By “physical probability” I mean the empirical concept of probability in ordinary language. It can be represented as a function of an experiment type and an outcome type, which explains how non-extreme physical probabilities are compatible with determinism. Two principles, called specification and independence, put restrictions on the existence of physical probabilities, while a principle of direct inference connects physical probability with inductive probability. This account avoids a variety of weaknesses in the theories of Levi and Lewis.

In October 2009 I decided to stop doing philosophy. This meant, in particular, stopping work on the book that I was writing on the nature of probability. At that time, I had no intention of making my unfinished draft available to others. However, I recently noticed how many people are reading the lecture notes and articles on my web site. Since this draft book contains some important improvements on those materials, I decided to make it available to anyone who wants (...) to read it. That is what you have in front of you. The account of Laplace’s theory of probability in Chapter 4 is very different to what I said in my seminar lectures, and also very different to any other account I have seen; it is based on a reading of important texts by Laplace that appear not to have been read by other commentators. The discussion of von Mises’ theory in Chapter 7 is also new, though perhaps less revolutionary. And the final chapter is a new attempt to come to grips with the popular, but amorphous, subjective theory of probability. The material in the other chapters has mostly appeared in previous articles of mine but things are sometimes expressed differently here. I would like to say again that this is an incomplete draft of a book, not the book I would have written if I had decided to finish it. It no doubt contains poor expressions, it may contain some errors or inconsistencies, and it doesn’t cover all the theories that I originally intended to discuss. Apart from this preface, I have done no work on the book since October 2009. (shrink)

Bayesian Epistemology is a general framework for thinking about agents who have beliefs that come in degrees. Theories in this framework give accounts of rational belief and rational belief change, which share two key features: (i) rational belief states are represented with probability functions, and (ii) rational belief change results from the acquisition of evidence. This dissertation focuses specifically on the second feature. I pose the Evidence Question: What is it to have evidence? Before addressing this question we must have (...) an understanding of Bayesian Epistemology. The first chapter argues that we should understand Bayesian Epistemology as giving us theories that are evaluative and not action-guiding. I reach this verdict after considering the popular ‘ought’-implies-‘can’ objection to Bayesian Epistemology. The second chapter argues that it is important for theories in Bayesian Epistemology to answer the Evidence Question, and distinguishes between internalist and externalist answers. The third and fourth chapters present and defend a specific answer to the Evidence Question. The account is inspired by reliabilist accounts of justification, and attempts to understand what it is to have evidence by appealing solely to considerations of reliability. Chapter 3 explains how to understand reliability, and how the account fits with Bayesian Epistemology, in particular, the requirement that an agent’s evidence receive probability 1. Chapter 4 responds to objections, which maintain that the account gives the wrong verdict in a variety of situations including skeptical scenarios, lottery cases, scientific cases, and cases involving inference. After slight modifications, I argue that my account has the resources to answer the objections. The fifth chapter considers the possibility of losing evidence. I show how my account can model these cases. To do so, however, we require a modification to Conditionalization, the orthodox principle governing belief change. I present such a modification. The sixth and seventh chapters propose a new understanding of Dutch Book Arguments, historically important arguments for Bayesian principles. The proposal shows that the Dutch Book Arguments for implausible principles are defective, while the ones for plausible principles are not. The final chapter is a conclusion. (shrink)

In response to two trenchant objections, radical subjective Bayesianism has been widely rejected. In this paper, I seek, if not to rehabilitate subjectivism, at least to show its critic what is attractive about the position. I argue that what is at stake in the subjectivism/anti-subjectivism debate is not, as is commonly thought, which norms of rationality are true, but rather, the conception of rationality that we adopt: there is an alternative approach to the widespread telic approach to rationality, which I (...) call the poric approach, on which subjectivism is an attractive position. I then argue that the poric approach—and therefore subjectivism—cannot be easily dismissed. (shrink)

We introduce a family of rules for adjusting one's credences in response to learning the credences of others. These rules have a number of desirable features. 1. They yield the posterior credences that would result from updating by standard Bayesian conditionalization on one's peers' reported credences if one's likelihood function takes a particular simple form. 2. In the simplest form, they are symmetric among the agents in the group. 3. They map neatly onto the familiar Condorcet voting results. 4. They (...) preserve shared agreement about independence in a wide range of cases. 5. They commute with conditionalization and with multiple peer updates. Importantly, these rules have a surprising property that we call synergy - peer testimony of credences can provide mutually supporting evidence raising an individual's credence higher than any peer's initial prior report. At first, this may seem to be a strike against them. We argue, however, that synergy is actually a desirable feature and the failure of other updating rules to yield synergy is a strike against them. (shrink)

How should probabilities be interpreted in causal models in the social and health sciences? In this paper we take a step towards answering this question by investigating the case of cancer in epidemiology and arguing that the objective Bayesian interpretation is most appropriate in this domain.

