Second or higher order probabilities have commonly been viewed with scepticism by those working within the realm of probability and decision theory. The aim of the present note is to show how the notion of second order probabilities can add to our understanding of judgmental and decision processes and how the traditional framework of Bayesian decision theory can be extended in a fruitful way by taking such entities into account. Section one consists of a brief (...) account of arguments put forth against higher order probabilities as well as of counterarguments. In order to provide an example of the applicability of second order probabilities a decision theory encompassing such probabilities will be presented in section two. In section three I will try to emphasize the value of second order probabilities for a deeper and more complete understanding of the notion of risk. (shrink)

Although it has often been claimed that all the information contained in second-order probabilities can be contained in first-order probabilities, no practical recipe for the elimination of second-order probabilities without loss of information seems to have been presented. Here, such an elimination method is introduced for repeatable events. However, its application comes at the price of losses in cognitive realism. In spite of their technical eliminability, second-order probabilities are useful because they can provide (...) models of important features of the world that are cognitively more plausible than those that can be obtained with single-level probabilities. (shrink)

The importance, legitimacy and role of second-order probabilities are discussed. Two descriptive models of the use of second-order probabilities in decisions are presented. The results of two empirical studies of the effects of second-order probabilities upon the rank orderings of bets are summarized briefly. The bets were of three basic types and involved a wide variety of first- and second-order probabilities as subjectively assessed by the subjects. Support was obtained for the assumption (...) that the majority of subjects make use of the one model or the other. It is suggested that greater attention should be paid to second-order probabilities, both from a normative and descriptive standpoint. (shrink)

The concepts of supportive evidence and of relevant evidence seem very closely related to each other. Supportive evidence is clearly always relevant as well. But must relevant evidence be defined as evidence which is either supportive or weakeking? In an explicit or implicit manner, this is indeed the position of many philosophers. The paradox of ideal evidence, however, shows us that this is to restrictive. Besides increasing or decreasing the probability attached to some hypothesis, evidence can alter or interact (...) with the background assumptions underlying the hypothesis.In most circumstances, the (post hoc) relevance of evidence can indeed be judged by its effect on the confidence one attaches to hypotheses. Occasionally, as in the circumstances described by Example I, and more generally called “the Paradox of Ideal Evidence”, the relevance of evidence to an hypothesis can only be understood by appeal to a broader sense. (shrink)

The last 20 years or so has seen an intense search carried out within Dempster–Shafer theory, with the aim of finding a generalization of the Shannon entropy for belief functions. In that time, there has also been much progress made in credal set theory—another generalization of the traditional Bayesian epistemic representation—albeit not in this particular area. In credal set theory, sets of probability functions are utilized to represent the epistemic state of rational agents instead of the single probability (...) function of traditional Bayesian theory. The Shannon entropy has been shown to uniquely capture certain highly intuitive properties of uncertainty, and can thus be considered a measure of that quantity. This article presents two measures developed with the purpose of generalizing the Shannon entropy for (1) unordered convex credal sets and (2) possibly non-convex credal sets ordered by second order probability, thereby providing uncertainty measures for such epistemic representations. There is also a comparison with the results of the measure AU developed within Dempster–Shafer theory in a few instances where unordered convex credal set theory and Dempster–Shafer theory overlap. (shrink)

We show that the 0-1 law does not hold for the class Σ 1 1 (∀∃∀ without =) by finding a sentence in this class which almost surely expresses parity. We also show that every recursive real in the unit interval is the asymptotic probability of a sentence in this class. This expands a result by Lidia Tendera, who in 1994 proved that every rational number in the unit interval is the asymptotic probability of a sentence in the (...) class Σ 1 1 ∀∃∀ with equality. (shrink)

Problems about the accidental properties of properties motivate us--force us, I think--not to identify properties with the sets of their instances. If we identify them instead with functions from worlds to extensions, we get a theory of properties that is neutral with respect to disputes over counterpart theory, and we avoid a problem for Lewis's theory of events. Similar problems about the temporary properties of properties motivate us--though this time they probably don't force us--to give up this theory as well, (...) and to identify properties with functions from ⟨world, time⟩ pairs to extensions. Again, the replacement theory is neutral with respect to a metaphysical dispute that the old theory (arguably) forces us to take a stand on--the dispute over whether objects have temporal parts. It also allows us to give a smoother semantics for predication, to better accommodate our intuitions about which objects temporary properties are properties of, and to make temporally self-locating beliefs genuinely self -locating. (shrink)

Problems about the accidental properties of properties motivate us--force us, I think--not to identify properties with the sets of their instances. If we identify them instead with functions from worlds to extensions, we get a theory of properties that is neutral with respect to disputes over counterpart theory, and we avoid a problem for Lewis's theory of events. Similar problems about the temporary properties of properties motivate us--though this time they probably don't force us--to give up this theory as well, (...) and to identify properties with functions from world, time pairs to extensions. Again, the replacement theory is neutral with respect to a metaphysical dispute that the old theory (arguably) forces us to take a stand on--the dispute over whether objects have temporal parts. It also allows us to give a smoother semantics for predication, to better accommodate our intuitions about which objects temporary properties are properties of, and to make temporally self-locating beliefs genuinely self -locating. (shrink)

We extend the framework of Inductive Logic to Second Order languages and introduce Wilmers' Principle, a rational principle for probability functions on Second Order languages. We derive a representation theorem for functions satisfying this principle and investigate its relationship to the first order principles of Regularity and Super Regularity.

