Branden Fitelson and Alan Hájek have suggested that it is finally time for a “revolution” in which we jettison Kolmogorov’s axiomatization of probability, and move to an alternative like Popper’s. According to these authors, not only did Kolmogorov fail to give an adequate analysis of conditional probability, he also failed to give an adequate account of another central notion in probability theory: probabilistic independence. This paper defends Kolmogorov, with a focus on this independence charge. I show that (...) Kolmogorov’s sophisticated theory of conditional distributions, advocated by authors such as Kenny Easwaran but often overlooked by philosophers, has the resources to define the notion of independence Fitelson and Hájek seek. I then argue that Kolmogorov’s original definition of independence, once suitably reinterpreted as capturing a slight logical weakening of this notion, is not so problematic after all. Indeed, rather than discarding Kolmogorov’s definition, we should make room for two distinct notions of probabilistic independence, one stronger and one weaker, which play different theoretical roles. (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)

It is widely held that, as Morgenbesser’s case is usually taken to show, considerations of causal or probabilistic dependence should enter into the evaluation of counterfactuals. This paper challenges that idea. I present a modified version of Morgenbesser’s case and show how probabilistic approaches to counterfactuals are in serious trouble. Specifically, I show how probabilistic approaches run into a dilemma in characterizing probabilistic independence. The modified case also illustrates a difficulty in defining causal independence. (...) I close with a suggestion for a strategy to handle this difficulty. (shrink)

Naive deductivist accounts of confirmation have the undesirable consequence that if E confirms H, then E also confirms the conjunction H·X, for any X—even if X is completely irrelevant to E and H. Bayesian accounts of confirmation may appear to have the same problem. In a recent article in this journal Fitelson (2002) argued that existing Bayesian attempts to resolve of this problem are inadequate in several important respects. Fitelson then proposes a new‐and‐improved Bayesian account that overcomes the problem of (...) irrelevant conjunction, and does so in a more general setting than past attempts. We will show how to simplify and improve upon Fitelson's solution. (shrink)

The main goal of this work is to present the proof-theoretical and model-theoretical approaches to probabilistic logics which allow reasoning about independence and probabilistic support. We extend the existing formalisms [14] to obtain several variants of probabilistic logics by adding the operators for independence and confirmation to the syntax. We axiomatize these logics, provide corresponding semantics, prove that the axiomatizations are sound and strongly complete, and discuss decidability issues.

Suppose several individuals (e.g., experts on a panel) each assign probabilities to some events. How can these individual probability assignments be aggregated into a single collective probability assignment? This article reviews several proposed solutions to this problem. We focus on three salient proposals: linear pooling (the weighted or unweighted linear averaging of probabilities), geometric pooling (the weighted or unweighted geometric averaging of probabilities), and multiplicative pooling (where probabilities are multiplied rather than averaged). We present axiomatic characterisations of each class of (...) pooling functions (most of them classic, but one new) and argue that linear pooling can be justified procedurally, but not epistemically, while the other two pooling methods can be justified epistemically. The choice between them, in turn, depends on whether the individuals' probability assignments are based on shared information or on private information. We conclude by mentioning a number of other pooling methods. (shrink)

The epistemic modal auxiliaries must and might are vehicles for expressing the force with which a proposition follows from some body of evidence or information. Standard approaches model these operators using quantificational modal logic, but probabilistic approaches are becoming increasingly influential. According to a traditional view, must is a maximally strong epistemic operator and might is a bare possibility one. A competing account—popular amongst proponents of a probabilisitic turn—says that, given a body of evidence, must \ entails that \\) (...) is high but non-maximal and might \ that \\) is significantly greater than 0. Drawing on several observations concerning the behavior of must, might and similar epistemic operators in evidential contexts, deductive inferences, downplaying and retractions scenarios, and expressions of epistemic tension, I argue that those two influential accounts have systematic descriptive shortcomings. To better make sense of their complex behavior, I propose instead a broadly Kratzerian account according to which must \ entails that \ = 1\) and might \ that \ > 0\), given a body of evidence and a set of normality assumptions about the world. From this perspective, must and might are vehicles for expressing a common mode of reasoning whereby we draw inferences from specific bits of evidence against a rich set of background assumptions—some of which we represent as defeasible—which capture our general expectations about the world. I will show that the predictions of this Kratzerian account can be substantially refined once it is combined with a specific yet independently motivated ‘grammatical’ approach to the computation of scalar implicatures. Finally, I discuss some implications of these results for more general discussions concerning the empirical and theoretical motivation to adopt a probabilisitic semantic framework. (shrink)

