Branden Fitelson

Forcing is a fundamental set-theoretic technique, with which many independence results can be established. A famous example is the independence of the continuum hypothesis in ZFC set theory. Forcing is also well-known to be complex, and therefore difficult to master. Here, we provide a gentle introduction of forcing, by developing forcing for second-order logic. Second-order logic can be interpreted as a rudimentary kind of set theory. Although very limited as a theory of sets, second-order logic is rich enough to capture a form of the continuum hypothesis, as well as the generalized continuum hypothesis, which is slightly easier to state. We develop forcing for second-order logic in the form of possibility semantics, and use this to show that a second-order version of the generalized continuum hypothesis cannot be derived in a standard proof system for second-order logic. Mathematically, the results we obtain in this way are much weaker than standard independence results in set theory. However, as a consequence, we are able to avoid many technical complexities in the presentation of forcing. In this way, we hope that forcing for second-order logic can serve an expository function, of providing an accessible route towards understanding some of the central ideas behind forcing.

Alexander Bain (1818–1903) was once Britain’s greatest philosopher of mind. Author of The Senses and the Intellect (1855) and The Emotions and the Will (1859), Bain articulated a comprehensive theory of the human mind, integrating cognitive, developmental, and evolutionary psychology with neuroscience and philosophy. Mill utilized Bain’s theory of moral judgment to defend utilitarianism. Darwin leaned on Bain’s theory of conscience to explain the evolution of morality. James used Bain’s texts to teach philosophy of mind at Harvard, launching psychology as an independent department of inquiry. Peirce said Bain’s theory of belief was the axiom of pragmatism. And Bain founded the journal Mind, launching the cognitive sciences into the field it has become. Mill is still celebrated for his analyses of liberalism and utilitarianism. Darwin remains famous for his theory of natural selection. And James is well known as the father of both pragmatism and empirical psychology in America. Bain stands behind these accomplishments, his role unacknowledged. Alexander Bain: Philosopher of Mind seeks to rectify the oversight. After introducing the reader to Bain’s early work, Zimmerman analyzes Bain’s solution to the mind–body problem, describes Bain’s contributions to logic and his disagreement with Mill over proper names, engages with Bain’s theories of freewill, conscience, and moral responsibility, and explains Bain’s theory of belief and its role in the genesis of pragmatism. Bain’s philosophy will engage philosophers of mind, cognitive scientists, historians of philosophy, and anyone interested in the “Darwinian revolution” in our understanding of life on Earth.

The standard Bayesian solution to the paradox of the ravens maintains that the degree of confirmation provided by seeing a nonblack nonraven is positive but negligible compared to that provided by seeing a black raven. I show that, unless we impose severe and unmotivated restrictions on the subject’s priors, this has the consequence that the cumulative confirmation provided by all the nonblack nonravens the subject expects to see is nonnegligible compared to the cumulative confirmation provided by all the black ravens the subject expects to see. If this is so, however, then the paradox retains its full force.

To _consequentialize_ a moral theory is to construct a consequentialist version of the theory. The new theory is a ‘version’ of the old in the sense that they deliver the same deontic verdicts on actions: permissible, impermissible, required. This chapter gives some reasons for consequentializing and assesses some obstacles to completing the project consonant with those reasons. It considers the difficulties posed by theoretical verdicts of moral worth, by potential differences between theories not traceable to their ranking of outcomes, and by worries about whether reasons for action are really teleological.

In this article, we pursue two goals. First, we argue that computable probability theory offers a fitting framework for modeling the credences of computably bounded—and, thus, more realistic—Bayesian reasoners. Second, we develop a Bayesian perspective on algorithmic randomness: a branch of computability theory that provides a formal account of what it takes for a sequence of observations (a data stream) to be probabilistically typical in an algorithmically specifiable way. In particular, we argue that adopting such a perspective leads to novel insights for one of the pillars of Bayesian epistemology: Bayesian convergence to the truth. In a companion article, we showed that, for Bayesian agents whose credences are given by computable probability measures, the data streams that guarantee convergence to the truth coincide with the algorithmically random ones. Here, we put these results to use to counter various skeptical arguments which target the philosophical significance of Bayesian convergence-to-the-truth theorems.

What should we believe? One plausible view is that we should believe what is true. Another is that we should believe what is rational to believe. I will argue that both these theses can be accounted for once we add an independently motivated contextualism about normative terms. According to contextualism, the content of ‘ought’ depends on two parameters – a goal and a modal base (or set of possible worlds). It follows that there is a sense in which we should believe truths and a sense in which we should believe what is rational to believe.

