An introduction to the motivations and mechanics of upper and lower probabilities, from a lecture given at the Northern Institute of Philosophy in 2010.

We generalize the concept of Nash equilibrium in mixed strategies for strategic form games to allow for ambiguity in the players' expectations. In contrast to other contributions, we model ambiguity by means of so-called lower probability measures or belief functions, which makes it possible to distinguish between a player's assessment of ambiguity and his attitude towards ambiguity. We also generalize the concept of trembling hand perfect equilibrium. Finally, we demonstrate that for certain attitudes towards ambiguity it is possible to (...) explain cooperation in the one-shot Prisoner's Dilemma in a way that is in accordance with some recent experimental findings. (shrink)

We present a propositional and a first-order logic for reasoning about higher-order upper and lower probabilities. We provide sound and complete axiomatizations for the logics and we prove decidability in the propositional case. Furthermore, we show that the introduced logics generalize some existing probability logics.

We present a propositional and a first-order logic for reasoning about higher-order upper and lower probabilities. We provide sound and complete axiomatizations for the logics and we prove decidability in the propositional case. Furthermore, we show that the introduced logics generalize some existing probability logics.

(1976). PROBABILITY, LIKELIHOOD AND SUPPORT: A METAMATHEMATICAL APPROACH TO A SYSTEM OF AXIOMS FOR UPPER AND LOWER DEGREES OF BELIEF. Philosophical Papers: Vol. 5, No. 2, pp. 153-161.

A common objection to probabilistic theories of causation is that there are prima facie causes that lower the probability of their effects. Among the many replies to this objection, little attention has been given to Mellor's (1995) indirect strategy to deny that probability-lowering factors are bona fide causes. According to Mellor, such factors do not satisfy the evidential, explanatory, and instrumental connotations of causation. The paper argues that the evidential connotation only entails an epistemically relativized form of causal attribution, (...) not causation itself, and that there are clear cases of explanation and instrumental reasoning that must appeal to negatively relevant factors. In the end, it suggests a more liberal interpretation of causation that restores its connotations. Una objeción común a las teorías probabilísticas de la causalidad es que aparentemente existen causas que disminuyen la probabilidad de sus efectos. Entre las muchas respuestas a esta objeción, se le ha dado poca atención a la estrategia indirecta de D. H. Mellor (1995) para negar que un factor que disminuya la probabilidad de un efecto sea una causa legítima. Según Mellor, tales factores no satisfacen las connotaciones evidenciales, explicativas e instrumentales de la causalidad. El artículo argumenta que la connotación evidencial sólo implica una forma epistémicamente relativizada de atribución causal y no la causalidad misma, y que hay casos claros de explicación y razonamiento instrumental que deben apelar a factores negativamente relevantes. Se sugiere una interpretación más liberal de la causalidad que reinstaura sus connotaciones. (shrink)

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

The theory of lower previsions is designed around the principles of coherence and sure-loss avoidance, thus steers clear of all the updating anomalies highlighted in Gong and Meng's "Judicious Judgment Meets Unsettling Updating: Dilation, Sure Loss, and Simpson's Paradox" except dilation. In fact, the traditional problem with the theory of imprecise probability is that coherent inference is too complicated rather than unsettling. Progress has been made simplifying coherent inference by demoting sets of probabilities from fundamental building blocks to (...) secondary representations that are derived or discarded as needed. (shrink)

We argue that indeterminate probabilities are not only rationally permissible for a Bayesian agent, but they may even be rationally required . Our first argument begins by assuming a version of interpretivism: your mental state is the set of probability and utility functions that rationalize your behavioral dispositions as well as possible. This set may consist of multiple probability functions. Then according to interpretivism, this makes it the case that your credal state is indeterminate. Our second argument begins with (...) our describing a world that plausibly has indeterminate chances. Rationality requires a certain alignment of your credences with corresponding hypotheses about the chances. Thus, if you hypothesize the chances to be indeterminate, your will inherit their indeterminacy in your corresponding credences. Our third argument is motivated by a dilemma. Epistemic rationality requires you to stay open-minded about contingent matters about which your evidence has not definitively legislated. Practical rationality requires you to be able to act decisively at least sometimes. These requirements can conflict with each other-for thanks to your open-mindedness, some of your options may have undefined expected utility, and if you are choosing among them, decision theory has no advice to give you. Such an option is playing Nover and Hájek’s Pasadena Game , and indeed any option for which there is a positive probability of playing the Pasadena Game. You can serve both masters, epistemic rationality and practical rationality, with an indeterminate credence to the prospect of playing the Pasadena game. You serve epistemic rationality by making your upper probability positive-it ensures that you are open-minded. You serve practical rationality by making your lower probability 0-it provides guidance to your decision-making. No sharp credence could do both. (shrink)