How can different individuals' probability functions on a given sigma-algebra of events be aggregated into a collective probability function? Classic approaches to this problem often require 'event-wise independence': the collective probability for each event should depend only on the individuals' probabilities for that event. In practice, however, some events may be 'basic' and others 'derivative', so that it makes sense first to aggregate the probabilities for the former and then to let these constrain the probabilities for the latter. We (...) formalize this idea by introducing a 'premise-based' approach to probabilistic opinion pooling, and show that, under a variety of assumptions, it leads to linear or neutral opinion pooling on the 'premises'. This paper is the second of two self-contained, but technically related companion papers inspired by binary judgment-aggregation theory. (shrink)

Moss (2018) argues that rational agents are best thought of not as having degrees of belief in various propositions but as having beliefs in probabilistic contents, or probabilistic beliefs. Probabilistic contents are sets of probability functions. Probabilistic belief states, in turn, are modeled by sets of probabilistic contents, or sets of sets of probability functions. We argue that this Mossean framework is of considerable interest quite independently of its role in Moss’ account of probabilistic (...) knowledge or her semantics for epistemic modals and probability operators. It is an extremely general model of uncertainty. Indeed, it is at least as general and expressively powerful as every other current imprecise probability framework, including lower probabilities, lower previsions, sets of probabilities, sets of desirable gambles, and choice functions. In addition, we partially answer an important question that Moss leaves open, viz., why should rational agents have consistent probabilistic beliefs? We show that an important subclass of Mossean believers avoid Dutch bookability iff they have consistent probabilistic beliefs. (shrink)

This paper discusses two versions of reliabilism: modal and probabilistic reliabilism. Modal reliabilism faces the problem of the missing closeness metric for possible worlds while probalistic reliabilism faces the problem of the relevant reference class. Despite the severity of these problems, reliabilism is still very plausible (also for independent reasons). I propose to stick with reliabilism, propose a contextualist (or, alternatively, harmlessly relativist) solution to the above problems and suggest that probabilistic reliabilism has the advantage over modal reliabilism.

Two recent and influential papers, van Rooij 2007 and Lassiter 2012, propose solutions to the proviso problem that make central use of related notions of independence—qualitative in the first case, probabilistic in the second. We argue here that, if these solutions are to work, they must incorporate an implicit assumption about presupposition accommodation, namely that accommodation does not interfere with existing qualitative or probabilistic independencies. We show, however, that this assumption is implausible, as updating beliefs with conditional (...) information does not in general preserve independencies. We conclude that the approach taken by van Rooij and Lassiter does not succeed in resolving the proviso problem. (shrink)

While one strand of ramified natural theology focuses on direct evidence for mira­cles, another avenue to investigate is the argument from prophecy. Events that appear to fulfill prophecy may not be miraculous in themselves, but they can provide confirmation, even substantial confirmation, for a supernatural hypothesis. I examine the details of a small set of passages from the Old Testament and evaluate the probabilistic impact of the occurrence of events surrounding the death of Jesus of Nazareth that appear to (...) fulfill these prophecies. The hypothesis under consideration is M—that Jesus of Nazareth was the prophesied Messiah. Using Psalm 22 and Isaiah 53, historical evidence concerning the death of Jesus, and background evidence concerning Roman and Jewish history and culture, I estimate a cumulative Bayes factor of 2.5 × 107 in favor of M from the fact of Jesus’s crucifixion and four further details concerning his death. Independent confirmation of M is pertinent to the prior probability of miraculous claims such as the claim that Jesus rose from the dead. The examination of Jesus’s putative fulfillment of prophecy thus is an example of an objective treatment of the religious context of a miracle which makes a given putative miracle something other than an isolated and arbitrary wonder. (shrink)

The aim of the paper is to develop the notion of partial probability distributions as being more realistic models of belief systems than the standard accounts. We formulate the theory of partial probability functions independently of any classical semantic notions. We use the partial probability distributions to develop a formal semantics for partial propositional calculi, with extensions to predicate logic and higher order languages. We give a proof theory for the partial logics and obtain soundness and completeness results.

