This book explores a question central to philosophy--namely, what does it take for a belief to be justified or rational? According to a widespread view, whether one has justification for believing a proposition is determined by how probable that proposition is, given one's evidence. In this book this view is rejected and replaced with another: in order for one to have justification for believing a proposition, one's evidence must normically support it--roughly, one's evidence must make the falsity of that proposition (...) abnormal in the sense of calling for special, independent explanation. This conception of justification bears upon a range of topics in epistemology and beyond. Ultimately, this way of looking at justification guides us to a new, unfamiliar picture of how we should respond to our evidence and manage our own fallibility. This picture is developed here. (shrink)

This paper develops an information-sensitive theory of the semantics and probability of conditionals and statements involving epistemic modals. The theory validates a number of principles linking probability and modality, including the principle that the probability of a conditional If A, then C equals the probability of C, updated with A. The theory avoids so-called triviality results, which are standardly taken to show that principles of this sort cannot be validated. To achieve this, we deny that rational (...) agents update their credences via conditionalization. We offer a new rule of update, Hyperconditionalization, which agrees with Conditionalization whenever nonmodal statements are at stake but differs for modal and conditional sentences. (shrink)

Richard Jeffrey is beyond dispute one of the most distinguished and influential philosophers working in the field of decision theory and the theory of knowledge. His work is distinctive in showing the interplay of epistemological concerns with probability and utility theory. Not only has he made use of standard probabilistic and decision theoretic tools to clarify concepts of evidential support and informed choice, he has also proposed significant modifications of the standard Bayesian position in order that it provide a (...) better fit with actual human experience. Probability logic is viewed not as a source of judgment but as a framework for explaining the implications of probabilistic judgments and their mutual compatability. This collection of essays spans a period of some 35 years and includes what have become some of the classic works in the literature. There is also one completely new piece, while in many instances Jeffrey includes afterthoughts on the older essays. (shrink)

This book offers a concise survey of basic probability theory from a thoroughly subjective point of view whereby probability is a mode of judgment. Written by one of the greatest figures in the field of probability theory, the book is both a summation and synthesis of a lifetime of wrestling with these problems and issues. After an introduction to basic probability theory, there are chapters on scientific hypothesis-testing, on changing your mind in response to generally uncertain (...) observations, on expectations of the values of random variables, on de Finetti's dissolution of the so-called problem of induction, and on decision theory. (shrink)

The problem of probabilities of conditionals is one of the long-standing puzzles in philosophy of language. We defend and update Adams' solution to the puzzle: the probability of an epistemic conditional is not the probability of a proposition, but a probability under a supposition. Close inspection of how a triviality result unfolds in a concrete scenario does not provide counterexamples to the view that probabilities of conditionals are conditional probabilities: instead, it supports the conclusion that probabilities of (...) conditionals violate standard probability theory. This does not call into question probability theory per se; rather, it calls for a more careful understanding of its role: probability theory is a theory of probabilities of propositions; but as conditionals do not express propositions, their probabilities are not subject to the standard laws. We argue that both conditional probabilities and probabilities of conditionals are best understood in terms of the dynamics of supposing, modeled as a restriction operation on a probability space. This version of the suppositionalist view allows us to connect Adams' Thesis to the widely held restrictor view of the semantics of conditionals. We address two common objections to Adams' view: that the relevant probabilities are ‘probabilities only in name’, and that giving up conditional propositions puts us at a disadvantage when it comes to interpreting compounds. Finally, we argue that some putative counterexamples to Adams' Thesis can be diagnosed as fallacies of probabilistic reasoning: they arise from applying to conditionals laws of standard probability theory which are invalid for them. (shrink)

I argue that when we use ‘probability’ language in epistemic contexts—e.g., when we ask how probable some hypothesis is, given the evidence available to us—we are talking about degrees of support, rather than degrees of belief. The epistemic probability of A given B is the mind-independent degree to which B supports A, not the degree to which someone with B as their evidence believes A, or the degree to which someone would or should believe A if they had (...) B as their evidence. My central argument is that the degree-of-support interpretation lets us better model good reasoning in certain cases involving old evidence. Degree-of-belief interpretations make the wrong predictions not only about whether old evidence confirms new hypotheses, but about the values of the probabilities that enter into Bayes’ Theorem when we calculate the probability of hypotheses conditional on old evidence and new background information. (shrink)

In this book Pollock deals with the subject of probabilistic reasoning, making general philosophical sense of objective probabilities and exploring their ...

