Kenny Easwaran

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.

Buchak’s risk-weighted expected utility considers not just the probability of an outcome, but also the probability of getting a strictly better outcome, when weighting the contribution that outcome gives to the evaluation of a gamble. It uses a risk-weighting function $R$ sending probabilities in $\left[ {0,1} \right]$ to decision weights $\left[ {0,1} \right]$. I adapt this to allow weights in any real interval. Finite intervals yield nothing new, but if the interval is infinite, then the resulting rule can incorporate maximin or maximax preferences (or both!) while still satisfying stochastic dominance. There are advantages to working with marginal risk-weighting, $R$ ’s derivative, $r$.

Greaves, H., & Wallace (Mind, 115(459), 607–632 2006) justify Bayesian conditionalization as the update plan that maximizes expected accuracy, for an agent considering finitely many possibilities, who is about to undergo a learning event where the potential propositions that she might learn form a partition. In recent years, several philosophers have generalized this argument to less idealized circumstances. Some authors (Easwaran, Thought: A Journal of Philosophy, 2(1), 53–61 2013; Nielsen, Statistics & Probability Letters, 185, 109412 2022) relax finiteness, while others (Carr, Synthese, 198(4), 3581–3601 2021; Gallow, Noûs, 55(3), 487–516 2021; Isaacs & Russell, 2022; Schultheis, 2023) relax partitionality. In this paper, we show how to do both at once. We give novel philosophical justifications of the use of $$\sigma $$ σ -algebras in the infinite setting, and argue for a different interpretation of the “signals” in the non-partitional setting. We show that the resulting update plan mitigates some problems that arise when only relaxing finiteness, but not partitionality, such as the Borel-Kolmogorov paradox.

Seidenfeld, Kadane, Schervish, and Stern (henceforth SKSS), in “Finite Additivity, Countable Additivity, and the Comparative Principle” (2024) raise some important mathematical issues around the principles I developed in “Why Countable Additivity” (2013) (and one developed by Stewart and Nielsen (2021), another response to my paper). Some of SKSS’s results are phrased in dense mathematical language. In this note, I try to explain the important aspects of their results in ways that are more understandable, to help clarify their dialectical force. I show why I am dialectically unmoved by many of the points that they make. I still accept merely countably-additive probability as the best theory of probability for epistemology. However, I take their objection to my attempted extension of this argument to a comparative (rather than numerical) framework as a very significant one.

At first pass, Evidential Decision Theory (EDT) recommends one-boxing in Newcomb’s Problem and Causal Decision Theory (CDT) recommends two-boxing. However, it has been acknowledged that concrete instances of the problem have messy features complicating their analyses. Recently, a third competitor, Functional Decision Theory (FDT) has emerged recommending one-boxing in some versions and two-boxing in others. This paper explores the verdicts of these competing theories in a few variations of the problem involving iteration and habit. We argue that this motivates the re-evaluation of case intuitions in decision theory and greater consideration of formal and structural relationships between CDT and FDT.

Ted Chiang’s story, “Division by Zero”, contrasts the kind of knowledge of mathematics one has through understanding and insight with the kind of knowledge of mathematics one has through mere symbol manipulation. This chapter aims to give the reader the experience of this contrast, and show how it makes sense of certain aspects of the history of mathematics. There was once a thought that mathematics could be protected from inconsistency by making it entirely about symbol manipulation. Chiang’s story can help us see that this project would have made math just as meaningless as inconsistency would.

Many axiological theories ground the goodness or badness of options in the aggregate of the goodness or the badness of these options for individuals. Most commonly, this works by summing (or averaging), and taking the expectation of this result if there is uncertainty. Such theories face problems dealing with infinite populations, for which sums or averages are infinite or undefined. They fetishize certain mathematical operations, in a subject that is not inherently mathematical. The fact that the sum is the target of maximization is said to mean that the theory ignores the separateness of persons. I propose an extension of my 2014 paper, ‘Decision Theory without Representation Theorems’. I illustrate how the resulting theory can account for a case involving an infinite population, dealing with the objections. I connect this to the theory of measurement, and explain why the mathematical operations are co-extensive with the result without grounding it. Because there is no sum in these infinite populations, it avoids the worry that aggregative theories treat individuals as secondary to some aggregate.

Bayesians standardly claim that there is rational pressure for agents’ credences to cohere across time because they face bad (epistemic or practical) consequences if they fail to diachronically cohere. But as David Christensen has pointed out, groups of individual agents also face bad consequences if they fail to interpersonally cohere, and there is no general rational pressure for one agent's credences to cohere with another’s. So it seems that standard Bayesian arguments may prove too much. Here, we agree with Christensen that there is no general rational pressure to diachronically cohere, but we argue that there are particular cases in which there is rational pressure to diachronically cohere, as well as particular cases in which interpersonal probabilistic coherence is rationally required. More generally, we suggest that Bayesian arguments for coherence apply whenever a collection (of agents or time slices) has a shared dimension of value and an ability to coordinate their actions in a range of cases relevant to that value. Typically, this shared value and ability to coordinate is very strong across the time slices of one human being, and very weak across different human beings, but there are special cases where these can switch—i.e., some groups of humans will have as much reason for their beliefs to cohere across a particular range of cases as the time slices of one human usually do, but some time slices of a human will have as much freedom to differ in their beliefs from the others as the members of a group usually do.