This work is in two parts. The main aim of part 1 is a systematic examination of deductive, probabilistic, inductive and purely inductive dependence relations within the framework of Kolmogorov probability semantics. The main aim of part 2 is a systematic comparison of (in all) 20 different relations of probabilistic (in)dependence within the framework of Popper probability semantics (for Kolmogorov probability semantics does not allow such a comparison). Added to this comparison is an examination of (in all) 15 (...) purely inductive dependence relations. ————Part 1 leads in an axiomatic step-by-step development from the elementary classical truth value semantics of a sentential-logical language, called ‘L’, (chapter 1) to the elementary Kolmogorov probability semantics of L (chapter 2), which is then extended to four axiomatic semantical theories of dependence relations between the formulae of L. First the elementary Kolmogorov probability semantics of L is extended to a theory, called ‘Kdd’, of the relations of deductive dependence and deductive independence between formulae of L (chapter 3). Then Kdd is extended to a theory, called ‘Kpd1’, of the degree to which formulae of L probabilistically depend on each other in regard to a given probability distribution on the set of all formulae of L (chapter 4). Kpd1, in its turn, gets extended to a theory, called ‘Kpd2’, of the relations of probabilistic dependence and independence, relativized to unary Kolmogorov probability functions defined on L (chapter 5). Then Kpd2 is extended to a theory, called ‘Kid’, of the relations of inductive dependence and inductive independence, again relativized to unary Kolmogorov probability functions defined on L (chapter 6). Finally, Kid is extended to a theory, called ‘Kpid’, of the relations of purely inductive positive and negative dependence, relativized to unary Kolmogorov probability functions defined on L (chapter 7). ——Chapter 1, which deals with the familiar notions of truth value functions, tautologies, consequence relations and relations of logical opposition, is naturally the shortest chapter of part 1.——In chapter 2, the elementary classical semantics of L is extended to the elementary Kolmogorov probability semantics of L, i.e. to an axiomatic theory of unary and of binary Kolmogorov probability functions defined on the set of formulae of L. Because of the elementary character of this theory, chapter 2 is also rather short.——Chapter 3 introduces the first theory on dependence relations, to wit: Kdd, the theory of deductive (in)dependence between formulae of L. I follow here the well-known idea of Popper and Miller, who have used it in a famous discussion on the nature of probabilistic support for their arguments that probabilistic support is deductive, not inductive. I develop Kdd in the form of about 100 theorems, making ample use of the fact that deductive independence is nothing but subcontrary opposition, and close with a remark on the fundamental difference between deductive and logical dependence—two relations the ideas of which are all too easily mixed up.——Chapters 4 and 5 deal extensively with the traditional ideas of probabilistic (in)dependence, applied to formulae rather than to events. As always, I proceed axiomatically in a step-by-step process under systematic viewpoints and obtain about 300 theorems in this way. In the formulation of the theorems, I took special care to state clearly and expressly so-called tacit assumptions, especially those concerning the probability values of the formulae said to be dependent on each other. These assumptions are usually missing in the literature, due either to economy of writing or to sloppiness of thinking. Presumably, both chapters contain little that is new, their value lying more in the systematic grouping and organic development of the theorems than in the newness of these.——In chapter 6, I extend the axiomatic theory about probabilistic (in)dependence which has been elaborated in chapter 5, to an axiomatic theory of inductive (in)dependence by requiring of the relation of inductive (in)dependence that it be probabilistic (in)dependence, but not also logical implication or logical opposition. I point out the differences between probabilistic and inductive (in)dependence by means of some 60 theorems and close my examination of inductive (in)dependence by considering its relationship to the notion of support in the philosophy of science.——Finally, in chapter 7, the last of part 1, I take the step from inductive dependence to what I call ‘purely inductive dependence’ by combining the idea of inductive dependence with that of deductive independence in a way which is suggested by writings of Popper and Miller. I arrive at two noteworthy theorems. Firstly, there is indeed no purely inductive support. But secondly, and perhaps amazingly, countersupport is purely inductive.————Whereas the probabilistic framework of part 1 of the present work is Kolmogorov probability semantics, the framework of part 2 is Popper probability semantics, which is not only worth examining as a fascinating alternative to orthodox Kolmogorov probability semantics, but also allows us to examine dependence relations more deeply, than Kolmogorov probability semantics does. Part 2 leads—again in an axiomatic step-by-step development—from the basic Popper probability semantics of L, called ‘Pb’, (chapter 8) via a probabilistic theory of logical attributes, called ‘Ps’, (chapter 9) to four axiomatic semantical theories of dependence relations between the formulae of L. First, Ps is extended to a theory, called ‘Pdd’, of the relations of deductive dependence and deductive independence between formulae of L (chapter 10). Then Pdd is extended to a theory, called ‘Ppd’, of (in all) 20 relations of probabilistic (in)dependence, relativized to binary Popper probability functions defined on L (chapter 11). Ppd, in its turn, is extended to a theory, called ‘Pid’, of (in all) 10 relations of inductive dependence, again relativized to binary Popper probability functions defined on L (first part of chapter 12). Finally, Pid is extended to a theory, called ‘Ppid’, of (in all) 15 relations of purely inductive positive and negative dependence, relativized to binary Popper probability functions defined on L (second part of chapter 12).——Chapter 8, the first chapter of part 2 of the present work, is entirely preparatory. It introduces the axioms and about 180 theorems (150 of them together with their proofs) of basic Popper probability semantics in order to set this kind of semantics under way.——Then, in chapter 9, basic Popper probability semantics is extended to a probabilistic theory of logical properties of and relations between the formulae of L. Although I think that the way I did this extension is of some interest in itself, the main task of chapter 9 is again a preparatory one: to yield the indispensable lemmata (about 90 in number) for the theorems concerning probabilistic dependence relations in chapter 11 and concerning inductive dependence relations in chapter 12.——Chapter 10 brings the extension of Ps to the theory Pdd of deductive (in)dependence. Only half a dozen theorems are noted here for later use in the Pdd-extensions Ppd and Ppid. In view of the over 100 theorems already gained on this topic in the Kolmogorovian framework (cf. chapter 3), a similar extensive elaboration of Pdd would have been superfluous.——Chapter 11 is the most important one of part 2. It consists of a systematic comparison of 20 probabilistic (in)dependence concepts by means of about 230 theorems, obtained within the axiomatic theory Ppd, which is built up as an extension of Pdd. The main points of comparison were: differences in logical strength; reflexivity and symmetry; behaviour under the condition that the probability values of the formulae in question are extreme. It turned out that each of the examined concepts violates a strong and straightforward version of the intuitive requirement that probabilistic dependence should go with logical dependence. Whereas the corresponding chapter 5 in part 1 of the present work may not have led to new theorems, chapter 11 yields dozens of them in the process of comparison of concepts of dependence and independence which had—as far as I know—never before been treated in a single theoretical framework. With Popper probability semantics, this framework has become available, and here I have simply made full use of it.——In chapter 12, I extend the theory Ppd of probabilistic (in)dependence to the theories Pid and Ppid of inductive and purely inductive dependence, in a way very similar to that in which I have extended the theory Kpd2 to the theories Kid and Kpid in chapters 6 and 7. The first main result of Kpid (roughly: there is no purely inductive support) could be repeated for four of the five purely inductive positive dependence relations considered in chapter 12, whereas the second main result of Kpid (roughly: there is purely inductive countersupport ) could be repeated for each of the five examined purely inductive negative dependence relations. Chapter 12 closes with a brief recapitulation and critical discussion of the main results. (shrink)

