant in terms of contemporary Bayesian confirmation theory. Our main formal result is a confirmation-theoretic account of the conjuntion fallacy which is proven robust (i.e., not depending on various alternative ways of measuring degrees of confirmation). The proposed analysis is shown distinct from contentions that the conjunction effect is in fact not a fallacy and is compared with major competing explanations of the phenomenon, including earlier references to a confirmation-theoretic account.

Carnap's inductive logic (or confirmation) project is revisited from an "increase in firmness" (or probabilistic relevance) point of view. It is argued that Carnap's main desiderata can be satisfied in this setting, without the need for a theory of "logical probability." The emphasis here will be on explaining how Carnap's epistemological desiderata for inductive logic will need to be modified in this new setting. The key move is to abandon Carnap's goal of bridging confirmation and credence, in favor of bridging confirmation and evidential support.