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

Recently, Wilcox (JGPS 51: 65–87, 2020) argued against the so-called wide interval view and in favor of the principle of indifference as the correct response to unspecific evidence. Embedded in a formal model of the beliefs of an agent, the former presupposes imprecise probabilities and the latter numerically precise degrees of belief. His argument is illustrated by a thought experiment that comes with a fundamental intuition. According to Wilcox, the wide interval view is incompatible with this intuition and, thus, undermined. (...) In contrast, I show that the intuition behind the thought experiment is, in fact, compatible with the wide interval view if it is embedded into a specific conception of imprecise probabilities as model of belief. This conception is an extension of a framework which I call modified supervaluationism (MSV) and which I recently presented elsewhere (Karge 2021, 175–191). To accommodate the thought experiment’s fundamental intuition, it introduces a notion of second-order beliefs. (shrink)

Second-order uncertainty, also known as model uncertainty and Knightian uncertainty, arises when decision-makers can (partly) model the parameters of their decision problems. It is widely believed that subjective probability, and more generally Bayesian theory, are ill-suited to represent a number of interesting second-order uncertainty features, especially “ignorance” and “ambiguity”. This failure is sometimes taken as an argument for the rejection of the whole Bayesian approach, triggering a Bayes vs anti-Bayes debate which is in many ways (...) analogous to what the classical vs non-classical debate used to be in logic. This paper attempts to unfold this analogy and suggests that the development of non-standard logics offers very useful lessons on the contextualisation of justified norms of rationality. By putting those lessons to work I will flesh out an epistemological framework suitable for extending the expressive power of standard Bayesian norms of rationality to second- order uncertainty in a way which is both formally and foundationally conservative. Contents. (shrink)

We propose an original use of techniques from random graph theory to find a Monadic ∑ 1 1 sentence without an asymptotic probability. Our result implies that the 0-1 law fails for the logics ∑ 1 1 and ∑ 1 1 . Therefore we complete the classification of first-order prefix classes with or without equality, according to the existence of the 0-1 law for the corresponding ∑ 1 1 fragment. In addition, our counterexample can be viewed as a (...) single explanation of the failure of the 0-1 law of all the fragments of existential second-order logic for which the failure is already known. (shrink)

In 2001, Le Bars proved that there exists an existential monadic second-order sentence such that the probability that it is true on [Formula: see text] does not converge and conjectured that, for EMSO sentences with two first-order variables, the zero–one law holds. In this paper, we prove that the conjecture fails for [Formula: see text], and give new examples of sentences with fewer variables without convergence.

In this paper we consider the conditions of second-order exchangeability and second-order partially exchangeability together with some applications.The first application concerns a general linear state-space model in which the noise process is assumed to be second-order partially exchangeable. Other applications of the second-order exchangeability condition regard statistical procedures in which imprecise prior probability assessments are employed.

The Bayesian model is used in psychology as the reference for the study of dynamic probability judgment. The main limit induced by this model is that it confines the study of revision of degrees of belief to the sole situations of revision in which the universe is static (revising situations). However, it may happen that individuals have to revise their degrees of belief when the message they learn specifies a change of direction in the universe, which is considered as (...) changing with time (updating situations). We analyze the main results of the experimental literature with regard to elementary qualitative properties of these two situations of revision. First, the order effect phenomenon is confronted with the commutative property. Second, an apparent new phenomenon is presented: the redundancy effect that is confronted with the idempotence property. Finally, results obtained in this kind of experimental situations are reinterpreted in the light of pragmatic analysis. (shrink)

When combining information from multiple sources and attempting to estimate the probability of a conclusion, we often find ourselves in the position of knowing the probability of the conclusion conditional on each of the individual sources, but we have no direct information about the probability of the conclusion conditional on the combination of sources. The probability calculus provides no way of computing such joint probabilities. This paper introduces a new way of combining probabilistic information to estimate (...) joint probabilities. It is shown that on a particular conception of objective probabilities, clear sense can be made of second-order probabilities (probabilities of probabilities), and these can be related to combinatorial theorems about proportions in finite sets as the sizes of the sets go to infinity. There is a rich mathematical theory consisting of such theorems, and the theorems generate corresponding theorems about secondorder probabilities. Among the latter are a number of theorems to the effect that certain inferences from probabilities to probabilities, although not licensed by the probability calculus, have probability 1 of producing correct results. This does not mean that they will always produce correct results, but the set of cases in which the inferences go wrong form a set of measure 0. Among these inferences are some enabling us to reasonably estimate the values of joint probabilities in a wide range of cases. A function called the Y-function is defined. The central theorem is the Y-Theorem, which tells us that if we know the individual probabilities for the different information sources and estimate the joint probability using the Y-function, the second-order probability of getting the right answer is 1. This mathematical result is tested empirically using a simple multi-sensor example. The Y-theorem agrees with Dempster's rule of combination in special cases, but not in general. (shrink)