One of the many intriguing features of the axiomatic systems of probability investigated in Popper (1959), appendices _iv, _v, is the different status of the two arguments of the probability functor with regard to the laws of replacement and commutation. The laws for the first argument, (rep1) and (comm1), follow from much simpler axioms, whilst (rep2) and (comm2) are independent of them, and have to be incorporated only when most of the important deductions have been accomplished. It is plain that, (...) in the presence of (comm1), the principle (sub), which says that terms that are intersubstitutable in the first argument are intersubstitutable also in the second argument, implies (comm2), and in Popper’s systems the converse implication obtains. It is naturally asked what is needed in an axiomatic theory of probability in order to enforce this equivalence. Leblanc (1981) offered a rather weak set of axioms, containing (comm1) and (comm2), that suffice for the derivation of (sub). In this paper Leblanc’s result is improved in a number of different ways. Three weaker systems, one of which is incomparable with the other two, are shown to admit the same implication. DOI:10.5007/1808-1711.2011v15n2p271. (shrink)

On explanationist accounts of genealogical debunking, roughly, a belief is debunked when its explanation is not suitably related to its content. We argue that explanationism cannot accommodate cases in which beliefs are explained by factors unrelated to their contents but are nonetheless independently justified. Justification-specific versions of explanationism face an iteration of the problem. The best account of debunking is a probabilistic account according to which subject S’s justification J for their belief that P is debunked when S learns (...) that J is no more likely to be true on the hypothesis that P than on the hypothesis that ¬P. The probabilistic criterion is fully general, applying not only to cases where the learned undercutting defeater is a proposition about our beliefs or other mental states but to any case of undercutting defeat, providing the grounds for a debunking argument against the existence of a special, metacognitive debunking principle. (shrink)

We use the method of fixed points to describe a form of probabilistic truth approximation which we illustrate by means of three examples. We then contrast this form of probabilistic truth approximation with another, more familiar kind, where no fixed points are used. In probabilistic truth approximation with fixed points the events are dependent on one another, but in the second kind they are independent. The first form exhibits a phenomenon that we call ‘fading origins’, the second (...) one is subject to a phenomenon known as ‘the washing out of the prior’. We explain that the two phenomena may seem very similar, but are in fact quite different. (shrink)

Robustness arguments hold that hypotheses are more likely to be true when they are confirmed by diverse kinds of evidence. Robustness arguments require the confirming evidence to be independent. We identify two kinds of independence appealed to in robustness arguments: ontic independence —when the multiple lines of evidence depend on different materials, assumptions, or theories—and probabilistic independence. Many assume that OI is sufficient for a robustness argument to be warranted. However, we argue that, as typically construed, (...) OI is not a sufficient independence condition for warranting robustness arguments. We show that OI evidence can collectively confirm a hypothesis to a lower degree than individual lines of evidence, contrary to the standard assumption undergirding usual robustness arguments. We employ Bayesian networks to represent the ideal empirical scenario for a robustness argument and a variety of ways in which empirical scenarios can fall short of this ideal. (shrink)