Handbook of the History of Logic, vol. 10, eds. Dov Gabbay, Stephan Hartmann, and John Woods, forthcoming.

In the contemporary philosophical debate about probability, one of the main problems con‐ cerns the relation between objective probability and determinism. Is it possible for objective probability and determinism to co‐exist? this is one of the questions this dispute tries to answer. the scope of discussion is conducted between advocates of a positive answer (com‐ patibilist) and co‐existence opponents (incompatibilist). In the early twentieth century, many logicians also developed topics regarding probability and determinism. One of them was the outstanding Polish (...) logician and philosopher — Jan Łukasiewicz. the general purpose of this paper is to analyse and implement Łukasiewicz’s views regarding determinism and probability in the contemporary eld of this problem. I will try to show the relation between his interpre‐ tations of these concepts and in consequence his attempt to con ont them. As a result of the above analysis, I present some di erent positions (located in the elds of logic and semantics) in the contemporary discourse about the relation between objective probability and determi‐ nism. moreover, I will present Łukasiewicz’s views about this relation and the consequence of these solutions in the eld of logic. (shrink)

I respond to some challenges to Humean laws deriving from the claimed non-resilience of such laws under counterfactual assumptions.

Proponents of Bayesian confirmation theory believe that they have the solution to a significant, recalcitrant problem in philosophy of science. It is the identification of the logic that governs evidence and its inductive bearing in science. That is the logic that lets us say that our catalog of planetary observations strongly confirms Copernicus’ heliocentric hypothesis; or that the fossil record is good evidence for the theory of evolution; or that the 3oK cosmic background radiation supports big bang cosmology. The definitive (...) solution to this problem would be a significant achievement. The problem is of central importance to philosophy of science, for, in the end, what distinguishes science from myth making is that we have good evidence for the content of science, or at least of mature sciences, whereas myths are evidentially ungrounded fictions. The core ideas shared by all versions of Bayesian confirmation theory are, at a good first approximation, that a scientist’s beliefs are or should conform to a probability measure; and that the incorporation of new evidence is through conditionalization using Bayes’ theorem. While the burden of this chapter will be to inventory why critics believe this theory may not be the solution after all, it is worthwhile first to summarize here the most appealing virtues of this simple account. There are three. First, the theory reduces the often nebulous notion of a logic of.. (shrink)

Reprinted in Philosophy of Religion: An Anthology, Wadsworth, 2013, 6th edition, eds. Michael Rea and Louis Pojman. In this essay, I argue that the moral skepticism objection to what is badly named "skeptical theism" fails.

David Lewis' "Principal Principle" is a purported principle of rationality connecting credence and objective chance. Almost all of the discussion of the Principal Principle in the philosophical literature assumes classical probability theory, which is unfortunate since the theory of modern physics that, arguably, speaks most clearly of objective chance is the quantum theory, and quantum probabilities are not classical probabilities. Given the generally accepted updating rule for quantum probabilities, there is a straight forward sense in which the Principal Principle is (...) a theorem of quantum probability theory for any credence function satisfying a suitable additivity requirement. No additional principle of rationality is needed to bring credence into line with objective chance. (shrink)

In this paper, a concept of chance is introduced that is compatible with deterministic physical laws, yet does justice to our use of chance-talk in connection with typical games of chance. We take our cue from what Poincaré called "the method of arbitrary functions," and elaborate upon a suggestion made by Savage in connection with this. Comparison is made between this notion of chance, and David Lewis' conception.

Epistemic decision theory aims to defend a variety of epistemic norms in terms of their facilitation of epistemic ends. One of the most important components of EpDT is known as a scoring rule. This thesis addresses some problems about scoring rules in EpDT. I consider scoring rules both for precise credences and for imprecise credences. For scoring rules in the context of precise credences, I examine the rationale for requiring a scoring rule to be strictly proper, and argue that no (...) satisfactory justification has been given. I then investigate one possible response to my argument and show the problems with this response. The conclusion is that there is a further need for justifying the requirement that a scoring rule should be strictly proper for precise credences. For scoring rules in the context of imprecise credences, an impossibility result has been established in the literature purporting to show that no strictly proper, continuous and real-valued scoring rule exists. However, a precise statement of the impossibility result requires precise definitions both of strict propriety and of continuity in the context of imprecise credences. Moreover, the result implies that we need to drop one of the three properties - strict propriety, continuity or being real-valued - to have a scoring rule for applying EpDT to imprecise credences. So, firstly, I offer definitions of strict propriety and of continuity and clarify the impossibility result. Then, I investigate what will happen if we drop one of the three properties. I argue that we should drop the property of being real-valued and I offer the general forms of two kinds of strictly proper, continuous and lexicographic scoring rules for imprecise credences. (shrink)