Jeffrey conditionalization is generalized to the case in which new evidence bounds the possible revisions of a prior below by a Dempsterian lower probability. Classical probability kinematics arises within this generalization as the special case in which the evidentiary focal elements of the bounding lower probability are pairwise disjoint.

In this paper I reconsider a standard counterexample to the chance-raising theory of singular causation. Extant versions of this theory are so different that it is difficult to formulate the core thesis that they all share, despite the guiding idea that causes raise the chance of their effects. At one extreme, ‘Humean’ theories – which can be traced to Reichenbach – say that a particular event of type C is the cause of a particular event of type E only if (...) P(E|C & K) > P(E|~C & K) where K is a set of background conditions and where the probabilities are interpreted as relative frequencies. At the other extreme, explicitly non-Humean theories take chance to be a physical, particular, local feature of the world. (shrink)

This paper addresses the problem of why the conditions under which standard proofs of the Dutch Book argument proceed should ever be met. In particular, the condition that there should be odds at which you would be willing to bet indifferently for or against are hardly plausible in practice, and relaxing it and applying Dutch book considerations gives only the theory of upper and lower probabilities. It is argued that there are nevertheless admittedly rather idealised circumstances in which (...) the classic form of the Dutch Book argument is valid. (shrink)

Hempel's high probability requirement asserts that any rationally acceptable answer to the question 'Why did event X occur?' must offer information which shows that X was to be expected at least with reasonable probability. Salmon rejected this requirement in his S-R model. This led to a series of paradoxical consequences, such as the assertion that an explanation of an event can both lower its probability and make it arbitrarily low, and the assertion that the explanation of an outcome would (...) have qualified as an explanation of its non-occurrence as well. We argue that if inductive explanations are to be seen as generalizations of the causal-deterministic model, or if they are to be seen as satisfying the requirement--fulfilled by the D-N model--that explanations ought to identify certain features of the universe that are nomically responsible for the explanadum event, then the high probability requirement seems to be unacceptable. If this is so, a realistically inspired theory of inductive explanation will be committed to the paradoxes that follow from Salmon's model. (shrink)

We analyze selected iterated conditionals in the framework of conditional random quantities. We point out that it is instructive to examine Lewis's triviality result, which shows the conditions a conditional must satisfy for its probability to be the conditional probability. In our approach, however, we avoid triviality because the import-export principle is invalid. We then analyze an example of reasoning under partial knowledge where, given a conditional if A then Cas information, the probability of A should intuitively increase. We explain (...) this intuition by making some implicit background information explicit. We consider several iterated conditionals, which allow us to formalize different kinds of latent information. We verify that for these iterated conditionals the prevision is greater than or equal to the probability of A. We also investigate the lower and upper bounds of the Affirmation of the Consequent inference. We conclude our study with some remarks on the supposed "independence" of two conditionals, and we interpret this property as uncorrelation between two random quantities. 2020 Elsevier Inc. All rights reserved. (shrink)

In his entry on "Quantum Logic and Probability Theory" in the Stanford Encyclopedia of Philosophy, Alexander Wilce (2012) writes that "it is uncontroversial (though remarkable) the formal apparatus quantum mechanics reduces neatly to a generalization of classical probability in which the role played by a Boolean algebra of events in the latter is taken over the 'quantum logic' of projection operators on a Hilbert space." For a long time, Patrick Suppes has opposed this view (see, for example, the paper collected (...) in Suppes and Zanotti (1996). Instead of changing the logic and moving from a Boolean algebra to a non-Boolean algebra, one can also 'save the phenomena' by weakening the axioms of probability theory and work instead with upper and lower probabilities. However, it is fair to say that despite Suppes' efforts upper and lower probabilities are not particularly popular in physics as well as in the foundations of physics, at least so far. Instead, quantum logic is booming again, especially since quantum information and computation became hot topics. Interestingly, however, imprecise probabilities are becoming more and more popular in formal epistemology as recent work by authors such as James Joye (2010) and Roger White (2010) demonstrates. (shrink)