The aim of the paper is to draw a connection between a semantical theory of conditional statements and the theory of conditional probability. First, the probability calculus is interpreted as a semantics for truth functional logic. Absolute probabilities are treated as degrees of rational belief. Conditional probabilities are explicitly defined in terms of absolute probabilities in the familiar way. Second, the probability calculus is extended in order to provide an interpretation for counterfactual probabilities--conditional probabilities where the condition (...) has zero probability. Third, conditional propositions are introduced as propositions whose absolute probability is equal to the conditional probability of the consequent on the antecedent. An axiom system for this conditional connective is recovered from the probabilistic definition. Finally, the primary semantics for this axiom system, presented elsewhere, is related to the probabilistic interpretation. (shrink)

In this influential study of central issues in the philosophy of science, Paul Horwich elaborates on an important conception of probability, diagnosing the failure of previous attempts to resolve these issues as stemming from a too-rigid conception of belief. Adopting a Bayesian strategy, he argues for a probabilistic approach, yielding a more complete understanding of the characteristics of scientific reasoning and methodology. Presented in a fresh twenty-first-century series livery, and including a specially commissioned preface written by Colin Howson, illuminating (...) its enduring importance and relevance to philosophical enquiry, this engaging work has been revived for a new generation of readers. (shrink)

Probability plays a key role in modern physics: an international team of philosophers illuminate this role by exploring the epistemology and metaphysics of probabilities. They discuss statistical physics, probabilistic modelling, and quantum mechanics and critically assess objectivist and subjectivist views of probabilities in these fields.

According to what is now commonly referred to as “the Equation” in the literature on indicative conditionals, the probability of any indicative conditional equals the probability of its consequent of the conditional given the antecedent of the conditional. Philosophers widely agree in their assessment that the triviality arguments of Lewis and others have conclusively shown the Equation to be tenable only at the expense of the view that indicative conditionals express propositions. This study challenges the correctness of that (...) assessment by presenting data that cast doubt on an assumption underlying all triviality arguments. (shrink)

In response to cases that involve tiny probabilities of huge payoffs, some argue that we ought to discount small probabilities down to zero. However, this paper shows that doing so violates Independence and Continuity, and as a result of these violations, those who discount small probabilities can be exploited by money pumps. Various possible ways of avoiding exploitation will be discussed. This paper concludes that the money pump for Independence undermines the plausibility of discounting small probabilities. Much of the discussion (...) on Independence generalizes to other views that also violate Independence. (shrink)

Why is understanding causation so important in philosophy and the sciences? Should causation be defined in terms of probability? Whilst causation plays a major role in theories and concepts of medicine, little attempt has been made to connect causation and probability with medicine itself. Causality, Probability, and Medicine is one of the first books to apply philosophical reasoning about causality to important topics and debates in medicine. Donald Gillies provides a thorough introduction to and assessment of competing (...) theories of causality in philosophy, including action-related theories, causality and mechanisms, and causality and probability. Throughout the book he applies them to important discoveries and theories within medicine, such as germ theory; tuberculosis and cholera; smoking and heart disease; the first ever randomized controlled trial designed to test the treatment of tuberculosis; the growing area of philosophy of evidence-based medicine; and philosophy of epidemiology. This book will be of great interest to students and researchers in philosophy of science and philosophy of medicine, as well as those working in medicine, nursing and related health disciplines where a working knowledge of causality and probability is required. (shrink)

David Wallace has given a decision-theoretic argument for the Born Rule in the context of Everettian quantum mechanics. This approach promises to resolve some long-standing problems with probability in EQM, but it has faced plenty of resistance. One kind of objection charges that the requisite notion of decision-theoretic uncertainty is unavailable in the Everettian picture, so that the argument cannot gain any traction; another kind of objection grants the proof’s applicability and targets the premises. In this article I propose (...) some novel principles connecting the physics of EQM with the metaphysics of modality, and argue that in the resulting framework the incoherence problem does not arise. These principles also help to justify one of the most controversial premises of Wallace’s argument, ‘branching indifference’. Absent any a priori reason to align the metaphysics with the physics in some other way, the proposed principles can be adopted on grounds of theoretical utility. The upshot is that Everettians can, after all, make clear sense of objective probability. 1 Introduction2 Setup3 Individualism versus Collectivism4 The Ingredients of Indexicalism5 Indexicalism and Incoherence5.1 The trivialization problem5.2 The uncertainty problem6 Indexicalism and Branching Indifference6.1 Introducing branching indifference6.2 The pragmatic defence of branching indifference6.3 The non-existence defence of branching indifference6.4 The indexicalist defence of branching indifference7 Conclusion. (shrink)