Probability plays an essential role in many branches of AI, where it is typically assumed that we have a complete probability distribution when addressing a problem. But this is unrealistic for problems of real-world complexity. Statistical investigation gives us knowledge of some probabilities, but we generally want to know many others that are not directly revealed by our data. For instance, we may know prob(P/Q) (the probability of P given Q) and prob(P/R), but what we really want (...) is prob(P/Q&R), and we may not have the data required to assess that directly. The probability calculus is of no help here. Given prob(P/Q) and prob(P/R), it is consistent with the probability calculus for prob(P/Q&R) to have any value between 0 and 1. Is there any way to make a reasonable estimate of the value of prob(P/Q&R)? A related problem occurs when probability practitioners adopt undefended assumptions of statistical independence simply on the basis of not seeing any connection between two propositions. This is common practice, but its justification has eluded probability theorists, and researchers are typically apologetic about making such assumptions. Is there any way to defend the practice? This paper shows that on a certain conception of probability — nomic probability — there are principles of “probable probabilities” that license inferences of the above sort. These are principles telling us that although certain inferences from probabilities to probabilities are not deductively valid, nevertheless the second-order probability of their yielding correct results is 1. This makes it defeasibly reasonable to make the inferences. Thus I argue that it is defeasibly reasonable to assume statistical independence when we have no information to the contrary. And I show that there is a function Y(r,s:a) such that if prob(P/Q) = r, prob(P/R) = s, and prob(P/U) = a (where U is our background knowledge) then it is defeasibly reasonable to expect that prob(P/Q&R) = Y(r,s:a).. (shrink)

In concrete applications of probability, statistical investigation gives us knowledge of some probabilities, but we generally want to know many others that are not directly revealed by our data. For instance, we may know prob(P/Q) (the probability of P given Q) and prob(P/R), but what we really want is prob(P/Q&R), and we may not have the data required to assess that directly. The probability calculus is of no help here. Given prob(P/Q) and prob(P/R), it is consistent with (...) the probability calculus for prob(P/Q&R) to have any value between 0 and 1. Is there any way to make a reasonable estimate of the value of prob(P/Q&R)? A related problem occurs when probability practitioners adopt undefended assumptions of statistical independence simply on the basis of not seeing any connection between two propositions. This is common practice, but its justification has eluded probability theorists, and researchers are typically apologetic about making such assumptions. Is there any way to defend the practice? This paper shows that on a certain conception of probability — nomic probability — there are principles of “probable probabilities” that license inferences of the above sort. These are principles telling us that although certain inferences from probabilities to probabilities are not deductively valid, nevertheless the second-order probability of their yielding correct results is 1. This makes it defeasibly reasonable to make the inferences. Thus I argue that it is defeasibly reasonable to assume statistical independence when we have no information to the contrary. And I show that there is a function Y(r,s:a) such that if prob(P/Q) = r, prob(P/R) = s, and prob(P/U) = a (where U is our background knowledge) then it is defeasibly reasonable to expect that prob(P/Q&R) = Y(r,s:a). Numerous other defeasible inferences are licensed by similar principles of probable probabilities.. (shrink)

This paper is a comparison of how first-order Kyburgian Evidential Probability (EP), second-order EP, and objective Bayesian epistemology compare as to the KLM system-P rules for consequence relations and the monotonic / non-monotonic divide.

Logical argument forms are investigated by second order probability density functions. When the premises are expressed by beta distributions, the conclusions usually are mixtures of beta distributions. If the shape parameters of the distributions are assumed to be additive (natural sampling), then the lower and upper bounds of the mixing distributions (Polya-Eggenberger distributions) are parallel to the corresponding lower and upper probabilities in conditional probability logic.