A variety of widely-held epistemological theories are committed to one or more of the following theses: that you objectively epistemically ought to believe all and only truths, or have maximally accurate credences; that the epistemic value of a person’s beliefs is determined by their distance from the truth; and that rationality roughly consists in minimizing this distance to the extent possible, given one’s limited information. These commitments form the foundation of accuracy-based epistemology and veritist epistemology generally. This paper focuses on the first claim. Call this the ‘truth norm.’ This paper argues against the truth norm. The norm is open to a variety of interpretations, and each interpretation, once made precise, yields unacceptable verdicts about what we ought to believe. The worries I raise concern higher-order beliefs. Many epistemologists presuppose that puzzles generated by higher-order beliefs can be quarantined off, or idealized away, without impugning broad epistemological theories. One lesson of this paper: higher-orderdom is infectious. It pervades more of our beliefs—beliefs that matter to us—than is generally acknowledged. This fact will create problems for the truth norm and for many truth- and accuracy-based theories of epistemic rationality.

Robert Aumann presents his Agreement Theorem as the key conditional: “if two people have the same priors and their posteriors for an event A are common knowledge, then these posteriors are equal” (Aumann, 1976, p. 1236). This paper focuses on four assumptions which are used in Aumann’s proof but are not explicit in the key conditional: (1) that agents commonly know, of some prior μ, that it is the common prior; (2) that agents commonly know that each of them updates on the prior by conditionalization; (3) that agents commonly know that if an agent knows a proposition, she knows that she knows that proposition (the “K K” principle); (4) that agents commonly know that they each update only on true propositions. It is shown that natural weakenings of any one of these strong assumptions can lead to countermodels to Aumann’s key conditional. Examples are given in which agents who have a common prior and commonly know what probability they each assign to a proposition nevertheless assign that proposition unequal probabilities. To alter Aumann’s famous slogan: people can “agree to disagree”, even if they share a common prior. The epistemological significance of these examples is presented in terms of their role in a defense of the Uniqueness Thesis: If an agent whose total evidence is E is fully rational in taking doxastic attitude D to P, then necessarily, any subject with total evidence E who takes a different attitude to P is less than fully rational.

A core tenet of Bayesian epistemology is that Bayesian conditionalization is the rule of rational credal revision. But it has been pointed out in the recent literature that if learning can be nontransparent, then Bayesian conditionalization does not universally maximize expected accuracy. This result raises an explanatory challenge for any externalist Bayesian who does not want to give up on a connection between accuracy and epistemic rationality: Why is Bayesian conditionalization the rule of rational credal revision, despite the fact that it does not universally maximize expected accuracy? This article proposes an answer to this challenge. The article argues that an updating rule can be evaluated, not only on the basis of its expected accuracy in particular learning situations, but also on the basis of its total expected accuracy across different learning situations. The article proposes a notion of global evidential constancy, and shows that, out of all globally evidentially constant rules, Bayesian conditionalization has the highest total expected accuracy across all learning situations, including nontransparent ones. Bayes is back!

It is widely accepted that knowledge is necessary for proper assertion. More controversial is the thesis that omega-knowledge – infinitely higher-order knowledge – is necessary for proper assertion. Proponents of the ‘KK’ principle take the former thesis to entail the latter, since they take knowledge to entail omega-knowledge. But in Iterated Knowledge, Simon Goldstein argues against the KK principle and in favor of the thesis that omega-knowledge is necessary for proper assertion. This paper argues that reflection on some independently plausible principles about the relationship between knowledge, assertion, and vagueness shows that both Goldstein and the proponents of KK are mistaken. First: if (as is widely accepted) vagueness is a barrier to knowledge, then one can't omega-know that it's vague what one omega-knows. But if omega-knowledge is plentiful, then it's clearly assertable (and thus knowable) that sometimes it's vague what one omega-knows. So if vagueness is a barrier to knowledge, then omega-knowledge can't be necessary for proper assertion, and knowledge can't entail omega-knowledge. Second: even if vagueness isn't a barrier to knowledge, it's a barrier to proper assertion. But this presents a dilemma for Goldstein: either being in a position to assert something entails being in a position to assert that one is in a position to assert it, or it doesn't. If it does, then one can't assert that it's sometimes vague what one is in a position to assert. But that looks like a problem, because it seems clear that one can assert that sometimes it's vague what one is in a position to assert. But if being in a position to assert something doesn't entail being in a position to assert that one is in a position to assert it, then the principle that assertion requires omega-knowledge fails to do the explanatory work Goldstein needs it to do. Either way, we should reject arguments for omega-knowledge anti-skepticism grounded in judgments about proper assertion.