By virtue of what do alarm calls and facial expressions carry natural information? The answer I defend in this paper is that they carry natural information by virtue of changing the probabilities of various states of affairs, relative to background data. The Probabilistic Difference Maker Theory of natural information that I introduce here is inspired by Dretske's [1981] seminal analysis of natural information, but parts ways with it by eschewing the requirements that information transmission must be nomically underwritten, mind-independent, (...) and knowledge-yielding. PDMT includes both a qualitative account of information transmission and a measure of natural information in keeping with the basic principles of Shannon's communication theory and Bayesian confirmation theory. It also includes a new account of the informational content of a signal, understood as the combination of the incremental and overall support that the signal provides for all states of affairs at the source. Finally, I compare and.. (shrink)

This paper calls for a re-appraisal of McGee's analysis of the semantics, logic and probabilities of indicative conditionals presented in his 1989 paper Conditional probabilities and compounds of conditionals. The probabilistic measures introduced by McGee are given a new axiomatisation built on the principle that the antecedent of a conditional is probabilistically independent of the conditional and a more transparent method of constructing such measures is provided. McGee's Dutch book argument is restructured to more clearly reveal that it introduces (...) a novel contribution to the epistemology of semantic indeterminacy, and shows that its more controversial implications are unavoidable if we want to maintain the Ramsey Test along with the standard laws of probability. Importantly, it is shown that the counterexamples that have been levelled at McGee's analysis|generating a rather wide consensus that it yields `unintuitive' or `wrong' probabilities for compounds |fail to strike at their intended target; for to honour the intuitions of the counterexamples one must either give up the Ramsey Test or the standard laws of probability. It will be argued that we need to give up neither if we take the counterexamples as further evidence that the indicative conditional sometimes allows for a non-epistemic `causal' interpretation alongside its usual epistemic interpretation. (shrink)

Bonjour (1985: 101 and 1999: 124) and other coherence theorists of justification before him (e.g. Ewing, 1934: 246) have complained that we do not have a satisfactory analysis of the notion of coherence. The problem with existing accounts of coherence is that they try to bring precision to our intuitive notion of coherence independently of the particular role that it is meant to play within the coherence theory of justification (e.g Lewis, 1946: 338). This is a mistake: it does not (...) make any sense to ask what precisely makes for a more coherent information set independently of the particular role that coherence is supposed to play within the context in question. What is this context and what is this role? The coherence theory of justification rides on a particular common sense intuition: when we gather information from less than fully reliable sources, then the more coherent the story that materializes is, the more confident we may be, ceteris paribus. Within the context of information gathering from certain types of sources, coherence is a property of stories which plays a confidence boosting role. But what features should the information sources have, so that the coherence of the information set is indeed a determinant of our degree of confidence in question? And what goes into the ceteris paribus clause? In other words, what other factors affect our confidence in the information set in question? (shrink)

Democratic decision-making is often defended on grounds of the ‘wisdom of crowds’: decisions are more likely to be correct if they are based on many independent opinions, so a typical argument in social epistemology. But what does it mean to have independent opinions? Opinions can be probabilistically dependent even if individuals form their opinion in causal isolation from each other. We distinguish four probabilistic notions of opinion independence. Which of them holds depends on how individuals are causally affected (...) by environmental factors such as commonly perceived evidence. In a general theorem, we identify causal conditions guaranteeing each kind of opinion independence. These results have implications for whether and how ‘wisdom of crowds’ arguments are possible, and how truth-conducive institutions can be designed. (shrink)

We consider two ways one might use algorithmic randomness to characterize a probabilistic law. The first is a generative chance* law. Such laws involve a nonstandard notion of chance. The second is a probabilistic* constraining law. Such laws impose relative frequency and randomness constraints that every physically possible world must satisfy. While each notion has virtues, we argue that the latter has advantages over the former. It supports a unified governing account of non-Humean laws and provides independently motivated (...) solutions to issues in the Humean best-system account. On both notions, we have a much tighter connection between probabilistic laws and their corresponding sets of possible worlds. Certain histories permitted by traditional probabilistic laws are ruled out as physically impossible. As a result, such laws avoid one variety of empirical underdetermination, but the approach reveals other varieties of underdetermination that are typically overlooked. (shrink)