‘Probability dynamics’ (PD) is a second-order probabilistic theory in which probability distribution d X = (P(X 1), . . . , P(X m )) on partition U m X of sample space Ω is weighted by ‘credence’ (c) ranging from −∞ to +∞. c is the relative degree of certainty of d X in ‘α-evidence’ α X =[c; d X ] on U m X . It is shown that higher-order probabilities cannot provide a theory (...) of PD. PD applies to both subjectivist and frequentist theories. ‘Straight PD’ (SPD) produces associative and commutative mergers of evidence, in which evidences of positive credence are mutually reinforcing. ‘Offsetting PD’ (OPD) sets off conflicting evidences against each other. Subjectivist PD is a quantified second-order logic of action. Frequentist PD relates to descriptions of physical states of affairs. Acceptance of evidence α 1 X = [c 1; d 1 X ] at t 1 updates α 0 X = [c 0; d 0 X ] at t 0 into SPD-merger or OPD-merger . Given ‘co-evidence’ at t 0 < t 1, ‘indirect’ PD accepts evidence at t 1 and produces support for update α 10 X = [c10;d 10 X α 0 X such that in SPD and in OPD. For binary X and Y, with α 0 X = [c 0; P 0(X)] at t 0 (short-hand for ) and the accepted evidence, the support is ; ; where ρ0(X, Y) is the correlation coefficient of X and Y, and update α 10 X of α 0 X is with ‘accord’ λ = 1 in SPD and in OPD. As , tends toward update P 1(Y) of P 0(Y), but P 10(X) does not converge toward ‘ ’ of ‘probability kinematics’. Therefore PD is not compatible with probability kinematics. A process of ‘normalization’ interprets ‘β-evidence’ [[c 1; q 1]& . . . &[c m ; q m ]; U m X ]] with [c j ; q j ] on and m ≥ 3, as an α-evidence on {X 1, . . . , X m }. It is shown that SPD- and OPD-updates can be derived from updated cumulative functions. Time-biased updates are discussed. A PD-based theory of confirmation (PDCT) is presented. (shrink)

An assertion of high conditional probability or, more briefly, an HCP assertion is a statement of the type: The conditional probability of B given A is close to one. The goal of this paper is to construct logics of HCP assertions whose conclusions are highly likely to be correct rather than certain to be correct. Such logics would allow useful conclusions to be drawn when the premises are not strong enough to allow conclusions to be reached with certainty. (...) This goal is achieved by taking Adams' (1966) logic, changing its intended application from conditionals to HCP assertions, and then weakening its criterion for entailment. According to the weakened entailment criterion, called the Criterion of Near Surety and which may be loosely interpreted as a Bayesian criterion, a conclusion is entailed if and only if nearly every model of the premises is a model of the conclusion. The resulting logic, called NSL, is nonmonotonic. Entailment in this logic, although not as strict as entailment in Adams' logic, is more strict than entailment in the propositional logic of material conditionals. Next, NSL was modified by requiring that each HCP assertion be scaled; this means that to each HCP assertion was associated a bound on the deviation from 1 of the conditional probability that is the subject of the assertion. Scaling of HCP assertions is useful for breaking entailment deadlocks. For example, it it is known that the conditional probabilities of C given A and of - C given B are both close to one but the bound on the former's deviation from 1 is much smaller than the latter's, then it may be concluded that in all likelihood the conditional probability of C given A ∧ B is close to one. The resulting logic, called NSL-S, is also nonmonotonic. Despite great differences in their definitions of entailment, entailment in NSL is equivalent to Lehmann and Magidor's rational closure and, disregarding minor differences concerning which premise sets are considered consistent, entailment in NSL-S is equivalent to entailment in Goldszmidt and Pearl's System-Z⁺. Bacchus, Grove, Halpern, and Koller proposed two methods of developing a predicate calculus based on the Criterion of Near Surety. In their randomstructures method, which assumed a prior distribution similar to that of NSL, it appears possible to define an entailment relation equivalent to that of NSL. In their random-worlds method, which assumed a prior distribution dramatically different from that of NSL, it is known that the entailment relation is different from that of NSL. (shrink)

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)

The generalized Bayes’ rule (GBR) can be used to conduct ‘quasi-Bayesian’ analyses when prior beliefs are represented by imprecise probability models. We describe a procedure for deriving coherent imprecise probability models when the event space consists of a finite set of mutually exclusive and exhaustive events. The procedure is based on Walley’s theory of upper and lower prevision and employs simple linear programming models. We then describe how these models can be updated using Cozman’s linear programming formulation of (...) the GBR. Examples are provided to demonstrate how the GBR can be applied in practice. These examples also illustrate the effects of prior imprecision and prior-data conflict on the precision of the posterior probability distribution. (shrink)

Some have argued that chance and determinism are compatible in order to account for the objectivity of probabilities in theories that are compatible with determinism, like Classical Statistical Mechanics (CSM) and Evolutionary Theory (ET). Contrarily, some have argued that chance and determinism are incompatible, and so such probabilities are subjective. In this paper, I argue that both of these positions are unsatisfactory. I argue that the probabilities of theories like CSM and ET are not chances, but also that they (...) are not subjective probabilities either. Rather, they are a third type of probability, which I call counterfactual probability. The main distinguishing feature of counterfactual-probability is the role it plays in conveying important counterfactual information in explanations. This distinguishes counterfactual probability from chance as a second concept of objective probability. (shrink)