State of art paper on the topic laws of nature, around the problem of identification what is to be a law of nature. The most prominent theories of contemporary philosophical literature are discussed and analysed, such as: the simple regularity theory, from Hume; the Mill-Ramsey-Lewis best systems theory; the Dretske-Tooley-Armstrong theory of laws as relations among universals; Ellis’s essentialist theory; Cartwright’s theory of laws as ceteris paribus laws; the anti-reductionist theories of Lange, Maudlin and Carroll, the anti-realist theories of Mumford, (...) van Fraassen and Giere; etc. (shrink)

An extended analysis is given of the program, originally suggested by Deutsch, of solving the probability problem in the Everett interpretation by means of decision theory. Deutsch's own proof is discussed, and alternatives are presented which are based upon different decision theories and upon Gleason's Theorem. It is argued that decision theory gives Everettians most or all of what they need from `probability'. Contact is made with Lewis's Principal Principle linking subjective credence with objective chance: an Everettian Principal Principle is (...) formulated, and shown to be at least as defensible as the usual Principle. Some consequences of (Everettian) quantum mechanics for decision theory itself are also discussed. -/- [NB: this this long (70 pages) and occasionally rambling online-only paper has been almost entirely superseded by material in subsequent; if something is not included in them it usually means that I have had second thoughts. I include it for completeness only. (shrink)

It is well known that de se (or ‘self-locating’) propositions complicate the standard picture of how we should respond to evidence. This has given rise to a substantial literature centered around puzzles like Sleeping Beauty, Dr. Evil, and Doomsday—and it has also sparked controversy over a style of argument that has recently been adopted by theoretical cosmologists. These discussions often dwell on intuitions about a single kind of case, but it’s worth seeking a rule that can unify our treatment of (...) all evidence, whether de dicto or de se. -/- This paper is about three candidates for such a rule, presented as replacements for the standard updating rule. Each rule stems from the idea that we should treat ourselves as a random sample, a heuristic that underlies many of the intuitions that have been pumped in treatments of the standard puzzles. But each rule also yields some strange results when applied across the board. This leaves us with some difficult options. We can seek another way to refine the random-sample heuristic, e.g. by restricting one of our rules. We can try to live with the strange results, perhaps granting that useful principles can fail at the margins. Or we can reject the random-sample heuristic as fatally flawed—which means rethinking its influence in even the simplest cases. (shrink)

Bayesian confirmation theory—abbreviated to in these notes—is the predominant approach to confirmation in late twentieth century philosophy of science. It has many critics, but no rival theory can claim anything like the same following. The popularity of the Bayesian approach is due to its flexibility, its apparently effortless handling of various technical problems, the existence of various a priori arguments for its validity, and its injection of subjective and contextual elements into the process of confirmation in just the places where (...) critics of earlier approaches had come to think that subjectivity and sensitivity to context were necessary. (shrink)

Some philosophers have claimed that it is meaningless or paradoxical to consider the probability of a probability. Others have however argued that second-order probabilities do not pose any particular problem. We side with the latter group. On condition that the relevant distinctions are taken into account, second-order probabilities can be shown to be perfectly consistent. May the same be said of an infinite hierarchy of higher-order probabilities? Is it consistent to speak of a probability of a probability, and of a (...) probability of a probability of a probability, and so on, {\em ad infinitum}? We argue that it is, for it can be shown that there exists an infinite system of probabilities that has a model. In particular, we define a regress of higher-order probabilities that leads to a convergent series which determines an infinite-order probability value. We demonstrate the consistency of the regress by constructing a model based on coin-making machines. (shrink)

In discussions of the foundations of statistical mechanics, it is widely held that the Gibbsian and Boltzmannian approaches are incompatible but empirically equivalent; the Gibbsian approach may be calculationally preferable but only the Boltzmannian approach is conceptually satisfactory. I argue against both assumptions. Gibbsian statistical mechanics is applicable to a wide variety of problems and systems, such as the calculation of transport coefficients and the statistical mechanics and thermodynamics of mesoscopic systems, in which the Boltzmannian approach is inapplicable. And the (...) supposed conceptual problems with the Gibbsian approach are either misconceived, or apply only to certain versions of the Gibbsian approach, or apply with equal force to both approaches. I conclude that Boltzmannian statistical mechanics is best seen as a special case of, and not an alternatve to, Gibbsian statistical mechanics. (shrink)

This is a preliminary version of an article to appear in the forthcoming Ashgate Companion to the New Philosophy of Physics.In it, I aim to review, in a way accessible to foundationally interested physicists as well as physics-informed philosophers, just where we have got to in the quest for a solution to the measurement problem. I don't advocate any particular approach to the measurement problem (not here, at any rate!) but I do focus on the importance of decoherence theory to (...) modern attempts to solve the measurement problem, and I am fairly sharply critical of some aspects of the "traditional" formulation of the problem. (shrink)