Two of the probabilistic measures of coherence discussed in this paper take probabilistic dependence into account and so depend on prior probabilities in a fundamental way. An example is given which suggests that this prior-dependence can lead to potential problems. Another coherence measure is shown to be independent of prior probabilities in a clearly defined sense and consequently is able to avoid such problems. The issue of prior-dependence is linked to the fact that the first two measures can (...) be understood as measures of coherence as striking agreement, while the third measure represents coherence as agreement. Thus, prior (in)dependence can be used to distinguish different conceptions of coherence. (shrink)

We introduce an independence notion for choice functions, which we call ‘epistemic independence’ following the work by De Cooman et al. [17] for lower previsions, and study it in a multivariate setting. This work is a continuation of earlier work of one of the authors [29], and our results build on the characterization of choice functions in terms of sets of binary preferences recently established by De Bock and De Cooman [11]. We obtain the many-to-one independent natural extension (...) in this framework. Given the generality of choice functions, our expression for the independent natural extension is the most general one we are aware of, and we show how it implies the independent natural extension for sets of desirable gambles, and therefore also for less expressive imprecise-probabilistic models. Once this is in place, we compare this concept of epistemic independence to another independence concept for choice functions proposed by Seidenfeld [28], which De Bock and De Cooman [2] have called S-independence. We show that neither is more general than the other. (shrink)

One of the many intriguing features of the axiomatic systems of probability investigated in Popper (1959), appendices _iv, _v, is the different status of the two arguments of the probability functor with regard to the laws of replacement and commutation. The laws for the first argument, (rep1) and (comm1), follow from much simpler axioms, whilst (rep2) and (comm2) are independent of them, and have to be incorporated only when most of the important deductions have been accomplished. It is plain that, (...) in the presence of (comm1), the principle (sub), which says that terms that are intersubstitutable in the first argument are intersubstitutable also in the second argument, implies (comm2), and in Popper’s systems the converse implication obtains. It is naturally asked what is needed in an axiomatic theory of probability in order to enforce this equivalence. Leblanc (1981) offered a rather weak set of axioms, containing (comm1) and (comm2), that suffice for the derivation of (sub). In this paper Leblanc’s result is improved in a number of different ways. Three weaker systems, one of which is incomparable with the other two, are shown to admit the same implication. (shrink)

According to orthodox (Kolmogorovian) probability theory, conditional probabilities are by definition certain ratios of unconditional probabilities. As a result, orthodox conditional probabilities are undefined whenever their antecedents have zero unconditional probability. This has important ramifications for the notion of probabilistic independence. Traditionally, independence is defined in terms of unconditional probabilities (the factorization of the relevant joint unconditional probabilities). Various “equivalent” formulations of independence can be given using conditional probabilities. But these “equivalences” break down if conditional probabilities (...) are permitted to have conditions with zero unconditional probability. We reconsider probabilistic independence in this more general setting. We argue that a less orthodox but more general (Popperian) theory of conditional probability should be used, and that much of the conventional wisdom about probabilistic independence needs to be rethought. (shrink)

The aim of the paper is to explicate the concept of causal independence between sets of factors and Reichenbach's screening-off-relation in probabilistic terms along the lines of Suppes' probabilistic theory of causality (1970). The probabilistic concept central to this task is that of conditional stochastic independence. The adequacy of the explication is supported by proving some theorems about the explicata which correspond to our intuitions about the explicanda.

Experimental studies suggest that individuals exhibit more risk aversion in choices among prospects when the payment and resolution of uncertainty are immediate relative to when it is delayed. This leads to preference reversals that cannot be attributed to discounting. When data suggest that utility is time-independent, probability weighting functions, such as those used to model prospect theory preferences, can accommodate such reversals. We propose a simple descriptive model with a two-parameter probability weighting function where one of these parameters depends on (...) the time at which a prospect is resolved. The time-dependent parameter is responsible for the curvature of the probability weighting function and is regarded as an index of sensitivity towards changes in probabilities. We provide conditions that characterize increased sensitivity towards more distant probabilities; this can account for the observed relatively less risk aversion towards delayed prospects. In our framework, the discount function is unrestricted, such that the model is compatible with empirical findings of non-constant discounting. In a simple application to bargaining we illustrate when it is advantageous for an individual to advance or delay the bargaining resolution time if an opponent displays increased sensitivity towards probability changes with delay. (shrink)

We investigate epistemic independence for choice functions in a multivariate setting. This work is a continuation of earlier work of one of the authors [23], and our results build on the characterization of choice functions in terms of sets of binary preferences recently established by De Bock and De Cooman [7]. We obtain the independent natural extension in this framework. Given the generality of choice functions, our expression for the independent natural extension is the most general one we are (...) aware of, and we show how it implies the independent natural extension for sets of desirable gambles, and therefore also for less informative imprecise-probabilistic models. Once this is in place, we compare this concept of epistemic independence to another independence concept for choice functions proposed by Seidenfeld [22], which De Bock and De Cooman [1] have called S-independence. We show that neither is more general than the other. (shrink)