Question order effects are commonly observed in self-report measures of judgment and attitude. This article develops a quantum question order model (the QQ model) to account for four types of question order effects observed in literature. First, the postulates of the QQ model are presented. Second, an a priori, parameter-free, and precise prediction, called the QQ equality, is derived from these mathematical principles, and six empirical data sets are used to test the prediction. Third, a new (...) index is derived from the model to measure similarity between questions. Fourth, we show that in contrast to the QQ model, Bayesian and Markov models do not generally satisfy the QQ equality and thus cannot account for the reported empirical data that support this equality. Finally, we describe the conditions under which order effects are predicted to occur, and we review a broader range of findings that are encompassed by these very same quantum principles. We conclude that quantum probability theory, initially invented to explain order effects on measurements in physics, appears to be a powerful natural explanation for order effects of self-report measures in social and behavioral sciences, too. (shrink)

Many proponents of libertarian freedom assume that the free choices we might make have particular objective probabilities of occurring. In this paper, I examine two common motivations for positing such probabilities: first, to account for the phenomenal character of decision-making, in which our reasons seem to have particular strengths to incline us to act, and second, to naturalize the role of reasons in influencing our decisions, such that they have a place in the causal order as we know (...) it. I argue, however, that neither introspective reflection nor the metaphysics of causation gives us reason for thinking there are such particular objective probabilities of our free choices. (shrink)

As is clear from the other articles in this volume, logic has applications in a broad range of areas of philosophy. If logic is taken to include the mathematical disciplines of set theory, model theory, proof theory, and recursion theory (as well as first-order logic, second-order logic, and modal logic), then the only other area of mathematics with such wide-ranging applications in philosophy is probability theory.

Alan Hájek has recently argued that certain assignments of vague probability defeat Pascals Wager. In particular, he argues that skeptical agnostics – those whose probability for God''s existence is vague over an interval containing zero – have nothing to fear from Pascal. In this paper, I make two arguments against Hájek: (1) that skeptical agnosticism is a form of dogmatism, and as such should be rejected; (2) that in any case, choice situations with vague probability assignments ought (...) to be treated as second-order cases of choice under uncertainty, with the result that belief in God is the favored option in a very wide range of cases. (shrink)

We prove a 0-1 law for the fragment of second order logic SO over parametric classes of finite structures which allow only one unary atomic type. This completes the investigation of 0-1 laws for fragments of second order logic defined in terms of first order quantifier prefixes over, e.g., simple graphs and tournaments. We also prove a low oscillation law, and establish the 0-1 law for Σ14 without any restriction on the number of unary types.

In recent papers, Peter Carruthers and others have argued that the feeling of uncertainty is not metacognitive (i.e., it is not elicited by second-order cognitive appraisals) but is elicited solely by first-order likelihood estimates—a probability account of the feeling of uncertainty. In this paper, I make a case for why a probability account is sufficient to explain neither the feeling of uncertainty nor the feeling of certainty in self-reflecting humans. I argue first that humans’ feelings (...) of (un)certainty vary in ways that their probability estimates on the matter do not, and second that probability accounts elide the essentially epistemic nature of epistemic feelings. (shrink)