We propose two different characterizations for preference relations representable by lower (upper) expectations with the aim of removing either fair price or completeness requirements. Moreover, we give an explicit characterization for comparative degrees of belief on a finite algebra of events representable by lower probabilities.

This approach does not define a probability measure by syntactical structures. It reveals a link between modal logic and mathematical probability theory. This is shown (1) by adding an operator (and two further connectives and constants) to a system of lower predicate calculus and (2) regarding the models of that extended system. These models are models of the modal system S₅ (without the Barcan formula), where a usual probability measure is defined on their set of possible worlds. Mathematical probability (...) models can be seen as models of S₅. (shrink)

Quantum mechanical entangled configurations of particles that do not satisfy Bell’s inequalities, or equivalently, do not have a joint probability distribution, are familiar in the foundational literature of quantum mechanics. Nonexistence of a joint probability measure for the correlations predicted by quantum mechanics is itself equivalent to the nonexistence of local hidden variables that account for the correlations (for a proof of this equivalence, see Suppes and Zanotti, 1981). From a philosophical standpoint it is natural to ask what sort of (...) concept can be used to provide a “joint” analysis of such quantum correlations. In other areas of application of probability, similar but different problems arise. A typical example is the introduction of upper and lower probabilities in the theory of belief. A person may feel uncomfortable assigning a precise probability to the occurrence of rain tomorrow, but feel comfortable saying the probability should be greater than ½ and less than ⅞. Rather extensive statistical developments have occurred for this framework. A thorough treatment can be found in Walley (1991) and an earlier measurement-oriented development in Suppes (1974). It is important to note that this focus on beliefs, or related Bayesian ideas, is not concerned, as we are here, with the nonexistence of joint probability distributions. Yet earlier work with no relation to quantum mechanics, but focused on conditions for existence has been published by many people. For some of our own work on this topic, see Suppes and Zanotti (1989). Still, this earlier work naturally suggested the question of whether or not upper and lower measures could be used in quantum mechanics, as a generalization of.. (shrink)

The modus ponens (A -> B, A :. B) is, along with modus tollens and the two logically not valid counterparts denying the antecedent (A -> B, ¬A :. ¬B) and affirming the consequent, the argument form that was most often investigated in the psychology of human reasoning. The present contribution reports the results of three experiments on the probabilistic versions of modus ponens and denying the antecedent. In probability logic these arguments lead to conclusions with imprecise probabilities. In (...) the modus ponens tasks the participants inferred probabilities that agreed much better with the coherent normative values than in the denying the antecedent tasks, a result that mirrors results found with the classical argument versions. For modus ponens a surprisingly high number of lower and upper probabilities agreed perfectly with the conjugacy property (upper probabilities equal one complements of the lower probabilities). When the probabilities of the premises are imprecise the participants do not ignore irrelevant (“silent”) boundary probabilities. The results show that human mental probability logic is close to predictions derived from probability logic for the most elementary argument form, but has considerable difficulties with the more complex forms involving negations. (shrink)

Given a few assumptions, the probability of a conjunction is raised, and the probability of its negation is lowered, by conditionalising upon one of the conjuncts. This simple result appears to bring Bayesian confirmation theory into tension with the prominent dogmatist view of perceptual justification – a tension often portrayed as a kind of ‘Bayesian objection’ to dogmatism. In a recent paper, David Jehle and Brian Weatherson observe that, while this crucial result holds within classical probability theory, it fails within (...) intuitionistic probability theory. They conclude that the dogmatist who is willing to take intuitionistic logic seriously can make a convincing reply to the Bayesian objection. In this paper, I argue that this conclusion is premature – the Bayesian objection can survive the transition from classical to intuitionistic probability, albeit in a slightly altered form. I shall conclude with some general thoughts about what the Bayesian objection to dogmatism does and doesn’t show. (shrink)