In the Bayesian approach to quantum mechanics, probabilities—and thus quantum states—represent an agent’s degrees of belief, rather than corresponding to objective properties of physical systems. In this paper we investigate the concept of certainty in quantum mechanics. Particularly, we show how the probability-1 predictions derived from pure quantum states highlight a fundamental difference between our Bayesian approach, on the one hand, and Copenhagen and similar interpretations on the other. We first review the main arguments for the general claim that (...) probabilities always represent degrees of belief. We then argue that a quantum state prepared by some physical device always depends on an agent’s prior beliefs, implying that the probability-1 predictions derived from that state also depend on the agent’s prior beliefs. Quantum certainty is therefore always some agent’s certainty. Conversely, if facts about an experimental setup could imply agent-independent certainty for a measurement outcome, as in many Copenhagen-like interpretations, that outcome would effectively correspond to a preexisting system property. The idea that measurement outcomes occurring with certainty correspond to preexisting system properties is, however, in conflict with locality. We emphasize this by giving a version of an argument of Stairs [(1983). Quantum logic, realism, and value-definiteness. Philosophy of Science, 50, 578], which applies the Kochen–Specker theorem to an entangled bipartite system. (shrink)

This paper offers a metaphysics of physical probability in (or if you prefer, truth conditions for probabilistic claims about) deterministic systems based on an approach to the explanation of probabilistic patterns in deterministic systems called the method of arbitrary functions. Much of the appeal of the method is its promise to provide an account of physical probability on which probability assignments have the ability to support counterfactuals about frequencies. It is argued that the eponymous arbitrary functions are (...) of little philosophical use, but that they can be substituted for facts about frequencies without losing the ability to provide counterfactual support. The result is an account of probability in deterministic systems that has a “propensity-like” look and feel, yet which requires no supplement to the standard modern empiricist tool kit of particular matters of fact and principles of physical dynamics. (shrink)

As a survey of many technical results in probability theory and probability logic, this monograph by two widely respected scholars offers a valuable compendium of the principal aspects of the formal study of probability. Hugues Leblanc and Peter Roeper explore probability functions appropriate for propositional, quantificational, intuitionistic, and infinitary logic and investigate the connections among probability functions, semantics, and logical consequence. They offer a systematic justification of constraints for various types of probability functions, in (...) particular, an exhaustive account of probability functions adequate for first-order quantificational logic. The relationship between absolute and relative probability functions is fully explored and the book offers a complete account of the representation of relative functions by absolute ones. The volume is designed to review familiar results, to place these results within a broad context, and to extend the discussions in new and interesting ways. Authoritative, articulate, and accessible, it will interest mathematicians and philosophers at both professional and post-graduate levels. (shrink)

Probabilities figure centrally in much of the literature on the semantics of conditionals. I find this surprising: it accords a special status to conditionals that other parts of language apparently do not share. I critically discuss two notable ‘probabilities first’ accounts of counterfactuals, due to Edgington and Leitgeb. According to Edgington, counterfactuals lack truth values but have probabilities. I argue that this combination gives rise to a number of problems. According to Leitgeb, counterfactuals have truth conditions-roughly, a counterfactual is true (...) when the corresponding conditional chance is sufficiently high. I argue that problems arise from the disparity between truth and high chance, between approximate truth and high chance, and from counterfactuals for which the corresponding conditional chances are undefined. However, Edgington, Leitgeb and I can unite in opposition to Stalnaker and Lewis-style ‘similarity’ accounts of counterfactuals. (shrink)

Probability theory promises to deliver an exact and unified foundation for inquiry in epistemology and philosophy of science. But philosophy of religion is also fertile ground for the application of probabilistic thinking. This volume presents original contributions from twelve contemporary researchers, both established and emerging, to offer a representative sample of the work currently being carried out in this potentially rich field of inquiry. Grouped into five parts, the chapters span a broad range of traditional issues in religious epistemology. (...) The first three parts discuss the evidential impact of various considerations that have been brought to bear on the question of the existence of God. These include witness reports of the occurrence of miraculous events, the existence of complex biological adaptations, the apparent 'fine-tuning' for life of various physical constants and the existence of seemingly unnecessary evil. The fourth part addresses a number of issues raised by Pascal's famous pragmatic argument for theistic belief. A final part offers probabilistic perspectives on the rationality of faith and the epistemic significance of religious disagreement. (shrink)