The Everett interpretation of quantum mechanics - better known as the Many-Worlds Theory - has had a rather uneven reception. Mainstream philosophers have scarcely heard of it, save as science fiction. In philosophy of physics it is well known but has historically been fairly widely rejected. Among physicists, it is taken very seriously indeed, arguably tied for first place in popularity with more traditional operationalist views of quantum mechanics. In this article, I provide a fairly short and self-contained introduction to (...) the Everett interpretation as it is currently understood. I use little technical machinery, although I do assume the reader has encountered the measurement problem already. (shrink)

I develop the decision-theoretic approach to quantum probability, originally proposed by David Deutsch, into a mathematically rigorous proof of the Born rule in (Everett-interpreted) quantum mechanics. I sketch the argument informally, then prove it formally, and lastly consider a number of proposed ``counter-examples'' to show exactly which premises of the argument they violate. (This is a preliminary version of a chapter to appear --- under the title ``How to prove the Born Rule'' --- in Saunders, Barrett, Kent and Wallace, "Many (...) worlds? Everett, quantum theory and reality", forthcoming from Oxford University Press.). (shrink)

En el presente trabajo se realiza una exposición breve del punto de vista filosófico de la teoría de la decisión, profundizando específicamente en el trabajo que propone David Lewis al respecto. Este autor sustenta que la teoría de la decisión puede sostenerse sobre bases causales o no causales, siendo la Teoría Causal de la Decisión un modelo necesario para el análisis de este tipo de problemas. En casos particulares como en la Paradoja de Newcomb, las teorías no causales de la (...) decisión serían insuficientes. A continuación se presentarán brevemente las nociones básicas del análisis de decisiones, para en segundo lugar explicar en detalle algunos aspectos de la Paradoja deNewcomb. Finalmente se presentan las tesis principales de la teoría de Lewis y algunas posibles objeciones de la misma. -/- In this work a brief account of the philosophical point of view on decision theory is shown by studying David Lewis’s work in particular. This author states that a theory of decision can have either causal or non causal grounds and that causal decision theory is a necessary model for the analysis of this kind ofproblems. In particular cases like Newcomb’s Paradox, non causal decision theories would be insufficient. I will show some basic notions of decision analysis, in order to explain in detail some aspects of Newcomb’s Paradox. Finally, the fundamental theses of Lewis’s Theory and some possible objections are presented. (shrink)

In what sense are the special sciences autonomous of fundamental physics? Autonomy is an enduring theme in discussions of the relationship between the special sciences and fundamental physics or, more generally, between higher and lower-level facts. Discussion of ‘autonomy’ often fails to recognise that autonomy admits of degrees; consequently, autonomy is either taken to require full independence, or risk relegation to mere apparent autonomy. In addition, the definition of autonomy used by Fodor, the most famous proponent of the autonomy of (...) the special sciences, has been robustly criticised by Loewer. In this paper I develop a new account of autonomy following Woodward which I dub ‘generalised autonomy’ since it unifies dynamical, causal and nomic autonomy. Autonomy, on this account, can be partial: some lower-level details matter while others do not. To summarise: whilst the detailed lower level is unconditionally relevant, conditionalising on the higher-level facts renders some lower-level details irrelevant. The macrodependencies that the higher-level facts enter into — be they dynamical, causal or nomic — screen off the underlying microdetails. This account helps resolve an explanatory puzzle: if the lower-level facts in some way underpin the higher-level facts, why don’t the lower-level details matter more for the day-to-day practice of the special sciences? The answer will be: the facts uncovered by the special sciences are autonomous in my sense, and so practitioners of these special sciences need not study more fundamental sciences, since these underlying facts are genuinely irrelevant. (shrink)

Everyday life reasoning and argumentation is defeasible and uncertain. I present a probability logic framework to rationally reconstruct everyday life reasoning and argumentation. Coherence in the sense of de Finetti is used as the basic rationality norm. I discuss two basic classes of approaches to construct measures of argument strength. The first class imposes a probabilistic relation between the premises and the conclusion. The second class imposes a deductive relation. I argue for the second class, as the first class is (...) problematic if the arguments involve conditionals. I present a measure of argument strength that allows for dealing explicitly with uncertain conditionals in the premise set. (shrink)

This chapter presents a new semantics for inductive empirical knowledge. The epistemic agent is represented concretely as a learner who processes new inputs through time and who forms new beliefs from those inputs by means of a concrete, computable learning program. The agent’s belief state is represented hyper-intensionally as a set of time-indexed sentences. Knowledge is interpreted as avoidance of error in the limit and as having converged to true belief from the present time onward. Familiar topics are re-examined within (...) the semantics, such as inductive skepticism, the logic of discovery, Duhem’s problem, the articulation of theories by auxiliary hypotheses, the role of serendipity in scientific knowledge, Fitch’s paradox, deductive closure of knowability, whether one can know inductively that one knows inductively, whether one can know inductively that one does not know inductively, and whether expert instruction can spread common inductive knowledge—as opposed to mere, true belief—through a community of gullible pupils. (shrink)

I offer an explicit account of the underdetermination thesis as well as of the many challenges it poses to scientific realism; a way to answer to these challenges is explored and outlined, by shifting attention to the content of theories. I argue that, even if we have solid grounds (as I contend we do) to support that some varieties of the underdetermination thesis are true, scientific realism can still offer an adequate picture of the aims and achievements of science.