Pascal’s Wager holds that one has pragmatic reason to believe in God, since that course of action has infinite expected utility. The mixed strategy objection holds that one could just as well follow a course of action that has infinite expected utility but is unlikely to end with one believing in God. Monton (2011. Mixed strategies can’t evade Pascal’s Wager. Analysis 71: 642–45.) has argued that mixed strategies can’t evade Pascal’s Wager, while Robertson (2012. Some mixed strategies can evade Pascal’s Wager: a reply to Monton. Analysis 72: 295–98.) has argued that Monton is mistaken. We show that Monton is correct, highlight the crucial assumptions that he relies on, and shed some light on the role of mixed strategies in decision theory

Many philosophers have become worried about the use of standard real numbers for the probability function that represents an agent's credences. They point out that real numbers can't capture the distinction between certain extremely unlikely events and genuinely impossible ones—they are both represented by credence 0, which violates a principle known as “regularity.” Following Skyrms 1980 and Lewis 1980, they recommend that we should instead use a much richer set of numbers, called the “hyperreals.” This essay argues that this popular view is the result of two mistakes. The first mistake, which this essay calls the “numerical fallacy,” is to assume that a distinction that isn't represented by different numbers isn't represented at all in a mathematical representation. In this case, the essay claims that although the real numbers do not make all relevant distinctions, the full mathematical structure of a probability function does. The second mistake is that the hyperreals make too many distinctions. They have a much more complex structure than credences in ordinary propositions can have, so they make distinctions that don't exist among credences. While they might be useful for generating certain mathematical models, they will not appear in a faithful mathematical representation of credences of ordinary propositions

Many philosophers have argued that "degree of belief" or "credence" is a more fundamental state grounding belief. Many other philosophers have been skeptical about the notion of "degree of belief", and take belief to be the only meaningful notion in the vicinity. This paper shows that one can take belief to be fundamental, and ground a notion of "degree of belief" in the patterns of belief, assuming that an agent has a collection of beliefs that isn't dominated by some other collection in terms of the overall balance of truth and falsity that it could contain.

In my () I argued that a central component of mathematical practice is that published proofs must be “transferable” — that is, they must be such that the author's reasons for believing the conclusion are shared directly with the reader, rather than requiring the reader to essentially rely on testimony. The goal of this paper is to explain this requirement of transferability in terms of a more general norm on defeat in mathematical reasoning that I will call “convertibility”. I begin by discussing two types of epistemic defeat: “rebutting” and “undercutting”. I give examples of both of these kinds of defeat from the history of mathematics. I then argue that an important requirement in mathematics is that published proofs be detailed enough to allow the conversion of rebutting defeat into undercutting defeat. Finally, I show how this sort of convertibility explains the requirement of transferability, and contributes to the way mathematics develops by the pattern referred to by Lakatos () as “lemma incorporation”.

Belief and credence are often characterized in three different ways—they ought to govern our actions, they ought to be governed by our evidence, and they ought to aim at the truth. If one of these roles is to be central, we need to explain why the others should be features of the same mental state rather than separate ones. If multiple roles are equally central, then this may cause problems for some traditional arguments about what belief and credence must be like. I read the history of formal and traditional epistemology through the lens of these functional roles, and suggest that considerations from one literature might have a role in the other. The similarities and differences between these literatures may suggest some more general ideas about the nature of epistemology in abstraction from the details of credence and belief in particular.

Naive versions of decision theory take probabilities and utilities as primitive and use expected value to give norms on rational decision. However, standard decision theory takes rational preference as primitive and uses it to construct probability and utility. This paper shows how to justify a version of the naive theory, by taking dominance as the most basic normatively required preference relation, and then extending it by various conditions under which agents should be indifferent between acts. The resulting theory can make all the decisions of classical expected utility theory, plus more in cases where expected utilities are infinite or undefined. Although the theory requires similarly strong assumptions to classical expected utility theory, versions of the theory can be developed with slightly weaker assumptions, without having to prove a new representation theorem for the weaker theory. This alternate foundation is particularly useful if probability is prior to preference, as suggested by the recent program to base probabilism on accuracy and alethic considerations rather than pragmatic ones