This is a study in the meaning of natural language probability operators, sentential operators such as probably and likely. We ask what sort of formal structure is required to model the logic and semantics of these operators. Along the way we investigate their deep connections to indicative conditionals and epistemic modals, probe their scalar structure, observe their sensitivity to contex- tually salient contrasts, and explore some of their scopal idiosyncrasies.

Within the τ-Logic program, this paper reconstructs scalar probability as a licensed projection/readout from a declared normalized τ phase regime. It uses the Layer-0 τ-identity reference and the zero/projection-nullity analysis as public continuity references, without treating either work as a hidden premise for the probability theorems. Its formal core is conservative: once τ-identity is represented in a declared native normalized compact phase domain, S¹ ≅ U(1) supplies normalized Haar phase measure, measurable τ-readout maps induce pushforward laws, and (...) class='Hi'>probability laws on standard Borel spaces are representable as such pushforwards. The paper preserves Kolmogorov probability as the ordinary measure-theoretic readout layer; it does not claim that named distributions are uniquely forced from bare τ. Instead, τ-Probability supplies structural provenance only under declared phase-completion, compact phase-domain, Haar-measure, measurable-readout, and local-licensing assumptions, so probability values remain late scalar readouts whose licensing structure includes a source space, σ-algebra, measure, readout/projection map, and local constraint regime. Universal representability is global; distributional necessity is regime-specific. Probability-zero is treated as measure-nullity under a declared regime, not as impossibility, event absence, or ontological nullity. The Borel–Kolmogorov boundary confirms this discipline: E = 0, C = 0, and E = C are phase-coordinate null-readout conditions, not intrinsic conditional events or carrier-nullities, and exact conditioning on them is licensed only by a declared conditioning/readout regime. (shrink)

The term probability can be used in two main senses. In the frequency interpretation it is a limiting ratio in a sequence of repeatable events. In the Bayesian view, probability is a mental construct representing uncertainty. This 2002 book is about these two types of probability and investigates how, despite being adopted by scientists and statisticians in the eighteenth and nineteenth centuries, Bayesianism was discredited as a theory of scientific inference during the 1920s and 1930s. Through the (...) examination of a dispute between two British scientists, the author argues that a choice between the two interpretations is not forced by pure logic or the mathematics of the situation, but depends on the experiences and aims of the individuals involved. The book should be of interest to students and scientists interested in statistics and probability theories and to general readers with an interest in the history, sociology and philosophy of science. (shrink)

This collection of essays is on the relation between probabilities, especially conditional probabilities, and conditionals. It provides negative results which sharply limit the ways conditionals can be related to conditional probabilities. There are also positive ideas and results which will open up areas of research. The collection is intended to honour Ernest W. Adams, whose seminal work is largely responsible for creating this area of inquiry. As well as describing, evaluating, and applying Adams's work the contributions extend his ideas in (...) directions he may or may not have anticipated, but that he certainly inspired. In addition to a wide range of philosophers of science, the volume should interest computer scientists and linguists. (shrink)

This chapter will review selected aspects of the terrain of discussions about probabilities in statistical mechanics (with no pretensions to exhaustiveness, though the major issues will be touched upon), and will argue for a number of claims. None of the claims to be defended is entirely original, but all deserve emphasis. The first, and least controversial, is that probabilistic notions are needed to make sense of statistical mechanics. The reason for this is the same reason that convinced Maxwell, Gibbs, and (...) Boltzmann that probabilities would be needed, namely, that the second law of thermodynamics, which in its original formulation says that certain processes are impossible, must, on the kinetic theory, be replaced by a weaker formulation according to which what the original version deems impossible is merely improbable. Second is that we ought not take the standard measures invoked in equilibrium statistical mechanics as giving, in any sense, the correct probabilities about microstates of the system. We can settle for a much weaker claim: that the probabilities for outcomes of experiments yielded by the standard distributions are effectively the same as those yielded by any distribution that we should take as a representing probabilities over microstates. Lastly, (and most controversially): in asking about the status of probabilities in statistical mechanics, the familiar dichotomy between epistemic probabilities (credences, or degrees of belief) and ontic (physical) probabilities is insufficient; the concept of probability that is best suited to the needs of statistical mechanics is one that combines epistemic and physical considerations. (shrink)

The so-called "non-commutativity" of probability kinematics has caused much unjustified concern. When identical learning is properly represented, namely, by identical Bayes factors rather than identical posterior probabilities, then sequential probability-kinematical revisions behave just as they should. Our analysis is based on a variant of Field's reformulation of probability kinematics, divested of its (inessential) physicalist gloss.