Taking as starting point two familiar interpretations of probability, we develop these in a perhaps unfamiliar way to arrive ultimately at an improbable claim concerning the proper axiomatization of probability theory: the domain of definition of a point-valued probability distribution is an orthomodular partially ordered set. Similar claims have been made in the light of quantum mechanics but here the motivation is intrinsically probabilistic. This being so the main task is to investigate what light, if any, this sheds on quantum (...) mechanics. In particular it is important to know under what conditions these point-valued distributions can be thought of as derived from distribution-pairs of upper and lower probabilities on boolean algebras. Generalising known results this investigation unsurprisingly proves unrewarding. In the light of this failure the next topic investigated is how these generalized probability distributions are to be interpreted. (shrink)

There are a number of reasons for being interested in uncertainty, and there are also a number of uncertainty formalisms. These formalisms are not unrelated. It is argued that they can all be reflected as special cases of the approach of taking probabilities to be determined by sets of probability functions defined on an algebra of statements. Thus, interval probabilities should be construed as maximum and minimum probabilities within a set of distributions, Glenn Shafer's belief functions should (...) be construed as lower probabilities, etc. Updating probabilities introduces new considerations, and it is shown that the representation of belief as a set of probabilities conflicts in this regard with the updating procedures advocated by Shafer. The attempt to make subjectivistic probability plausible as a doctrine of rational belief by making it more flowery — i.e., by adding new dimensions — does not succeed. But, if one is going to represent beliefs by sets of distributions, those sets of distributions might as well be based in statistical knowledge, as they are in epistemological or evidential probability. (shrink)

The field of of imprecise probability has matured, in no small part because of Teddy Seidenfeld’s decades of original scholarship and essential contributions to building and sustaining the ISIPTA community. Although the basic idea behind imprecise probability is (at least) 150 years old, a mature mathematical theory has only taken full form in the last 30 years. Interest in imprecise probability during this period has also grown, but many of the ideas that the mature theory serves can be difficult to (...) apprehend to those new to the subject. Although these fundamental ideas are common knowledge in the ISIPTA community, they are expressed, when they are expressed at all, obliquely, over the course of years with students and colleagues. -/- A single essay cannot convey the store of common knowledge from any research community, let alone the ISIPTA community. But, this essay nevertheless is an attempt to guide those familiar with the basic Bayesian framework to appreciate some of the elegant and powerful ideas that underpin the contemporary theory of lower previsions, which is the theory that most people associate with the term ‘imprecise probabilities’. (shrink)

I examine Hume’s proposal about rationally considering testimonial evidence for miracles. He proposes that we compare the probability of the miracle (independently of the testimony) with the probability that the testimony is false, rejecting whichever has the lower probability. However, this superficially plausible proposal is massively ignored in our treatment of testimonial evidence in nonreligious contexts. I argue that it should be ignored, because in many cases, including the resurrection of Jesus, neither we nor Hume have any experience which (...) is at all relevant to assigning a prior probability to the alleged event. (shrink)

There are a number of reasons for being interested in uncertainty, and there are also a number of uncertainty formalisms. These formalisms are not unrelated. It is argued that they can all be reflected as special cases of the approach of taking probabilities to be determined by sets of probability functions defined on an algebra of statements. Thus, interval probabilities should be construed as maximum and minimum probabilities within a set of distributions, Glenn Shafer's belief functions should (...) be construed as lower probabilities, etc. Updating probabilities introduces new considerations, and it is shown that the representation of belief as a set of probabilities conflicts in this regard with the updating procedures advocated by Shafer. The attempt to make subjectivistic probability plausible as a doctrine of rational belief by making it more flowery -- i.e., by adding new dimensions -- does not succeed. But, if one is going to represent beliefs by sets of distributions, those sets of distributions might as well be based in statistical knowledge, as they are in epistemological or evidential probability. (shrink)