Conditional probability is one of the central concepts in probability theory. Some notion of conditional probability is part of every interpretation of probability. The basic mathematical fact about conditional probability is that p(A |B) = p(A ∧B)/p(B) where this is defined. However, while it has been typical to take this as a definition or analysis of conditional probability, some (perhaps most prominently Hájek, 2003) have argued that conditional probability should instead be taken as the primitive notion, so that this formula is at best coextensive, and at worst sometimes gets it wrong. Section 1.1 considers the concept of conditional probability in each of the major families of interpretation of probability. Section 1.2 considers a conceptual argument for the claim that conditional probability is prior to unconditional probability, while Section 1.3 considers a family of mathematical arguments for this claim, leading to consideration specifically of the question of how to understand probability conditional on events of probability 0. Section 1.4 discusses several mathematical principles that have been alleged to be important for understanding how probability 0 behaves, and raises a dilemma for probability conditional on events of probability 0. Section 2 and Section 3 take the two horns of this dilemma and describe the two main competing families of mathematical accounts of conditional probability for events of probability 0. Section 4 summarizes the results, and their significance for the two arguments that conditional probability is prior to unconditional probability.

I defend a causal reductionist account of the nature of rates of change like velocity and acceleration. This account identifies velocity with the past derivative of position and acceleration with the future derivative of velocity. Unlike most reductionist accounts, it can preserve the role of velocity as a cause of future positions and acceleration as the effect of current forces. I show that this is possible only if all the fundamental laws are expressed by differential equations of the same order. Consideration of the continuity of time explains why the differential equations are all second order. This explanation is not available on non-causal or non-reductionist accounts of rates of change. Finally, I argue that alleged counterexamples to the reductionist account involving physically impossible worlds are irrelevant to an analysis of the properties that play a causal role in the actual world. 1 Background2 Grounding3 Causation4 The Proposal5 Why No Third Derivatives?6 Why Any Derivatives?7 Counterexamples?

Fine has shown that assigning any value to the Pasadena game is consistent with a certain standard set of axioms for decision theory. However, I suggest that it might be reasonable to believe that the value of an individual game is constrained by the long-run payout of repeated plays of the game. Although there is no value that repeated plays of the Pasadena game converges to in the standard strong sense, I show that there is a weaker sort of convergence it exhibits, and use this to define a notion of ‘weak expectation’ that can give values to the Pasadena game and many others, though not to all games that fail to have a strong expectation in the standard sense. CiteULike    Connotea    Del.icio.us    What's this?

In a series of papers, Don Fallis points out that although mathematicians are generally unwilling to accept merely probabilistic proofs, they do accept proofs that are incomplete, long and complicated, or partly carried out by computers. He argues that there are no epistemic grounds on which probabilistic proofs can be rejected while these other proofs are accepted. I defend the practice by presenting a property I call ‘transferability’, which probabilistic proofs lack and acceptable proofs have. I also consider what this says about the similarities between mathematics and, on the one hand natural sciences, and on the other hand philosophy

To answer the question of whether mathematics needs new axioms, it seems necessary to say what role axioms actually play in mathematics. A first guess is that they are inherently obvious statements that are used to guarantee the truth of theorems proved from them. However, this may neither be possible nor necessary, and it doesn’t seem to fit the historical facts. Instead, I argue that the role of axioms is to systematize uncontroversial facts that mathematicians can accept from a wide variety of philosophical positions. Once the axioms are generally accepted, mathematicians can expend their energies on proving theorems instead of arguing philosophy. Given this account of the role of axioms, I give four criteria that axioms must meet in order to be accepted. Penelope Maddy has proposed a similar view in Naturalism in Mathematics, but she suggests that the philosophical questions bracketed by adopting the axioms can in fact be ignored forever. I contend that these philosophical arguments are in fact important, and should ideally be resolved at some point, but I concede that their resolution is unlikely to affect the ordinary practice of mathematics. However, they may have effects in the margins of mathematics, including with regards to the controversial “large cardinal axioms” Maddy would like to support.

There is general agreement in mathematics about what continuity is. In this paper we examine how well the mathematical definition lines up with common sense notions. We use a recent paper by Hud Hudson as a point of departure. Hudson argues that two objects moving continuously can coincide for all but the last moment of their histories and yet be separated in space at the end of this last moment. It turns out that Hudson’s construction does not deliver mathematically continuous motion, but the natural question then is whether there is any merit in the alternative definition of continuity that he implicitly invokes.

To the extent that we have reasons to avoid these “bad B -properties”, these arguments provide reasons not to have an incoherent credence function b — and perhaps even reasons to have a coherent one. But, note that these two traditional arguments for probabilism involve what might be called “pragmatic” reasons (not) to be (in)coherent. In the case of the Dutch Book argument, the “bad” property is pragmatically bad (to the extent that one values money). But, it is not clear whether the DBA pinpoints any epistemic defect of incoherent agents. The same can be said for Representation Theorem arguments, since they involve the structure of an agent’s preferences