This paper begins with a puzzle regarding Lewis' theory of radical interpretation. On the one hand, Lewis convincingly argued that the facts about an agent's sensory evidence and choices will always underdetermine the facts about her beliefs and desires. On the other hand, we have several representation theorems—such as those of (Ramsey 1931) and (Savage 1954)—that are widely taken to show that if an agent's choices satisfy certain constraints, then those choices can suffice to determine her beliefs and desires. In (...) this paper, I will argue that Lewis' conclusion is correct: choices radically underdetermine beliefs and desires, and representation theorems provide us with no good reasons to think otherwise. Any tension with those theorems is merely apparent, and relates ultimately to the difference between how 'choices' are understood within Lewis' theory and the problematic way that they're represented in the context of the representation theorems. For the purposes of radical interpretation, representation theorems like Ramsey's and Savage's just aren't very relevant after all. (shrink)

Taking Savage's subjective expected utility theory as a starting point, this thesis distinguishes three types of uncertainty which are incompatible with Savage's theory for small worlds: ambiguity, option uncertainty and state space uncertainty. Under ambiguity agents cannot form a unique and additive probability function over the state space. Option uncertainty exists when agents cannot assign unique consequences to every state. Finally, state space uncertainty arises when the state space the agent constructs is not exhaustive, such that unforeseen contingencies can occur. (...) Chapter 2 explains Savage's notions of small and large worlds, and shows that ambiguity, option and state space uncertainty are incompatible with the small world representation. The chapter examines whether it is possible to reduce these types of uncertainty to one another. Chapter 3 suggests a definition of objective ambiguity by extending Savage's framework to include an exogenous likelihood ranking over events. The definition allows for a precise distinction between ambiguity and ambiguity attitude. The chapter argues that under objective ambiguity, ambiguity aversion is normatively permissible. Chapter 4 gives a model of option uncertainty. Using the two weak assumptions that the status quo is not uncertain, and that agents are option uncertainty averse, we derive status quo bias, the empirical tendency for agents to choose the status quo over other available alternatives. The model can be seen as rationalising status quo bias. Chapter 5 gives an axiomatic characterisation and corresponding representation theorem for the priority heuristic, a heuristic which predicts binary decisions be- tween lotteries particularly well. The chapter analyses the normative implications of this descriptive model. Chapter 6 defends the pluralist view of decision theory this thesis assumes. The chapter discusses possible applications of the types of uncertainty defined in the thesis, and concludes. (shrink)

The paper uses the concept of typicality to spell out an argument against Humean supervenience and the best system account of laws. It proves that, in a very general and robust sense, almost all possible Humean worlds have no Humean laws. They are worlds of irreducible complexity that do not allow for any systematization. After explaining typicality reasoning in general, the implications of this result for the metaphysics of laws are discussed in detail.