This study extends experimental tests of (cumulative) prospect theory (PT) over prospects with more than three outcomes and tests second-order stochastic dominance principles (Levy and Levy, Management Science 48:1334–1349, 2002; Baucells and Heukamp, Management Science 52:1409–1423, 2006). It considers choice behavior of people facing prospects of three different types: gain prospects (losing is not possible), loss prospects (gaining is not possible), and mixed prospects (both gaining and losing are possible). The data supports the distinction of risk behavior into (...) these three categories of prospects, Further, probability weighting and diminishing sensitivity of utility as predicted by PT are observed. Loss aversion is, however, less pronounced, except for choices where one prospect is degenerate. The data suggests that the probability of losing may be relevant for loss aversion. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Logic of Probabilities in Hume's Argument against Miracles Fred Wilson The position is often stated that Hume's discussion of miracles is inconsistent with his views on the logical or ontological status oflaws ofnature and with his more general scepticism. Broad, for one, has so argued.1 Hume's views on induction are assumed to go somethinglike this. Any attempt to demonstrate knowledge ofmatters offact presupposes causal reasoning, but the latter (...) is based not on any perception of necessary connections, but on an unreasoned expectation that because events have been constantly conjoined in the past they will be constantly conjoined in the future. The fact that our past experience gives rise to certain expectations provides absolutely no reason to think that these expectations will be fulfilled. But this is forgotten as soon as he turns to the topic of miracles where the argument requires certain assumptions about laws of nature. Here he claims that the laws of nature arebased upon afirm andunalterable experience and dismisses what he apparently admits to be strong evidence for miracles on the basis of the claim that such events are absolutely impossible. The inconsistency between the discussion of miracles and the earlier discussions of induction and causality is clear. Nonetheless, there is a certain implausibility to this claim that makes it a difficult one to entertain seriously, since it is unlikely that a philosopher as careful as Hume would have failed to recognize the inconsistency if it existed. What I propose here to argue is that there is in fact no inconsistency, and that this becomes clear once one places the discussion ofmiracles in the broader context ofthe overall argument ofthe first Enquiry. In particular, the charge ofinconsistency disappears once one eliminates the caricature ofHume's views on causal reasoning upon which it rests. To come to grips with the discussion of miracles in Hume it is necessary to place that discussion in its historical context. In the generally empiricist atmosphere that developed in Britain following the mid-seventeenth century civil wars, the general defence of the reasonableness of Christianity consisted of two parts. First, it washeldthatrational argument, whether causal or teleological orboth, could yield a reasonable belief that a deity ofa more or less traditional sort existed. Then, second, in order to establish further that Jesus was the Son ofGod it was held that one could rely upon a chain oftestimony Volume XV Number 2 255 FRED WILSON leading back from the present to the past to infer the existence of miracles testifying to the divinity ofJesus. Hume attacks this second inference in the essay on miracles andthe first in the essay that follows in the Enquiries on God's particular providence (the themes of which are developed more fully in the Dialogues on Natural Religion). Given the central place that the appeal to miracles held in Christian apologetics in the 17th and 18th centuries, Hume's systematic attack on the argument from miracles constituted a clear attack on Christianity. To be sure, Hume's argument would never have persuaded JohnWesley, but that would hardly have surprised Hume himself, nor should it surprise us: those in the grips of religious enthusiasm, whether it be John Wesley or Jimmy Swaggart, will hardly be persuaded by rational argument, however much it is still true that they ought to be persuaded by it. The appeal to miracles tojustify specifically Christian beliefs had a long history. One can find it already in the late Roman period in Eusebius. Muslim thinkers were to challenge this traditional defence by arguing that chains oftestimony become more unreliable the longer they become. As for their own Islamic beliefs, these they admitted had also to be justified by an appeal to miracles, but there was, it was claimed, a unique self-justifying miracle in the Koran, the specially revealed Word of God. The critical theme of the Muslim apologists, that the longer a chain of testimony the less assurance it gives of the fact that a miracle had occurred, was developed in Britain in the 17th and 18th centuries.6 It seems to have been introduced by the Oxford orientalist Edward Pocock who had discovered this line of Muslim apologetics while... (shrink)

In a fair finite lottery with n tickets, the probability assigned to each ticket winning is 1/n and no other answer. That is, 1/n is unique. Now, consider a fair lottery over the natural numbers. What probability is assigned to each ticket winning in this lottery? Well, this probability value must be smaller than 1/n for all natural numbers n. If probabilities are real-valued, then there is only one answer: 0, as 0 is the only real and (...) non-negative value that is smaller than 1/n for all natural numbers n. However, this answer seems counter-intuitive, as it violates Regularity: for all possible events A that are assigned a probability, A is assigned a positive probability. It is possible for ticket i to win in the second lottery. So, the set {i} should be assigned a positive probability and not 0. Consequently, in order to satisfy Regularity, probabilities must be allowed to take on `non-real' values, e.g. a positive infinitesimal x that is smaller than 1/n for all natural numbers n. So, what regular probability is assigned to {i} in the second lottery? This probability value must be positive but smaller than 1/n for all natural numbers n. Given the definition of x, x is a correct answer. But x isn't unique, because every other positive infinitesimal is a correct answer as well and there are uncountably many positive infinitesimals. After all, every positive infinitesimal is strictly between 0 and 1/n for all natural numbers n. Nothing about a fair lottery over the natural numbers uniquely determines x as the correct answer and nothing about the lottery rules out other positive infinitesimals as a correct answer as well. What probability to assign to {i} is underdetermined. -/- In this thesis, I explore this difference between fair finite lotteries and fair infinite lotteries. The constraints governing the former uniquely determine the probabilities assigned to events in those lotteries, but this isn't true for the latter lotteries, when Regularity is a constraint on probabilities. Now, although there are uncountably many correct probability values that can be assigned to {i} in the second lottery, there are some applications of probability theory, for which at least one correct probability value must be chosen in order to accomplish that application, e.g. calculation of expected utilities. In this thesis, I spell out sufficient conditions for when a choice like that is problematic. These conditions apply to precise probabilities, imprecise probabilities, and qualitative probabilities. So, I will consider Benci et al. (2018)'s and Pruss (2021)'s suggestion of allowing probabilities to be imprecise, as well as resorting to qualitative probabilities, to overcome the choice problem. I will argue that both don't solve the problem. Furthermore, there's a hitherto unnoticed difficulty in deriving imprecise probabilities by supervaluating over precise `non-real' probability values. To avoid this difficulty, precise probability values should be real-valued. Thus, Regularity cannot be a constraint on probabilities. But as it turns out, dropping Regularity as a constraint and requiring precise probabilities to be real-valued solve the choice problem. (shrink)

Our uncertainty about matters of fact can often be adequately represented by probabilities, but there are also cases in which we, intuitively speaking, know too little even to assign meaningful probabilities. In many of these cases, other formal representations can be used to capture some of the prominent features of our uncertainty. This is a non-technical overview of some of these representations, including probability intervals, belief functions, fuzzy sets, credal sets, weighted credal sets, and second order probabilities.