Here I critically assess an argument put forward by Kuorikoski et al. (Br J Philos Sci, 61(3):541–567, 2010) for the epistemic import of model-based robustness analysis. I show that this argument is not sound since the sort of probabilistic independence on which it relies is unfeasible. By revising the notion of probabilistic independence imposed on the models’ results, I introduce a prima-facie more plausible argument. However, despite this prima-facie plausibility, I show that even this new argument (...) is unsound in most if not all cases of model-based robustness analysis. This I do to demonstrate that the epistemic import of model-based robust analysis cannot be satisfactorily defended on the basis of probabilistic independence. (shrink)

The probabilistic corroboration of two or more hypotheses or series of observations may be performed additively or multiplicatively . For additive corroboration (e.g. by Laplace's rule of succession), stochastic independence is needed. Inferences, based on overwhelming numbers of observations without unexplained counterinstances permit hyperinduction , whereby extremely high probabilities, bordering on certainty for all practical purposes may be achieved. For multiplicative corroboration, the error probabilities (1 - Pr) of two (or more) hypotheses are multiplied. The probabilities, obtained by (...) reconverting the product, are valid for both of the hypotheses and indicate the gain by corroboration.. This method is mathematically correct, no probabilities > 1 can result (as in some conventional methods) and high probabilities with fewer observations may be obtained, however, semantical independence is a prerequisite. The combined method consists of (1) the additive computation of the error probabilities (1 - Pr) of two or more single hypotheses, whereby arbitrariness is avoided or at least reduced and (2) the multiplicative procedure . The high reliability of Empirical Counterfactual Statements is explained by the possibility of multiplicative corroboration of "all-no" statements due to their strict semantical independence. (shrink)

Elementary results concerning the connections between deductive relations and probabilistic support are given. These are used to show that Popper-Miller's result is a special case of a more general result, and that their result is not very unexpected as claimed. According to Popper-Miller, a purely inductively supports b only if they are deductively independent — but this means that a b. Hence, it is argued that viewing induction as occurring only in the absence of deductive relations, as Popper-Miller sometimes (...) do, is untenable. Finally, it is shown that Popper-Miller's claim that deductive relations determine probabilistic support is untrue. In general, probabilistic support can vary greatly with fixed deductive relations as determined by the relevant Lindenbaum algebra. (shrink)

A Bayesian account of independent evidential support is outlined. This account is partly inspired by the work of C. S. Peirce. I show that a large class of quantitative Bayesian measures of confirmation satisfy some basic desiderata suggested by Peirce for adequate accounts of independent evidence. I argue that, by considering further natural constraints on a probabilistic account of independent evidence, all but a very small class of Bayesian measures of confirmation can be ruled out. In closing, another application (...) of my account to the problem of evidential diversity is also discussed. (shrink)

As stochastic independence is essential to the mathematical development of probability theory, it seems that any foundational work on probability should be able to account for this property. Bayesian decision theory appears to be wanting in this respect. Savage’s postulates on preferences under uncertainty entail a subjective expected utility representation, and this asserts only the existence and uniqueness of a subjective probability measure, regardless of its properties. What is missing is a preference condition corresponding to stochastic independence. To (...) fill this significant gap, the article axiomatizes Bayesian decision theory afresh and proves several representation theorems in this novel framework. (shrink)

This paper derives a mathematical expression giving the development of the probability of a scientific hypothesis with the number of confirming tests, as determined by Bayes's theorem, in a special case in which all the tests are "independent" of one another. The simple expression obtained shows clearly how the various factors influence the growth of the probability. The result is used to set a numerical lower bound on the probabilities representing the a priori beliefs of humans in generalizations that become (...) accepted. By making a comparison with the predictions of a "logical atomic" model in the case of physical laws, it is argued that humans have significant a priori "knowledge" in a weak sense. (shrink)