The two main features of this thesis are (i) an account of contextualized (context indexed) counterfactuals, and (ii) a non-vacuist account of counterpossibles. Experience tells us that the truth of the counterfactual is contingent on what is meant by the antecedent, which in turn rests on what context is assumed to underlie its reading (intended meaning). On most conditional analyses, only the world of evaluation and the antecedent determine which worlds are relevant to determining the truth of a conditional, and (...) consequently what its truth value is. But that results in the underlying context being fixed, when evaluating distinct counterfactuals with the same antecedent on any single occasion, even when the context underlying the evaluation of each counterfactual may vary. Alternative approaches go some of the way toward resolving this inadequacy by appealing to a difference in the consequents associated with counterfactuals with the same antecedent. That is, in addition to the world of evaluation and the antecedent, the consequent contributes to the counterfactual’s evaluation. But these alternative approaches nevertheless give a single, determinate truth value to any single conditional (same antecedent and consequent), despite the possibility that this value may vary with context. My reply to these shortcomings (chapter 4) takes the form of an analysis of a language that makes appropriate explicit access to the intended context available. That is, I give an account of a contextualized counterfactual of the form ‘In context C: If it were the case that … , then it would be the case that …’. Although my proposal is largely based on Lewis’ (1973, 1981) analyses of counterfactuals (the logic VW and its ordering semantics), it does not require that any particular logic of counterfactuals should serve as its basis – rather, it is a general prescription for contextualizing a conditional language. The advantage of working with ordering semantics stems from existing results (which I apply and develop) concerning the properties of ordering frames that facilitate fashioning and implementing a notion of contextual information preservation. Analyses of counterfactuals, such as Lewis’ (1973), that cash out the truth of counterfactuals in terms of the corresponding material conditional’s truth at possible worlds result in all counterpossibles being evaluated as vacuously true. This is because antecedents of counterpossibles are not true at any possible world, by definition. Such vacuist analyses have already been identified and challenged by a number of authors. I join this critical front, and drawing on existing proposals, I develop an impossible world semantics for a non-vacuist account of counterpossibles (chapter 5), by modifying the same system and semantics that serve the basis of the contextualized account offered in chapter 4, i.e. Lewis’ (1986) ordering semantics for the logic VW. I critically evaluate the advantages and disadvantages of key conditions on the ordering of worlds on the extended domain and show that there is a sense in which all of Lewis’ analysis of mere counterfactuals can be preserved, whilst offering an analysis of counterpossibles that meets our intuitions. The first part of chapter 1 consists of an outline of the usefulness of impossible worlds across philosophical analyses and logic. That outline in conjunction with a critical evaluation of Lewis’ logical arguments in favour of vacuism in chapter 2, and his marvellous mountain argument against impossible worlds in chapter 3, serves to motivate and justify the impossible world semantics for counterpossibles proposed in chapter 5. The second part of chapter 1 discusses the limitations that various conditional logics face when tasked to give an adequate treatment of the influence of context. That introductory discussion in conjunction with an overview of conditional logics and their various semantics in chapter 2 – which includes an in-depth exposition of Stalnaker-Lewis similarity semantics for counterfactuals – serves as the motivation and conceptual basis for the contextualized account of counterfactuals proposed in chapter 4. (shrink)

Bayesian epistemology addresses epistemological problems with the help of the mathematical theory of probability. It turns out that the probability calculus is especially suited to represent degrees of belief (credences) and to deal with questions of belief change, confirmation, evidence, justification, and coherence. Compared to the informal discussions in traditional epistemology, Bayesian epis- temology allows for a more precise and fine-grained analysis which takes the gradual aspects of these central epistemological notions into account. Bayesian epistemology therefore complements traditional epistemology; it (...) does not re- place it or aim at replacing it. (shrink)

[1] You have a crystal ball. Unfortunately, it’s defective. Rather than predicting the future, it gives you the chances of future events. Is it then of any use? It certainly seems so. You may not know for sure whether the stock market will crash next week; but if you know for sure that it has an 80% chance of crashing, then you should be 80% confident that it will—and you should plan accordingly. More generally, given that the chance of a (...) proposition A is x%, your conditional credence in A should be x%. This is a chance-credence principle: a principle relating chance (objective probability) with credence (subjective probability, degree of belief). Let’s call it the Minimal Principle (MP). (shrink)

Starting from the sixties of the past century theory change has become a main concern of philosophy of science. Two of the best known formal accounts of theory change are the post-Popperian theories of verisimilitude (PPV for short) and the AGM theory of belief change (AGM for short). In this paper, we will investigate the conceptual relations between PPV and AGM and, in particular, we will ask whether the AGM rules for theory change are effective means for approaching the truth, (...) i.e., for achieving the cognitive aim of science pointed out by PPV. First, the key ideas of PPV and AGM and their application to a particular kind of propositional theories - the so called "conjunctive propositions" - will be illustrated. Afterwards, we will prove that, as far as conjunctive propositions are concerned, AGM belief change is an effective tool for approaching the truth. (shrink)

According to the traditional Bayesian view of credence, its structure is that of precise probability, its objects are descriptive propositions about the empirical world, and its dynamics are given by conditionalization. Each of the three essays that make up this thesis deals with a different variation on this traditional picture. The first variation replaces precise probability with sets of probabilities. The resulting imprecise Bayesianism is sometimes motivated on the grounds that our beliefs should not be more precise than the evidence (...) calls for. One known problem for this evidentially motivated imprecise view is that in certain cases, our imprecise credence in a particular proposition will remain the same no matter how much evidence we receive. In the first essay I argue that the problem is much more general than has been appreciated so far, and that it’s difficult to avoid without compromising the initial evidentialist motivation. The second variation replaces descriptive claims with moral claims as the objects of credence. I consider three standard arguments for probabilism with respect to descriptive uncertainty—representation theorem arguments, Dutch book arguments, and accuracy arguments—in order to examine whether such arguments can also be used to establish probabilism with respect to moral uncertainty. In the second essay, I argue that by and large they can, with some caveats. First, I don’t examine whether these arguments can be given sound non-cognitivist readings, and any conclusions therefore only hold conditional on cognitivism. Second, decision-theoretic representation theorems are found to be less convincing in the moral case, because there they implausibly commit us to thinking that intertheoretic comparisons of value are always possible. Third and finally, certain considerations may lead one to think that imprecise probabilism provides a more plausible model of moral epistemology. The third variation considers whether, in addition to conditionalization, agents may also change their minds by becoming aware of propositions they had not previously entertained, and therefore not previously assigned any probability. More specifically, I argue that if we wish to make room for reflective equilibrium in a probabilistic moral epistemology, we must allow for awareness growth. In the third essay, I sketch the outline of such a Bayesian account of reflective equilibrium. Given that this account gives a central place to awareness growth, and that the rationality constraints on belief change by awareness growth are much weaker than those on belief change by conditionalization, it follows that the rationality constraints on the credences of agents who are seeking reflective equilibrium are correspondingly weaker. (shrink)