Coherent-ambiguity aversion is defined within the smooth-ambiguity model as the combination of choice-ambiguity and value-ambiguity aversion. Five ambiguous decision tasks are analyzed theoretically, where an individual faces two-stage lotteries with binomial, uniform, or unknown second-order probabilities. Theoretical predictions are then tested through a 10-task experiment. In tasks 1–5, risk aversion is elicited through both a portfolio choice method and a BDM mechanism. In tasks 6–10, choice-ambiguity aversion is elicited through the portfolio choice method, while value-ambiguity aversion comes about (...) through the BDM mechanism. The behavior of over 75 % of classified subjects is in line with the KMM model in all tasks 6–10, independent of their degree of risk aversion. Furthermore, the percentage of coherent-ambiguity-averse subjects is lower in the binomial than in the uniform and in the unknown treatments, with only the latter difference being significant. The most part of coherent-ambiguity-loving subjects show a high risk aversion. (shrink)

We consider collective quantification in natural language. For many years the common strategy in formalizing collective quantification has been to define the meanings of collective determiners, quantifying over collections, using certain type-shifting operations. These type-shifting operations, i.e., lifts, define the collective interpretations of determiners systematically from the standard meanings of quantifiers. All the lifts considered in the literature turn out to be definable in second-order logic. We argue that second-order definable quantifiers are probably not expressive enough (...) to formalize all collective quantification in natural language. (shrink)

I present a puzzle which seems simple, but is found to have interesting implications for confirmation. Its dissolution also helps us to throw light on the relationship between first- and second-order probabilities construed as rational degrees of belief.

This article presents three experiments that examine the relation between order effects and frequency learning, with the following results. First, when frequencies of occurrence are presented as sequences of real events, base rates can be learned and used with a high degree of accuracy. However, conditional probabilities for multiple sequentially presented evidence items cannot be completely learned, due to the distortion of a recency order effect for actual decisions. Second, there is also a recency order effect (...) for belief evaluations, which cannot be eliminated even if base rates are used correctly. Third, base rates learned in one environment can be transferred to another environment, but the transfer soon diminishes due to learning in the new environment. However, belief evaluations are not transferred from one environment to another The existing models of frequency learning cannot explain the order effect for actual decisions because they do not consider sequential information. The existing models of belief updating can explain both types of order effects, but they do not have any mechanisms for frequency learning. To account for the complete spectrum of frequency learning and order effects, we outline our initial effort in developing a unified model that integrates frequency learning and order effects. (shrink)

Richard Swinburne, in his The Existence of God (2004), presents a cosmological argument in defence of theism (Swinburne 1991: 119, 135). God, Swinburne argues, is more likely to bring about an ordered universe than other states (ibid.: 144, 299). To defend this view, Swinburne presents the following arguments: (1) That this ordered universe is a priori improbable (2004: 49, 150, 1991: 304 et seq.), given the stringent requirements for life (cf. also Leslie 2000: 12), and the Second Law of (...) Thermodynamics (Giancoli 1990: 396); (2) That it seems as if this ordered universe can be explained by theism; (3) A theistic explanation for the universe is more probable because it is a simple explanation. To this end, Swinburne makes use of Bayes’ Theorem. Symbolically, this claim can be represented as (e) for the evidence of the existence of a complex universe, and (h) for a hypothesis. Swinburne’s argument is that theism has a higher prior probability, P(htheism) > P(hmaterialism), since theism is simpler than materialism. He concludes that P(e|htheism) > P(e|hmaterialism). In this paper I will address only this argument (3) above, and defend the view that it is false: theism is not simpler than materialism, nor it is more probably true. I conclude that theism is less probable than materialism, expressed by P(htheism) < P(hmaterialism) : 2/N(2n+1) < 1/n, where N is the number of possible universes and n the number of entities in existence. (shrink)

