We generalize the Kolmogorov axioms for probability calculus to obtain conditions defining, for any given logic, a class of probability functions relative to that logic, coinciding with the standard probability functions in the special case of classical logic but allowing consideration of other classes of "essentially Kolmogorovian" probability functions relative to other logics. We take a broad view of the Bayesian approach as dictating inter alia that from the perspective of a given logic, rational degrees of belief are those representable (...) by probability functions from the class appropriate to that logic. Classical Bayesianism, which fixes the logic as classical logic, is only one version of this general approach. Another, which we call Intuitionistic Bayesianism, selects intuitionistic logic as the preferred logic and the associated class of probability functions as the right class of candidate representions of epistemic states (rational allocations of degrees of belief). Various objections to classical Bayesianism are, we argue, best met by passing to intuitionistic Bayesianism—in which the probability functions are taken relative to intuitionistic logic—rather than by adopting a radically non-Kolmogorovian, for example, nonadditive, conception of (or substitute for) probability functions, in spite of the popularity of the latter response among those who have raised these objections. The interest of intuitionistic Bayesianism is further enhanced by the availability of a Dutch Book argument justifying the selection of intuitionistic probability functions as guides to rational betting behavior when due consideration is paid to the fact that bets are settled only when/if the outcome bet on becomes known. (shrink)

Bayes' Theorem is a simple mathematical formula used for calculating conditional probabilities. It figures prominently in subjectivist or Bayesian approaches to epistemology, statistics, and inductive logic. Subjectivists, who maintain that rational belief is governed by the laws of probability, lean heavily on conditional probabilities in their theories of evidence and their models of empirical learning. Bayes' Theorem is central to these enterprises both because it simplifies the calculation of conditional probabilities and because it clarifies significant features of subjectivist position. Indeed, (...) the Theorem's central insight — that a hypothesis is confirmed by any body of data that its truth renders probable — is the cornerstone of all subjectivist methodology. (shrink)

Recently a number of authors have tried to avoid the failures of traditional Dutch book arguments by separating them from pragmatic concerns of avoiding a sure loss. In this paper I examine defenses of this kind by Howson and Urbach, Hellman, and Christensen. I construct rigorous explications of their arguments and show that they are not cogent. I advocate abandoning Dutch book arguments in favor of a representation theorem.

Both Representation Theorem Arguments and Dutch Book Arguments support taking probabilistic coherence as an epistemic norm. Both depend on connecting beliefs to preferences, which are not clearly within the epistemic domain. Moreover, these connections are standardly grounded in questionable definitional/metaphysical claims. The paper argues that these definitional/metaphysical claims are insupportable. It offers a way of reconceiving Representation Theorem arguments which avoids the untenable premises. It then develops a parallel approach to Dutch Book Arguments, and compares the results. In each case (...) preferencedefects serve as a diagnostic tool, indicating purely epistemic defects. (shrink)

This dissertation looks at a set of interconnected questions concerning the foundations of probability, and gives a series of interconnected answers. At its core is a piece of old-fashioned philosophical analysis, working out what probability is. Or equivalently, investigating the semantic question of what is the meaning of ‘probability’? Like Keynes and Carnap, I say that probability is degree of reasonable belief. This immediately raises an epistemological question, which degrees count as reasonable? To solve that in its full generality would (...) mean the end of human inquiry, so that won’t be attempted here. Rather I will follow tradition and merely investigate which sets of partial beliefs are coherent. -/- The standard answer to this question, what is commonly called the Bayesian answer, says that degrees of belief are coherent iff they form a probability function. I disagree with the way this is usually justified, but subject to an important qualification I accept the answer. The important qualification is that degrees of belief may be imprecise, or vague. Part one of the dissertation, chapters 1 to 6, looks largely at the consequences of this qualification for the semantic and epistemological questions already mentioned. It turns out that when we allow degrees of belief to be imprecise, we can discharge potentially fatal objections to some philosophically attractive theses. Two of these, that probability is degree of reasonable belief and that the probability calculus provides coherence constraints on partial beliefs, have been mentioned. Others include the claim, defended in chapter 4, that chance is probability given total history. -/- As well as these semantic and epistemological questions, studies of the foundations of probability usually include a detailed discussion of decision theory. For reasons set out in chapter 2, I deny we can gain epistemological insights from decision theory. Nevertheless, it is an interesting field to study on its own, and it might be expected that there would be decision theoretic consequences of allowing imprecise degrees of belief. As I show in part two, this expectation seems to be mistaken. Chapter 9 shows that there aren’t interesting consequences of this theory for decision theory proper, and chapters 10 and 11 show that Keynes’s attempt to use imprecision in degrees of belief to derive a distinctive theory of interest rates is unsound. -/- Chapters 7 and 8 provide a link between these two parts. In chapter 7 I look at some previous philosophical investigations into the effects of imprecision. In chapter 8 I develop what I take to be the best competitor to the theory defended here – a constructivist theory of probability. On this view degrees of belief are precise, but the relevant coherence constraint is a constructivist probability calculus. This view is, I think, mistaken, but the calculus has some intrinsic interest, and there are at least enough arguments for it to warrant a chapter-length examination. (shrink)