Decisions are made under uncertainty when there are distinct outcomes of a given action, and one is uncertain to which the act will lead. Decisions are made under indeterminacy when there are distinct outcomes of a given action, and it is indeterminate to which the act will lead. This paper develops a theory of (synchronic and diachronic) decision-making under indeterminacy that portrays the rational response to such situations as inconstant. Rational agents have to capriciously and randomly choose how to resolve (...) the indeterminacy relevant to a given choice-situation, but such capricious choices once made constrain how they will choose in the future. The account is illustrated by the case of self-interested action in situations where it is indeterminate whether you yourself will survive to benefit or suffer the consequences. The conclusion emphasizes some distinctive anti-hedging predictions of the account. (shrink)

In robustness analysis, hypotheses are supported to the extent that a result proves robust, and a result is robust to the extent that we detect it in diverse ways. But what precise sense of diversity is at work here? In this paper, I show that the formal explications of evidential diversity most often appealed to in work on robustness – which all draw in one way or another on probabilistic independence – fail to shed light on the notion of diversity (...) relevant to robustness analysis. I close by briefly outlining a promising alternative approach inspired by Horwich’s (1982) eliminative account of evidential diversity. (shrink)

There is a growing interest in the foundations as well as the application of imprecise probability in contemporary epistemology. This dissertation is concerned with the application. In particular, the research presented concerns ways in which imprecise probability, i.e. sets of probability measures, may helpfully address certain philosophical problems pertaining to rational belief. The issues I consider are disagreement among epistemic peers, complete ignorance, and inductive reasoning with imprecise priors. For each of these topics, it is assumed that belief can be (...) modeled with imprecise probability, and thus there is a non-classical solution to be given to each problem. I argue that this is the case for peer disagreement and complete ignorance. However, I discovered that the approach has its shortcomings, too, specifically in regard to inductive reasoning with imprecise priors. Nevertheless, the dissertation ultimately illustrates that imprecise probability as a model of rational belief has a lot of promise, but one should be aware of its limitations also. (shrink)

We review and discuss the recent monograph by David Wallace on Everettian Quantum Mechanics. This book is a high point of two decades of work on Everett in both physics and philosophy. It is also a beautiful and welcome exemplar of a modern way of doing metaphysics. We discuss certain aspects more critically, and take the opportunity to sketch an alternative pragmatist approach to probability in Everett, to be fully developed elsewhere.

I explore the thesis that the future is open, in the sense that future contingents are neither true nor false. The paper is divided into three sections. In the first, I survey how the thesis arises on a variety of contemporary views on the metaphysics of time. In the second, I explore the consequences for rational belief of the ‘Aristotelian’ view that indeterminacy is characterized by truth-value gaps. In the third, I outline one line of defence for the Aristotelian against (...) the puzzles this induces: treating opinion about future contingents as a matter of fictional belief rather than simple belief. (shrink)

*This is a project I hope to come back to one day. It stalled, a bit, on the absence of a positive theory of update I could be satisfied with* When should we believe a indicative conditional, and how much confidence in it should we have? Here’s one proposal: one supposes actual the antecedent; and sees under that supposition what credence attaches to the consequent. Thus we suppose that Oswald did not shot Kennedy; and note that under this assumption, Kennedy (...) was assassinated by someone other than Oswald. Thus we are highly confident in the indicative: if Oswald did not kill Kennedy, someone else did. (shrink)

The use of probability theory is widespread in our daily life as well as in scientific theories. In virtually all cases, calculations can be carried out within the framework of classical probability theory. A special exception is given by quantum mechanics, which gives rise to a new probability theory: quantum probability theory. This dissertation deals with the question of how this formalism can be understood from a philosophical and physical perspective. The dissertation is divided into three parts. In the first (...) part, the formalism of quantum probability theory and its relation to quantum mechanics is presented. The second part considers the possibility of reformulating quantum probability theory to embed it within classical probability. Mathematically, this possibility overlaps with the possibility of the existence of hidden variables theories of quantum mechanics: theories that attempt to solve fundamental problems in quantum mechanics by introducing additional physical properties of systems. The conclusion of this part is that, although classical reformulations of quantum probability theory are formally possible, they offer little insight into the formalism of quantum probability theory itself. The third part regards a more direct investigation of quantum probability theory. A reformulation of quantum probability theory is obtained by constructing a quantum logic on the basis of empirical non-probabilistic predictions of quantum mechanics. In this reformulation quantum probability functions are conditional probability functions on an algebra of propositions about measurements and measurement outcomes. (shrink)