I argue that normative formal epistemology (NFE) is best understood as modelling, in the sense that this is the reconstruction of its methodology on which NFE is doing best. I focus on Bayesianism and show that it has the characteristics of modelling. But modelling is a scientific enterprise, while NFE is normative. I thus develop an account of normative models on which they are idealised representations put to normative purposes. Normative assumptions, such as the transitivity of comparative credence, are characterised as modelling idealisations motivated normatively. I then survey the landscape of methodological options: what might formal epistemologists be up to? I argue the choice is essentially binary: modelling or theorising. If NFE is theorising it is doing very poorly: generating false claims with no clear methodology for separating out what is to be taken seriously. Modelling, by contrast, is a successful methodology precisely suited to the management of useful falsehoods. Regarding NFE as modelling is not costless, however. First, our normative inferences are less direct and are muddied by the presence of descriptive idealisations. Second, our models are purpose-specific and limited in their scope. I close with suggestions for how to adapt our practice.

This paper introduces a rank-dependent decision theory that allows for, explicitly characterizes, and separates probabilistic risk-aversion and ambiguity-aversion. While these phenomena have previously been given independent treatments by theories that extend expected utility in different ways, we provide a unified treatment that preserves the distinctness of each phenomenon. The unified theory holds that a decision-maker assigns ‘as-if’ probabilities to events: where she holds events to be unambiguous, these probabilities are additive, and where she holds events to be ambiguous, they are non-additive and represent her attitude towards ambiguity. The decision-maker then distorts these as-if probabilities according to her attitude towards probabilistic risk, and uses the new weights to assess acts according to rank-dependent utility. We axiomatize this theory and show that assessments of probability, attitudes towards ambiguity, attitudes towards probabilistic risk, and utilities are all distinct features of preference.

This essay gives an account of epistemic deference for agents with imprecise credences. I look at the two main imprecise deference principles in the literature, known as Identity Reflection and Pointwise Reflection. I show that Pointwise Reflection is strictly weaker than Identity Reflection, and argue that, if you are certain you will update by conditionalization, you should defer to your future self according to Identity Reflection. Then I give a more general justification for Pointwise and Identity Reflection from the assumption that you defer to someone whenever you consider their doxastic state to be better than yours, in the sense of leading to better decision-making.

Naive set theory, as practiced in the nineteenth century by Cantor, Dedekind, and Frege, consists of two basic principles: _extensionality_ and _naive comprehension_. According to the latter, _every_ condition determines a set. This is usually formalized by the schema \(\exists y \forall x\, (x \in y \leftrightarrow A(x))\), where \(A(x)\) is any condition on \(x\) in which \(y\) doesn’t occur free. Some philosophers have argued that, since this restriction is in place to avoid paradox, in a logic weak enough to handle paradoxes, the restriction on \(A(x)\) should be lifted in order to better capture the full generality of naive comprehension. I argue that, on the contrary, lifting the restriction doesn’t do more justice to naive comprehension but is the product of a serious misunderstanding of what the principle really says.

Philosophers of mind tend to hold one of two broad views about mental representation: they are either robustly realist about mental representations, taking them to have determinate, objective content independent of attributors’ explanatory interests and goals, or they embrace some form of anti-realism, holding that mental representations are at best useful fictions. It is becoming increasingly clear that neither view is satisfactory. Realists disagree about the basis for objective content, and computational neuroscientists and psychologists, while making widespread use of representational talk in articulating their theories, seem indifferent to the efforts of these philosophers to articulate a secure foundation for such talk, suggesting a disconnect between the preoccupations of philosophers of cognitive science and actual practice in these sciences. Fictionalists, for their part, must deny that representational mental states can be causes of cognition and behavior, and that appeal to mental representation can be genuinely explanatory, views that flatly contradict assumptions central to both science and everyday life. I develop and defend a distinctive ‘third way’ – a view I call a deflationary account of mental representation – that both resolves philosophical worries about content and best fits actual practice in science and everyday life. According to the deflationary account, appeal to mental representation does indeed purport to pick out causes of behavior, but the attribution of content to these causes is best understood as a pragmatically motivated gloss, justified in part by attributors’ explanatory interests and goals. Content plays an explanatory role in the deflationary account, but one quite different than that assumed by robust representational realists.