Uncertainty and vagueness/imprecision are not the same: one can be certain about events described using vague predicates and about imprecisely specified events, just as one can be uncertain about precisely specified events. Exactly because of this, a question arises about how one ought to assign probabilities to imprecisely specified events in the case when no possible available evidence will eradicate the imprecision (because, say, of the limits of accuracy of a measuring device). Modelling imprecision by rough sets over an (...) approximation space presents an especially tractable case to help get one’s bearings. Two solutions present themselves: the first takes as upper and lower probabilities of the event X the (exact) probabilities assigned X ’s upper and lower rough-set approximations; the second, motivated both by formal considerations and by a simple betting argument, is to treat X ’s rough-set approximation as a conditional event and assign to it a point-valued (conditional) probability. (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)

A probability forecast scored ex post using a probability scoring rule (e.g. Brier) is analogous to a risky financial security. With only superficial adaptation, the same economic logic by which securities are valued ex ante – in particular, portfolio theory and the capital asset pricing model (CAPM) – applies to the valuation of probability forecasts. Each available forecast of a given event is valued relative to each other and to the “market” (all available forecasts). A forecast is seen to be (...) more valuable the higher its expected score and the lower the covariance of its score with the market aggregate score. Forecasts that score highly in trials when others do poorly are appreciated more than those with equal success in “easy” trials where most forecasts score well. The CAPM defines economically rational (equilibrium) forecast prices at which forecasters can trade shares in each other’s ex post score – or associated monetary payoff – thereby balancing forecast risk against return and ultimately forming optimally hedged portfolios. Hedging this way offers risk averse forecasters an “honest” alternative to the ruse of reporting conservative probability assessments. (shrink)

We review de Finetti’s two coherence criteria for determinate probabilities: coherence1defined in terms of previsions for a set of events that are undominated by the status quo – previsions immune to a sure-loss – and coherence2 defined in terms of forecasts for events undominated in Brier score by a rival forecast. We propose a criterion of IP-coherence2 based on a generalization of Brier score for IP-forecasts that uses 1-sided, lower and upper, probability forecasts. However, whereas Brier score is (...) a strictly proper scoring rule for eliciting determinate probabilities, we show that there is no real-valuedstrictly proper IP-score. Nonetheless, with respect to either of two decision rules – Γ-maximin or E-admissibility-+-Γ-maximin – we give a lexicographic strictly proper IP-scoring rule that is based on Brier score. (shrink)

. Philosopher-theologian Bernard J. F. Lonergan defines emergence as the process in which “otherwise coincidental manifolds of lower conjugate acts invite the higher integration effected by higher conjugate forms” (Insight, [1957] 1992, 477). The meaning and implications of Lonergan’s concept of emergence are considered in the context of the problem of reductionism in the natural sciences. Examples are taken primarily from physics, chemistry, and biology.

Jonathan Cohen has claimed that in cases of witness agreement there is an inverse relationship between the prior probability and the posterior probability of what is being agreed: the posterior rises as the prior falls. As is demonstrated in this paper, this contention is not generally valid. In fact, in the most straightforward case exactly the opposite is true: a lower prior also means a lower posterior. This notwithstanding, there is a grain of truth to what Cohen is (...) saying, as there are special circumstances under which a thesis similar to his holds good. What characterises these circumstances is that they allow for the fact of agreement to be surprising. In making this precise, I draw on Paul Horwich's probabilistic analysis of surprise. I also consider a related claim made by Cohen concerning the effect of lowering the prior on the strength of corroboration. 1 Introduction 2 Cohen's claim 3 A counterexample 4 A weaker claim 5 A counterexample to the weaker claim 6 The grain of truth in Cohen's claim 7 Prior probability and strength of corroboration 8 Conclusion. (shrink)