Probabilism—the doctrine that ideally rational degrees of belief conform to the calculus of probabilities—is habitually defended by citing Dutch-book and accuracy-dominance arguments. Yet these arguments rely on premises that are not plausibly necessary truths. I argue that the premises in question can be understood as fixing a convention. Relaxing these premises leads to alternative calculi which—under permissive though not universal assumptions—are intertranslatable with the probability calculus. This applies to Dutch-book and accuracy-dominance arguments alike; a mathematical correspondence is shown to (...) hold between them. (shrink)

Recognizing that probability (the Greek doxa) was understood in pre-modern theories as the polar opposite of certainty (episteme), the author of this study elaborates the forms which these polar opposites have taken in some twentieth century writers and then, in greater detail, in the writings of Thomas Aquinas. Profiting from subsequent more sophisticated theories of probability, he examines how Aquinas’s judgments about everything from God to gossip depend on schematizations of the polarity between the systematic and the non-systematic: (...) revelation/reason, science/opinion, saints/philosophers, Aristotle/others. Post-medieval developments have provided the means to discern within Thomas’s thought both a logical theory and a kind of relative frequency theory of probability. The former depends on Aristotle’s theory of demonstration, the latter on his theory of an orderly cosmos; but each as used by Thomas lead to convictions that are sometimes naive, sometimes amusing or even terrifying. (shrink)

This book presents a novel theory of probability applicable to general reasoning, science, and the courts. Based on a strongly subjective starting-point, with probabilities viewed simply as the guarded beliefs one can reasonably hold, the theory shows how such beliefs are legitimately "projected" outwards as if they existed in the world independent of our judgements.

Quantum mechanics and probability theory share one peculiarity. Both have well established mathematical formalisms, yet both are subject to controversy about the meaning and interpretation of their basic concepts. Since probability plays a fundamental role in QM, the conceptual problems of one theory can affect the other. We first classify the interpretations of probability into three major classes: inferential probability, ensemble probability, and propensity. Class is the basis of inductive logic; deals with the frequencies of (...) events in repeatable experiments; describes a form of causality that is weaker than determinism. An important, but neglected, paper by P. Humphreys demonstrated that propensity must differ mathematically, as well as conceptually, from probability, but he did not develop a theory of propensity. Such a theory is developed in this paper. Propensity theory shares many, but not all, of the axioms of probability theory. As a consequence, propensity supports the Law of Large Numbers from probability theory, but does not support Bayes theorem. Although there are particular problems within QM to which any of the classes of probability may be applied, it is argued that the intrinsic quantum probabilities are most naturally interpreted as quantum propensities. This does not alter the familiar statistical interpretation of QM. But the interpretation of quantum states as representing knowledge is untenable. Examples show that a density matrix fails to represent knowledge. (shrink)

The two main psychological theories of the ordinary conditional were designed to account for inferences made from assumptions, but few premises in everyday life can be simply assumed true. Useful premises usually have a probability that is less than certainty. But what is the probability of the ordinary conditional and how is it determined? We argue that people use a two stage Ramsey test that we specify to make probability judgements about indicative conditionals in natural language, and (...) we describe experiments that support this conclusion. Our account can explain why most people give the conditional probability as the probability of the conditional, but also why some give the conjunctive probability. We discuss how our psychological work is related to the analysis of ordinary indicative conditionals in philosophical logic. (shrink)

We explore ways in which purely qualitative belief change in the AGM tradition throws light on options in the treatment of conditional probability. First, by helping see why it can be useful to go beyond the ratio rule defining conditional from one-place probability. Second, by clarifying what is at stake in different ways of doing that. Third, by suggesting novel forms of conditional probability corresponding to familiar variants of qualitative belief change, and conversely. Likewise, we explain how (...) recent work on the qualitative part of probabilistic inference leads to a very broad class of 'proto-probability' functions. (shrink)