De Finetti is one of the founding fathers of the subjective school of probability. He held that probabilities are subjective, coherent degrees of expectation, and he argued that none of the objective interpretations of probability make sense. While his theory has been influential in science and philosophy, it has encountered various objections. I argue that these objections overlook central aspects of de Finetti’s philosophy of probability and are largely unfounded. I propose a new interpretation of de Finetti’s theory that highlights (...) these aspects and explains how they are an integral part of de Finetti’s instrumentalist philosophy of probability. I conclude by drawing an analogy between misconceptions about de Finetti’s philosophy of probability and common misconceptions about instrumentalism. (shrink)

The arguments for Bayesianism in the literature fall into three broad categories. There are Dutch Book arguments, both of the traditional pragmatic variety and the modern ‘depragmatised’ form. And there are arguments from the so-called ‘representation theorems’. The arguments have many similarities, for example they have a common conclusion, and they all derive epistemic constraints from considerations about coherent preferences, but they have enough differences to produce hostilities between their proponents. In a recent paper, Maher (1997) has argued that the (...) pragmatised Dutch Book arguments are unsound and the depragmatised Dutch Book arguments question begging. He urges we instead use the representation theorem argument as in his (1993). In this paper I argue that Maher’s own argument is question-begging, though in a more subtle and interesting way than his Dutch Book wielding opponents. (shrink)

The foundations of probability are viewed through the lens of the subjectivist interpretation. This article surveys conditional probability, arguments for probabilism, probability dynamics, and the evidential and subjective interpretations of probability.

Wayne (1995) critiques the Bayesian explication of the confirmational significance of evidential diversity (CSED) offered by Horwich (1982). Presently, I argue that Wayne’s reconstruction of Horwich’s account of CSED is uncharitable. As a result, Wayne’s criticisms ultimately present no real problem for Horwich. I try to provide a more faithful and charitable rendition of Horwich’s account of CSED. Unfortunately, even when Horwich’s approach is charitably reconstructed, it is still not completely satisfying.

How does deductive logic constrain probability? This question is difficult for subjectivistic approaches, according to which probability is just strength of (prudent) partial belief, for this presumes logical omniscience. This paper proposes that the way in which probability lies always between possibility and necessity can be made precise by exploiting a minor theorem of de Finetti: In any finite set of propositions the expected number of truths is the sum of the probabilities over the set. This is generalized to apply (...) to denumerable languages. It entails that the sum of probabilities can neither exceed nor be exceeded by the cardinalities of all consistent and closed (within the set) subsets. In general any numerical function on sentences is said to be logically coherent if it satisfies this condition. Logical coherence allows the relativization of necessity: A function p on a language is coherent with respect to the concept T of necessity iff there is no set of sentences on which the sum of p exceeds or is exceeded by the cardinality of every T-consistent and T-closed (within the set) subset of the set. Coherence is easily applied as well to sets on which the sum of p does not converge. Probability should also be relativized by necessity: A T-probability assigns one to every T-necessary sentence and is additive over disjunctions of pairwise T-incompatible sentences. Logical T-coherence is then equivalent to T-probability: All and only T-coherent functions are T-probabilities. (shrink)

Many people regard utility theory as the only rigorous foundation for subjective probability, and even de Finetti thought the betting approach supplemented by Dutch Book arguments only good as an approximation to a utility-theoretic account. I think that there are good reasons to doubt this judgment, and I propose an alternative, in which the probability axioms are consistency constraints on distributions of fair betting quotients. The idea itself is hardly new: it is in de Finetti and also Ramsey. What is (...) new is that it is shown that probabilistic consistency and consequence can be defined in a way formally analogous to the way these notions are defined in deductive (propositional) logic. The result is a free-standing logic which does not pretend to be a theory of rationality and is therefore immune to, among other charges, that of “logical omniscience”. (shrink)