The present study assessed the degree to which probabilistic reasoning performance and thinking style influenced perception of risk and self-reported levels of terrorism-related behaviour change. A sample of 263 respondents, recruited via convenience sampling, completed a series of measures comprising probabilistic reasoning tasks (perception of randomness, base rate, probability, and conjunction fallacy), the Reality Testing subscale of the Inventory of Personality Organization (IPO-RT), the Domain-Specific Risk-Taking Scale, and a terrorism-related behaviour change scale. Structural equation modelling examined three progressive (...) models. Firstly, the Independence Model assumed that probabilistic reasoning, perception of risk and reality testing independently predicted terrorism-related behaviour change. Secondly, the Mediation Model supposed that probabilistic reasoning and reality testing correlated, and indirectly predicted terrorism-related behaviour change through perception of risk. Lastly, the Dual-Influence Model proposed that probabilistic reasoning indirectly predicted terrorism-related behaviour change via perception of risk, independent of reality testing. Results indicated that performance on probabilistic reasoning tasks most strongly predicted perception of risk, and preference for an intuitive thinking style (measured by the IPO-RT) best explained terrorism-related behaviour change. The combination of perception of risk with probabilistic reasoning ability in the Dual-Influence Model enhanced the predictive power of the analytical-rational route, with conjunction fallacy having a significant indirect effect on terrorism-related behaviour change via perception of risk. The Dual-Influence Model possessed superior fit and reported similar predictive relations between intuitive-experiential and analytical-rational routes and terrorism-related behaviour change. The discussion critically examines these findings in relation to dual-processing frameworks. This includes considering the limitations of current operationalisations and recommendations for future research that align outcomes and subsequent work more closely to specific dual-process models. (shrink)

Definitions of syncopation share two characteristics: the presence of a meter or analogous hierarchical rhythmic structure and a displacement or contradiction of that structure. These attributes are translated in terms of a Bayesian theory of syncopation, where the syncopation of a rhythm is inferred based on a hierarchical structure that is, in turn, learned from the ongoing musical stimulus. Several experiments tested its simplest possible implementation, with equally weighted priors associated with different meters and independence of auditory events, which (...) can be decomposed into two terms representing note density and deviation from a metric hierarchy. A computational simulation demonstrated that extant measures of syncopation fall into two distinct factors analogous to the terms in the simple Bayesian model. Next, a series of behavioral experiments found that perceived syncopation is significantly related to both terms, offering support for the general Bayesian construction of syncopation. However, we also found that the prior expectations associated with different metric structures are not equal across meters and that there is an interaction between density and hierarchical deviation, implying that auditory events are not independent from each other. Together, these findings provide evidence that syncopation is a manifestation of a form of temporal expectation that can be directly represented in Bayesian terms and offer a complementary, feature‐driven approach to recent Bayesian models of temporal prediction. (shrink)

The article is a reappraisal of the requirement of maximal specificity (RMS) proposed by the author as a means of avoiding "ambiguity" in probabilistic explanation. The author argues that RMS is not, as he had held in one earlier publication, a rough substitute for the requirement of total evidence, but is independent of it and has quite a different rationale. A group of recent objections to RMS is answered by stressing that the statistical generalizations invoked in probabilistic explanations (...) must be lawlike, and by arguing that predicates fit for occurrence in lawlike statistical probability statements must meet two conditions, at least one of which is violated in each of the counterexamples adduced in the objections. These considerations suggest the conception that probabilistic-statistical laws concern the long-run frequency of some characteristic within a reference class as characterized by some particular "description" or predicate expression, and that replacement of such a description by a coextensive one may turn a statement that is lawlike into another that is not. Finally, to repair a defect noted by Grandy, the author's earlier formulation of RMS is replaced by a modified version. (shrink)

Some propositions are structurally unknowable for certain agents. Let me call them ‘Moorean propositions’. The structural unknowability of Moorean propositions is normally taken to pave the way towards proving a familiar paradox from epistemic logic—the so-called ‘Knowability Paradox’, or ‘Fitch’s Paradox’—which purports to show that if all truths are knowable, then all truths are in fact known. The present paper explores how to translate Moorean statements into a probabilistic language. A successful translation should enable us to derive a version (...) of Fitch’s Paradox in a probabilistic setting. I offer a suitable schematic form for probabilistic Moorean propositions, as well as a concomitant proof of a probabilistic Knowability Paradox. Moreover, I argue that traditional candidates to play the role of probabilistic Moorean propositions will not do. In particular, we can show that violations of the so-called ‘Reflection Principle’ in probability need not yield structurally unknowable propositions. Among other things, this should lead us to question whether violating the Reflection Principle actually amounts to a clear case of epistemic irrationality, as it is often assumed. This result challenges the importance of the principle as a tool to assess both synchronic and diachronic rationality—a topic which is largely independent of Fitch’s Paradox—from a somewhat unexpected source. (shrink)

The probabilistic support relation is known to violate transitivity. But over the years, philosophers have identified various conditions under which it does not, most notably screening-off and weak screening-off. In this short discussion note, I wish to highlight another condition that, unfortunately, is often neglected in the literature. This condition is due to Mary Hesse who recognized its transitivity-ensuring property long before other conditions entered the stage. I show that her condition is logically independent of screening-off and weak screening-off, (...) but that, just like the two aforementioned, it entails the recently proposed very weak screening-off condition due to Jeanne Peijnenburg and David Atkinson. (shrink)