This paper argues that expected utility theory for actions in chancy environments should be formulated in terms of centered chances. The subjective expected utility of an option A may be seen as a weighted sum of the utilities of A in different possible worlds, with weights being the credences that the agent assigns to these worlds. The utility of A in a given world is then definable as a weighted sum of the values of A’s different possible outcomes, with weights (...) being the conditional chances of these outcomes if A were performed. On the centered-chance view, the chances to be used as weights in the definition of utility are centered. Unlike ordinary chances, centered chances depend not only on what happens prior to the agent’s choice but also on the events that occur after the choice. Thus, to give an example, suppose that the action under consideration results in a bad outcome due to some event whose ordinary chance of occurring was very low at the time of choice. Then the utility of that action in the actual world could be high on the non-centered view, but on the centered view that utility is negative, since the centered chance of the event in question given the action was one, given that it did actually take place. A precise definition of centered chances is not easy to frame, but the concept can be made intuitively clear. The resulting decision theory is, in my opinion, philosophically more satisfactory than the extant proposals, even though it doesn’t differ much in its practical recommendations, with the exception of some rather peculiar cases. (shrink)

There are at least three core principles that define the chance role: ill the Principal Principle, l21 the Basic Chance Principle, and l31 the Humean Principle. These principles seem mutually incompatible. At least, no extant account of chance meets more than one of them. I ofier an account of chance which meets all three: L~-chance. So the good news is that L~-chance meets ill — l31. The bad news is that L~-chance turns out unlawful and unstable. But perhaps this is (...) not such bad news: L~-chance turns out to at least approximate plausible additional core principles concerning lawfulness and stability. And perhaps there is better news: one may treat 'chance' as vague, in a way that allows every core principle of chance to be met. (shrink)

How can self-locating propositions be integrated into normal patterns of belief revision? Puzzles such as Sleeping Beauty seem to show that such propositions lead to violation of ordinary principles for reasoning with subjective probability, such as Conditionalization and Reflection. I show that sophisticated forms of Conditionalization and Reflection are not only compatible with self-locating propositions, but also indispensable in understanding how they can function as evidence in Sleeping Beauty and similar cases.

Much recent philosophical attention has been devoted to variants on the Best System Analysis of laws and chance. In particular, philosophers have been interested in the prospects of such Best System Analyses for yielding *high-level* laws and chances. Nevertheless, a foundational worry about BSAs lurks: there do not appear to be uniquely appropriate measures of the degree to which a system exhibits theoretical virtues, such as simplicity and strength. Nor does there appear to be a uniquely correct exchange rate at (...) which the theoretical virtues of simplicity, strength, and likelihood trade off against one another in the determination of a best system. Moreover, it may be that there is no *robustly* best system: no system that comes out best under *any* reasonable measures of the theoretical virtues and exchange rate between them. This worry has been noted by several philosophers, with some arguing that there is indeed plausibly a set of tied-for-best systems for our world. Some have even argued that this entails that there are no Best System laws or chances in our world. I argue that, while it *is* plausible that there is a set of tied-for-best systems for our world, it doesn't follow from this that there are no Best System chances. Rather, it follows that the Best System chances for our world are *unsharp*. (shrink)

Belief is a familiar attitude: taking something to be the case or regarding it as true. But we are more confident in some of our beliefs than in others. For this reason, many epistemologists appeal to a second attitude, called credence, similar to a degree of confidence. This raises the question: how do belief and credence relate to each other? On a belief-first view, beliefs are more fundamental and credences are a species of beliefs, e.g. beliefs about probabilities. On a (...) credence-first view, credences are more fundamental and beliefs are a species of credence, e.g. credence above some threshold. In this thesis, I develop and defend a third view that I call belief-credence dualism. On this view, belief and credence are independent, equally fundamental attitudes, and neither reduces to the other. I begin by motivating the project: why should we care about the relationship between belief and credence? I argue it has broad implications for many debates in epistemology and beyond. Then, I defend dualism, arguing that it can explain features of our mental lives that a credence-first view and a belief-first view cannot. I also argue that dualism has attractive, interesting implications when applied to the pragmatic encroachment debate. Finally, I explore implications of dualism, both for the nature of evidence and how faith might go beyond the evidence but nonetheless be epistemically rational. I conclude that the human mind is, in some ways, complex, but we should be happy with this conclusion also long as each mental state we posit has a clear role to play. (shrink)