When we're inquiring to find out whether p is true, knowing that we'll get better evidence in the future seems like a good reason to suspend judgment about p now. But, as Matt McGrath has recently argued, this natural thought is in deep tension with traditional accounts of justification. On traditional views of justification, which doxastic attitude you are justified in having now depends on your current evidence, not on what you might learn later. McGrath proposes to resolve this tension by distinguishing between different ways of having a neutral attitude. I argue that McGrath's account is unable to account for the full range of cases in which an agnostic attitude is warranted. We can remedy this by pairing his account with my theory of transitional and terminal attitudes, which claims that attitudes are justified in different ways depending on whether they are formed in intermediate stages of deliberation or as conclusions of deliberation. I compare my view with an alternative, more parsimonious one, according to which deliberation itself is a source of new evidence. I argue that this alternative proposal is faced with a dilemma: it either generates a vicious regress, or it fails to capture the relevant cases.

This book is a major contribution to decision theory, focusing on the question of when it is rational to accept scientific theories. The author examines both Bayesian decision theory and confirmation theory, refining and elaborating the views of Ramsey and Savage. He argues that the most solid foundation for confirmation theory is to be found in decision theory, and he provides a decision-theoretic derivation of principles for how many probabilities should be revised over time. Professor Maher defines a notion of accepting a hypothesis, and then shows that it is not reducible to probability and that it is needed to deal with some important questions in the philosophy of science. A Bayesian decision-theoretic account of rational acceptance is provided together with a proof of the foundations for this theory. A final chapter shows how this account can be used to cast light on such vexed issues as verisimilitude and scientific realism.

Knowledge-first evidentialism combines the view that it is rational to believe what is supported by one's evidence with the view that one's evidence is what one knows. While there is much to be said for the view, it is widely perceived to fail in the face of cases of reasonable error—particularly extreme ones like new Evil Demon scenarios (Wedgwood, 2002). One reply has been to say that even in such cases what one knows supports the target rational belief (Lord, 201x, this volume). I spell out two versions of the strategy. The direct one uses what one knows as the input to principles of rationality such as conditionalization, dominance avoidance, etc. I argue that it fails in hybrid cases that are Good with respect to one belief and Bad with respect to another. The indirect strategy uses what one knows to determine a body of supported propositions that is in turn the input to principles of rationality. I sketch a simple formal implementation of the indirect strategy and show that it avoids the difficulty. I conclude that the indirect strategy offers the most promising way for knowledge-first evidentialists to deal with the New Evil Demon problem.

Veritism is the idea that what makes a belief epistemically rational is that it is a fitting response to the value of truth. This idea promises to serve as the foundation for an elegant and systematic treatment of epistemic rationality, one that illuminates the importance of distinctively epistemic normative standards without sacrificing extensional adequacy. But this article proposes that veritism cannot fulfill this promise. It goes on to explain why not, in part by showing that three radically different developments of veritism—one consequentialist, one deontological, and one virtue-theoretic—face eerily similar problems. This article also attempts to provide a general explanation of why any version of veritism is doomed to fail. If these arguments are successful, their upshot is that we must look beyond the value of truth if we want to understand the nature and significance of epistemic rationality.

We use the normality framework of Goodman and Salow (2018, 2021, 2023b) to investigate of dynamics of rational belief. The guiding idea is that people are entitled to believe that their circumstances aren’t especially abnormal. More precisely, a rational agent’s beliefs rule out all and only those possibilities that are either (i) ruled out by their evidence or (ii) sufficiently less normal than some other possibility not ruled out by their evidence. Working within this framework, we argue that the logic of rational belief revision is much weaker than is usually supposed. We do so by isolating a natural family of orthodox principles about belief revision, describing realistic cases in which these principles seem to fail, and showing how these counterexamples are predicted by independently motivated models of the cases in question. In these models, whether one evidential possibility counts as sufficiently less normal than another is determined by underlying probabilities (together with a contextually determined question). We argue that the resulting probabilistic account of belief compares favorably with other such accounts, including Lockeanism (Foley, 1993), a ‘stability’ account inspired by Leitgeb (2017), the ‘tracking theory’ of Lin and Kelly (2012), and the influential precursor of Levi (1967). We show that all of these accounts yield subtly different but similarly heterodox logics of belief revision.