Bogen and Woodward (1988) advance adistinction between data and phenomena. Roughly, theformer are the observations reported by experimentalscientists, the latter are objective, stable featuresof the world to which scientists infer based onpatterns in reliable data. While phenomena areexplained by theories, data are not, and so theempirical basis for an inference to a theory consistsin claims about phenomena. McAllister (1997) hasrecently offered a critique of their version of thisdistinction, offering in its place a version on whichphenomena are theory laden, and hence (...) on which theempirical support for inferences to theories is also,unavoidably, theory laden. In this commentary I arguethat McAllister and Bogen and Woodward are mistaken inthinking that the distinction is necessary, and thatthe empirical support for inferences to theories isnot necessarily theory laden in the way McAllister'saccount entails they are. (shrink)

In this paper, I present a simple and straightforward logic of induction: a consequence relation characterized by a proof theory and a semantics. This system will be called LI. The premises will be restricted to, on the one hand, a set of empirical data and, on the other hand, a set of background generalizations. Among the consequences will be generalizations as well as singular statements, some of which may serve as predictions and explanations.

One justification for preregistering research hypotheses, methods, and analyses is that it improves the transparent evaluation of the severity of hypothesis tests. In this article, I consider two cases in which preregistration does not improve this evaluation. First, I argue that, although preregistration can facilitate the transparent evaluation of severity in Mayo’s error statistical philosophy of science, it does not facilitate this evaluation in Popper’s theory-centric approach. To illustrate, I show that associated concerns about Type I error rate inflation are (...) only relevant in the error statistical approach and not in a theory-centric approach. Second, I argue that a preregistered test procedure that allows deviations in its implementation does not provide a more transparent evaluation of Mayoian severity than a non-preregistered procedure. In particular, I argue that sample-based validity-enhancing deviations cause an unknown inflation of the test procedure’s Type I (familywise) error rate and, consequently, an unknown reduction in its capability to license inferences severely. I conclude that preregistration does not improve the transparent evaluation of severity in Popper’s philosophy of science or when deviations are allowed. (shrink)

En este trabajo hago una comparación sistemática entre las dos teorías de la confirmación más populares en la actualidad: el método hipotético-deductivo y el bayesianismo. En primer lugar, enumero los cinco problemas fundamentales de la teoría hipotético-deductivista. Estos son el problema de las hipótesis estadísticas, el del grado de confirmación, el de la conjunción irrelevante, el del holismo epistemológico y el de las hipótesis alternativas. Luego, hago una presentación general de la epistemología bayesiana y muestro de qué manera estos problemas (...) resultan tratables en el marco bayesiano. Finalmente, concluyo que los tres primeros admiten una solución satisfactoria, sobre la base de algunos supuestos razonables. Los dos últimos, en cambio, todavía son problemas abiertos para la teoría bayesiana de la confirmación. Bayesians often affirm they have produced the best available theory of confirmation. In particular, they claim to have superseded the venerable hypothetico-deductive method. In this paper I intend to assess that claim by making a systematic comparison between both theories of confirmation. I begin by listing the main problems of hypothetico-deductive confirmation. I then show that this conception is incompatible with Hempel's conditions of adequacy for qualitative confirmation. I expound the Bayesian theory of confirmation with some detail in order to show that this theory is able to solve three out of the five fundamental problems of hypothetico-deductive confirmation. largue that the remaining two problems are still open questions for the Bayesians. (shrink)

Against Hellman's (1997) recent claims, I argue that Bayesianism is unable to explain the value of generally successful aspects of scientific methodology, viz., deflecting blame from well-confirmed theories onto auxiliaries and preferring more-varied data. Such an explanation would require not just objectification of priors, but a reason to believe priors will generally fall on values that justify the practice. Given the track record on the objectification problem, adding further conditions on priors merely makes the Bayesian's problems even worse.

The notion of a severe test has played an important methodological role in the history of science. But it has not until recently been analyzed in any detail. We develop a generally Bayesian analysis of the notion, compare it with Deborah Mayo’s error-statistical approach by way of sample diagnostic tests in the medical sciences, and consider various objections to both. At the core of our analysis is a distinction between evidence and confirmation or belief. These notions must be kept separate (...) if mistakes are to be avoided; combined in the right way, they provide an adequate understanding of severity. Those who think that the weight of the evidence always enables you to choose between hypotheses “ignore one of the factors (the prior probability) altogether, and treat the other (the likelihood) as though it ...meant something other than it actually does. This is the same mistake as is made by someone who has scruples about measuring the arms of a balance (having only a tape measure at his disposal ...), but is willing to assert that the heavier load will always tilt the balance (thereby implicitly assuming, although without admitting it, that the arms are of equal length!). (Bruno de Finetti, Theory of Probability)2. (shrink)