This chapter presents probability logic as a rationality framework for human reasoning under uncertainty. Selected formal-normative aspects of probability logic are discussed in the light of experimental evidence. Specifically, probability logic is characterized as a generalization of bivalent truth-functional propositional logic (short “logic”), as being connexive, and as being nonmonotonic. The chapter discusses selected argument forms and associated uncertainty propagation rules. Throughout the chapter, the descriptive validity of probability logic is compared to logic, which was used (...) as the gold standard of reference for assessing the rationality of human reasoning in the 20th century. (shrink)

Emch, G.G., Liu, C.: The Logic of Thermostatistical Physics. Springer, Berlin/ Heidelberg (2002) 11. Frigg, R., Werndl, C.: Entropy – a guide for the perplexed. Forthcoming in: Beisbart, C., Hartmann, S. (eds.) Probabilities in Physics. Oxford...

The contributions to this special collection concern issues and problems discussed in or related to the work of Wesley C. Salmon. Salmon has long been noted for his important work in the philosophy of science, which has included research on the interpretation of probability, the nature of explanation, the character of reasoning, the justification of induction, the structure of space/time and the paradoxes of Zeno, to mention only some of the most prominent. During a time of increasing preoccupation with (...) historical and sociological approaches to under standing science (which characterize scientific developments as though they could be adequately analysed from the perspective of political movements, even mistaking the phenomena of conversion for the rational appraisal of scientific theories), Salmon has remained stead fastly devoted to isolating and justifying those normative standards distinguishing science from non-science - especially through the vindi cation of general principles of scientific procedure and the validation of specific examples of scientific theories - without which science itself cannot be (even remotely) adequately understood. In this respect, Salmon exemplifies and strengthens a splendid tradi tion whose most remarkable representatives include Hans Reichenbach, Rudolf Carnap and Carl G. Hempel, all of whom exerted a profound influence upon his own development. (shrink)

Many philosophers accept that, as well as having a right that others not harm us, we also have a right that others not subject us to a risk of harm. And yet, when we attempt to spell out precisely what this ‘right against risk imposition’ involves, we encounter a series of notorious puzzles. Existing attempts to deal with these puzzles have tended to focus on the nature of rights – but I propose an approach that focusses instead on the nature (...) of risk. The key move is to distinguish two different ways in which to conceptualise the risk that a given action presents – one of which is linked to the notion of probability and the other to the notion of normalcy. (shrink)

Once upon a time, some thought that indicative conditionals could be effectively analyzed as material conditionals. Later on, an alternative theoretical construct has prevailed and received wide acceptance, namely, the conditional probability of the consequent given the antecedent. Partly following critical remarks recently ap- peared in the literature, we suggest that evidential support—rather than conditional probability alone—is key to understand indicative conditionals. There have been motivated concerns that a theory of evidential conditionals (unlike their more tra- ditional counterparts) (...) cannot generate a sufficiently interesting logical system. Here, we will describe results dispelling these worries. Happily, and perhaps surprisingly, appropriate technical variations of Ernst Adams’s classical approach allow for the construction of a logic of evidential conditionals with distinctive fea- tures, which is also well-behaved and reasonably strong. (shrink)

It is well known that classical, aka ‘sharp’, Bayesian decision theory, which models belief states as single probability functions, faces a number of serious difficulties with respect to its handling of agnosticism. These difficulties have led to the increasing popularity of so-called ‘imprecise’ models of decision-making, which represent belief states as sets of probability functions. In a recent paper, however, Adam Elga has argued in favour of a putative normative principle of sequential choice that he claims to be (...) borne out by the sharp model but not by any promising incarnation of its imprecise counterpart. After first pointing out that Elga has fallen short of establishing that his principle is indeed uniquely borne out by the sharp model, I cast aspersions on its plausibility. I show that a slight weakening of the principle is satisfied by at least one, but interestingly not all, varieties of the imprecise model and point out that Elga has failed to motivate his stronger commitment. (shrink)