Disagreement is a common feature of a social world. For various reasons, however, we sometimes need to resolve a disagreement into a single set of opinions. This can be achieved by pooling the opinions of individuals that make up the group. This Element provides an opinionated survey on some ways of pooling opinions: linear pooling, multiplicative pooling (including geometric), and pooling through imprecise probabilities. While this Element gives significant attention to the axiomatic approach in evaluating pooling strategies, it also evaluates them in terms of the epistemic and practical goals they might meet. In doing so, this Element connects opinion pooling to some philosophical problems in social epistemology and the philosophy of action, illuminating different perspectives one might take when figuring out how to pool opinions for a given purpose. This title is also available as Open Access on Cambridge Core.

Conditionalization is a widely endorsed rule for updating one’s beliefs. But a sea of complaints have been raised about it, including worries regarding how the rule handles error correction, changing desiderata of theory choice, evidence loss, self-locating beliefs, learning about new theories, and confirmation. In light of such worries, a number of authors have suggested replacing Conditionalization with a different rule — one that appeals to what I’ll call “ur-priors”. But different authors have understood the rule in different ways, and these different understandings solve different problems. In this paper, I aim to map out the terrain regarding these issues. I survey the different problems that might motivate the adoption of such a rule, flesh out the different understandings of the rule that have been proposed, and assess their pros and cons. I conclude by suggesting that one particular batch of proposals, proposals that appeal to what I’ll call “loaded evidential standards”, are especially promising.

In ‘The logic of partial supposition’, Eva and Hartmann present a dilemma for a normative account of partial subjunctive supposition: the natural subjunctive analogue of Jeffrey conditionalization is Jeffrey imaging, but this rule violates a natural monotonicity constraint. This paper offers a partial defence of Jeffrey imaging against Eva and Hartmann’s objection. I show that, although Jeffrey imaging is non-monotonic in Eva and Hartmann’s sense, it is what I call status quo monotonic. A status quo monotonic credal revision rule is monotonic in Eva and Hartmann’s sense if it is conservative in the sense of Meehan and Zhang (‘Jeffrey meets Kolmogorov’), but Jeffrey imaging is in general non-conservative. On the other hand, Jeffrey imaging satisfies a different constraint that I call convexity, and the only rule that is both convex and conservative is Jeffrey conditionalization. To this extent, the real dilemma for a normative account of partial subjunctive supposition is not between monotonicity (broadly construed) and Jeffrey imaging, but between convexity and conservativeness.

No existing normative decision theory adequately handles risk. Expected Utility Theory is overly restrictive in prohibiting a range of reasonable preferences. And theories designed to accommodate such preferences (for example, Buchak's (2013) Risk‐Weighted Expected Utility Theory) violate the Betweenness axiom, which requires that you are indifferent to randomizing over two options between which you are already indifferent. Betweenness has been overlooked by philosophers, and we argue that it is a compelling normative constraint. Furthermore, neither Expected nor Risk‐Weighted Expected Utility Theory allow for stakes‐sensitive risk‐attitudes—they require that risk matters in the same way whether you are gambling for loose change or millions of dollars. We provide a novel normative interpretation of Weighted‐Linear Utility Theory that solves all of these problems.

Replicability is widely taken to ground the epistemic authority of science. However, in recent years, important published findings in the social, behavioral, and biomedical sciences have failed to replicate, suggesting that these fields are facing a “replicability crisis.” For philosophers, the crisis should not be taken as bad news but as an opportunity to do work on several fronts, including conceptual analysis, history and philosophy of science, research ethics, and social epistemology. This article introduces philosophers to these discussions. First, I discuss precedents and evidence for the crisis. Second, I discuss methodological, statistical, and social-structural factors that have contributed to the crisis. Third, I focus on the philosophical issues raised by the crisis. Finally, I discuss proposed solutions and highlight the gaps that philosophers could focus on.

How do multiple reasons combine to support a conclusion about what to do or believe? This question raises two challenges: How can we represent the strength of a reason? How do the strengths of multiple reasons combine? Analogous challenges about confirmation have been answered using probabilistic tools. Can reductive and nonreductive theories of reasons use these tools to answer their challenges? Yes, or more exactly: reductive theories can answer both challenges. Nonreductive theories, with the help of a result in confirmation theory, can answer one, and there are grounds for optimism that they can answer the other.