The Temporal-Order Judgment (TOJ) paradigm has been widely investigated in previous studies as an accurate measure of temporal resolution and sequencing abilities in the millisecond time range. Two auditory TOJ tasks are often used: (1) a spatial TOJ task, in which two identical stimuli are presented in rapid succession monaurally and the task is to indicate which ear received the first stimulus and which ear received the second one ( left-right or right-left ), and (2) a spectral TOJ (...) task, in which two tones of different frequencies are presented asynchronously to both ears binaurally and the task is to report the sequence of these tones ( low-high or high-low ). The previous literature studies conducted on young volunteers indicated that the measured temporal acuity on these two tasks depended on the procedure used. As considerable data are now available about age-related decline in temporal resolution ability, the aim of the present study was to compare in elderly subjects the pattern of performance on these two tasks. A total of 40 normal healthy volunteers aged from 62 to 78 years performed two TOJ tasks. The measurement was repeated in two consecutive sessions. Temporal resolution was indexed by the Auditory Temporal-Order Threshold (ATOT), i.e., the minimum time gap between successive stimuli necessary for a participant to report a before-after relation with 75% correctness. The main finding of the present study was the indication of differences in the elderly in performance on two tasks. In the spatial task, the distribution of obtained ATOT values did not deviate from the Gaussian distribution. In contrast, the distribution of data in the spectral task deviated significantly from the Gaussian and was spread more to the right. Although lower ATOT values were usually observed in Session 2 than in Session 1, such difference was significant only in the spectral task. We conclude that although temporal acuity and sequencing abilities in the millisecond time range are probably based in neuronal oscillatory activity, the measured ATOTs in the elderly seem to be stimulus-dependent, procedure-related, and influenced by the perceptual strategies used by participants. (shrink)

This paper begins with a problem stemming from Hume regarding credences about credences. Suppose one has a credence of .95 in p, and suppose one assesses the credence to be such. But suppose one’s second-order credence in this assessment is less than 1. Then, by a standard conditionalization rule, one’s credence in p becomes less than .95. Moreover, such “erosion” can iterate by considering one’s, third-, fourth-, fifth-order credences, etc. (In light of this, some have rejected higher- (...) class='Hi'>order credences; however, it is argued that Lewis’ “Principal Principle” forbids this.) The paper then offers a partial solution by describing circumstances under which a person has reason to assign credence 1 to a first-order credence. The solution takes the form of a “transparency” view of self-knowledge from Parent (2017, chapter 8). Briefly, it is argued that when one “reflex-like” utters a sentence “The probability of p is n”, then as a matter of psychological law, the utterance is an expression of one’s own judgment. The Principal Principle then necessitates that one’s credence in p is n, assuming one is being rational. Thus, upon perceiving such an utterance, one can thereby have a reason to assign a second-order credence of 1. (shrink)

Operational statistics is an operational theory of probability and statistics which generalizes classical probability and statistics and provides a formalism particularly suited to the needs of quantum mechanics. Within this formalism, statistical inference can be accomplished using the Bayesian inference strategy. In a hierarchical Bayesian approach, a second-order probability measure, or credibility, represents degrees of belief in statistical hypotheses. A credibility determines an assignment of simple and conditioned betting rates to events in a natural way. (...) In the setting of operational statistics, we show that a credibility is completely determined by the assignment of the betting rates it induces. This result suggests a certain unity between the Bayesian philosophy which deems that betting rates are central and the one which advocates the hierarchical approach. (shrink)

In the traditional decision theories the role of forecasts is to a large extent swept under the carpet. I believe that a recognition of the connections between forecasts and decisions will be of benefit both for decision theory and for the art of forecasting.In this paper I have tried to analyse which factors, apart from the utilities of the outcomes of the decision alternatives, determine the value of a decision. I have outlined two answers to the question why a decision (...) which is made on the basis of a forecast is better than a decision which is based on a guess. Neither of these answers is universally valid. An assumption which is necessary for the first answer, i.e. Good's result, is that Bayes' rule is accepted as a correct and generally applicable decision principle. The second answer, which was given with the aid of probability intervals, departed from a more general decision principle, the maximin criterion for expected utilities, which was formulated in order to evade some of the criticism against Bayes' rule. However, the argument leading to the ansser is based on the assumption that the probability intervals associated with the states of nature represent certain knowledge. For this reason this answer is only approximatively valid.As a number of quotations in the section on “the weight of evidence” show, it is not sufficient to describe the knowledge about the states of nature by a single number, representing the (subjective) probability of the state, but something else has to be invoked which measures the amount of information on which a decision is based. Several authors have tried to characterize this mysterious quantity, which here was called the weight of evidence. However, there seems to be little agreement as to how this quantity should be measured. (shrink)

Epistemic uncertainties are included in probabilistic risk assessment (PRA) as second-order probabilities that represent the degrees of belief of the scientists that a model is correct. In this article, I propose an alternative approach that incorporates the scientist’s confidence in a probability set for a given quantity. First, I give some arguments against the use of precise probabilities to estimate scientific uncertainty in risk analysis. I then extend the “confidence approach” developed by Brian Hill and Richard Bradley (...) to PRA. Finally, I claim that this approach represents model uncertainty better than the standard (Bayesian) model does. (shrink)