On an attractive, naturalistically respectable theory of intentionality, mental contents are a form of measurement system for representing behavioral and psychological dispositions. This chapter argues that a consequence of this view is that the content/attitude distinction is measurement system relative. As a result, there is substantial arbitrariness in the content/attitude distinction. Whether some measurement of mental states counts as characterizing the content of mental states or the attitude is not a question of empirical discovery but of theoretical utility. If correct, (...) this observation has ramifications in the theory of rationality. Some epistemologists and decision theorists have argued that imprecise credences are rationally impermissible, while others have argued that precise credences are rationally impermissible. If the measure theory of mental content is correct, however, then neither imprecise credences nor precise credences can be rationally impermissible. (shrink)

We discuss the meaning of probabilities in the many worlds interpretation of quantum mechanics. We start by presenting very briefly the many worlds theory, how the problem of probability arises, and some unsuccessful attempts to solve it in the past. Then we criticize a recent attempt by Deutsch to derive the quantum mechanical probabilities from the nonprobabilistic parts of quantum mechanics and classical decision theory. We further argue that the Born probability does not make sense even as an (...) additional probability rule in the many worlds theory. Our conclusion is that the many worlds theory fails to account for the probabilistic statements of standard (collapse) quantum mechanics. (shrink)

The Ramseyan thesis that the probability of an indicative conditional is equal to the corresponding conditional probability of its consequent given its antecedent is both widely confirmed and subject to attested counterexamples (e.g., McGee 2000, Kaufmann 2004). This raises several puzzling questions. For instance, why are there interpretations of conditionals that violate this Ramseyan thesis in certain contexts, and why are they otherwise very rare? In this paper, I raise some challenges to Stefan Kaufmann's account of why the (...) Ramseyan thesis sometimes fails, and motivate my own theory. On my theory, the proposition expressed by an indicative conditional is partially determined by a background partition, and hence its probability depends on the choice of such a partition. I hold that this background partition is contextually determined, and in certain conditions is set by a salient question under discussion in the context. I show how the resulting theory offers compelling answers to the puzzling questions raised by failures of the Ramseyan thesis. (shrink)

Different inferences in probabilistic logics of conditionals 'preserve' the probabilities of their premisses to different degrees. Some preserve certainty, some high probability, some positive probability, and some minimum probability. In the first case conclusions must have probability I when premisses have probability 1, though they might have probability 0 when their premisses have any lower probability. In the second case, roughly speaking, if premisses are highly probable though not certain then conclusions must also (...) be highly probable. In the third case conclusions must have positive probability when premisses do, and in the last case conclusions must be at least as probable as their least probable premisses. Precise definitions and well known examples are given for each of these properties, characteristic principles are shown to be valid and complete for deriving conclusions of each of these kinds, and simple trivalent truthtable 'tests' are described for determining which properties are possessed by any given inference. Brief comments are made on the application of these results to certain modal inferences such as "Jones may own a car, and if he does he will have a driver's license. Therefore, he may have a driver's license.". (shrink)

I describe a realist, ontologically objective interpretation of probability, "far-flung frequency (FFF) mechanistic probability". FFF mechanistic probability is defined in terms of facts about the causal structure of devices and certain sets of frequencies in the actual world. Though defined partly in terms of frequencies, FFF mechanistic probability avoids many drawbacks of well-known frequency theories and helps causally explain stable frequencies, which will usually be close to the values of mechanistic probabilities. I also argue that it's (...) a virtue rather than a failing of FFF mechanistic probability that it does not define single-case chances, and compare some aspects of my interpretation to a recent interpretation proposed by Strevens. (shrink)

The problem of inferring probability comparisons between events from an initial set of comparisons arises in several contexts, ranging from decision theory to artificial intelligence to formal semantics. In this paper, we treat the problem as follows: beginning with a binary relation ≥ on events that does not preclude a probabilistic interpretation, in the sense that ≥ has extensions that are probabilistically representable, we characterize the extension ≥+ of ≥ that is exactly the intersection of all probabilistically representable extensions (...) of ≥. This extension ≥+ gives us all the additional comparisons that we are entitled to infer from ≥, based on the assumption that there is some probability measure of which ≥ gives us partial qualitative information. We pay special attention to the problem of extending an order on states to an order on events. In addition to the probabilistic interpretation, this problem has a more general interpretation involving measurement of any additive quantity: e.g., given comparisons between the weights of individual objects, what comparisons between the weights of groups of objects can we infer? (shrink)

This note is a postscript to Ramsey's 'Truth and Probability'. It replaces that article's psychological reading of subjective probability with a reading of it as a consistency condition on the theory that we act to maximise expected utility.