A non-monotonic theory of probability is put forward and shown to have applicability in the quantum domain. It is obtained simply by replacing Kolmogorov's positivity axiom, which places the lower bound for probabilities at zero, with an axiom that reduces that lower bound to minus one. Kolmogorov's theory of probability is monotonic, meaning that the probability of A is less then or equal to that of B whenever A entails B. The new theory violates monotonicity, as its (...) name suggests; yet, many standard theorems are also theorems of the new theory since Kolmogorov's other axioms are retained. What is of particular interest is that the new theory can accommodate quantum phenomena (photon polarization experiments) while preserving Boolean operations, unlike Kolmogorov's theory. Although non-standard notions of probability have been discussed extensively in the physics literature, they have received very little attention in the philosophical literature. One likely explanation for that difference is that their applicability is typically demonstrated in esoteric settings that involve technical complications. That barrier is effectively removed for non-monotonic probability theory by providing it with a homely setting in the quantum domain. Although the initial steps taken in this paper are quite substantial, there is much else to be done, such as demonstrating the applicability of non-monotonic probability theory to other quantum systems and elaborating the interpretive framework that is provisionally put forward here. Such matters will be developed in other works. (shrink)

We find suggestive evidence that emotional balance has an impact on probability weighting incremental to demographic controls. Specifically, low negative affectivity (implying high emotional balance) tends to be a characteristic of those whose probability weighting functions exhibit lower curvature and more neutral elevation. In other words, emotional balance seems to push people in the direction of normative expected utility theory.

Unlike other classical arguments for the existence of God, Pascal’s Wager provides a pragmatic rationale for theistic belief. Its most popular version says that it is rationally mandatory to choose a way of life that seeks to cultivate belief in God because this is the option of maximum expected utility. Despite its initial attractiveness, this long-standing argument has been subject to various criticisms by many philosophers. What is less discussed, however, is the rationality of this choice in situations where the (...) decision-makers are confronted with greater uncertainty. In this paper, I examine the imprecise version of Pascal’s Wager: those scenarios where an agent’s credence that God exists is imprecise or vague rather than precise. After introducing some technical background on imprecise probabilities, I apply five different principles for decision-making to two cases of state uncertainty. In the final part of the paper, I argue that it is not rationally permitted to include zero as the lower probability of God’s existence. Although the conditions for what makes an act uniquely optimal vary significantly across those principles, I also show how the option of wagering for God can defeat any mixed strategy under two distinct interpretations of salvation. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:125, HUME'S PROBABILITY ARGUMENT OF?,??,? In the Treatise,?,??,?, Hume presents an follows:' argument which, in the barest of outlines, goes as 1 (Pl) Every proposition has a probability less than one. (P2) If reason were the basis of our beliefs, then we would have no beliefs. (follows from (Pl)) (P3) We in fact do have beliefs. Hence, (P4) Reason is not the basis of our beliefs. The argument has (...) not been particularly well received. D. C. Stove, for example, refers to it as being "not merely defective, but one of the worst 2 arguments ever to impose itself on a man of genius." While not everyone is as unsympathetic as Stove, it nonetheless is difficult to find commentators favorably disposed toward the argument. Various sections of the Treatise are notoriously unclear.?,??,? is such a section; as such, it is difficult to say just what Hume intended his argument to be. My contention in the present paper is that there are two reasonable ways of reconstructing Hume's argument, both consistent with what Hume writes in?,??,?. The first reconstruction is clearly unsound; the second reconstruction fares somewhat better. In particular, my contention will be that this second reconstruction, if not sound, is at least valid 4 and contains no obviously false premises. Allow me to make some preliminary comments before presenting these reconstructions. The reconstructions to follow will primarily be concerned with premise (P2) above; in particular, the 126. reconstructions will focus on how Hume can claim that (P2) follows from (Pl). Premise (Pl) is a common 5 sceptical claim, not obviously false, and certainly not unique to Hume. As such, it strikes me that the more interesting part of the argument is Hume's claim that (P2) follows from (Pl), and such will be the focus of the reconstructions (although (Pl) will be discussed toward the end of this paper). Concerning (P2), why does Hume think this follows from (Pl)? His reasoning is contained in the following passage; Having thus found in every probability, beside the original uncertainty inherent in the subject, a new uncertainty deriv'd from the weakness of that faculty, which judges, and having adjusted these two together, we are oblig'd by our reason to add a new doubt deriv'd from the possibility of error in the estimation we make of the truth and fidelity of our faculties. This is a doubt, which immediately occurs to us, and of which, if we wou'd closely pursue our reason, we cannot avoid giving a decision. But this decision, tho' it shou'd be favourable to our preceeding judgment, being founded only on probability, must weaken still further our first evidence, and must itself be weaken'd by a fourth doubt of the same kind, and so on in infinitum; till at last there remain nothing of the original probability, however great we may suppose it to have been, and however small the diminution by every new uncertainty. No finite object can subsist under a decrease repeated in infinitum; and even the vastest quantity, which can enter into human imagination, must in this manner be reduc'd to nothing. (T182) Not for nothing did I claim that this section of the Treatise is unclear. At best, I can discern only the outline of Hume's reasoning, which seems to go as follows: first, when we assign a probability to some 127, proposition, reason dictates that we re-evaluate this probability. In particular, reason dictates that we "add a new doubt deriv'd from the possibility of error in the estimation we make of the truth and fidelity of our faculties." Note that Hume never specifies the object of this 'new doubt.' He might mean a) that our doubt concerns whether the probability we assigned the proposition is correct, or b) that our doubt concerns the evidence on which we based the assignment of that probability. Whatever the case, this 'new doubt' must "weaken still further our first evidence," presumably resulting in a lowering of the probability we originally assigned to the proposition. We then repeat this doubting process, adding new doubts of the same kind in infinitum, "till at last there remain nothing of the original... (shrink)

In a recent paper, probabilistic processes are used to generate Borel probability measures on topological spaces X that are equipped with a representation in the sense of type-2 theory of effectivity. This gives rise to a natural representation of the set of Borel probability measures on X. We compare this representation to a canonically constructed representation which encodes a Borel probability measure as a lower semicontinuous function from the open sets to the unit interval. We show that this canonical (...) representation is admissible with respect to the weak topology on Borel probability measures. Moreover, we prove that for countably-based topological spaces the representation via probabilistic processes is equivalent to the canonical representation and thus admissible with respect to the weak topology. (shrink)

Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for first-order sentences. Given first-order sentences φ and θ, we consider the structures with domain {1,..., N} that satisfy θ, and compute the fraction of them in which φ is true. We then consider what happens to this fraction as N gets large. This extends the work on 0-1 laws that considers the limiting probability of first-order sentences, by considering asymptotic (...) conditional probabilities. As shown by Liogon'kii [24], if there is a non-unary predicate symbol in the vocabulary, asymptotic conditional probabilities do not always exist. We extend this result to show that asymptotic conditional probabilities do not always exist for any reasonable notion of limit. Liogon'kii also showed that the problem of deciding whether the limit exists is undecidable. We analyze the complexity of three problems with respect to this limit: deciding whether it is well-defined, whether it exists, and whether it lies in some nontrivial interval. Matching upper and lower bounds are given for all three problems, showing them to be highly undecidable. (shrink)

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 (P´olya-Eggenberger distributions) are parallel to the corresponding lower and upper probabilities in conditional probability logic.

ABSTRACTCurrent theories of probability recognise a distinction between external certainty and internal certainty. The present studies investigated this distinction in lay people's judgements of probability statements formulated to suggest either an internal or an external interpretation. These subtle differences in wording influenced participants' perceptions and endorsements of such statements, and their impressions of the speaker. External expressions were seen to signal more reliable task duration estimates, and a lower degree of external than internal certainty was deemed necessary to advise (...) a course of action. In conversations about football, internal expressions were perceived as signalling more personal interest, and were expected to be on the average 10% higher than corresponding external probabilities. Finally, people who reported their outcome expectations for two m... (shrink)

Several scholars, including Martin Hengel, R. Alan Culpepper, and Richard Bauckham, have argued that Papias had knowledge of the Gospel of John on the grounds that Papias’s prologue lists six of Jesus’s disciples in the same order that they are named in the Gospel of John: Andrew, Peter, Philip, Thomas, James, and John. In “A Note on Papias’s Knowledge of the Fourth Gospel” (JBL 129 [2010]: 793–794), Jake H. O’Connell presents a statistical analysis of this argument, according to which the (...) probability of this correspondence occurring by chance is lower than 1%. O’Connell concludes that it is more than 99% probable that this correspondence is the result of Papias copying John, rather than chance. I show that O’Connell’s analysis contains multiple mistakes, both substantive and mathematical: it ignores relevant evidence, overstates the correspondence between John and Papias, wrongly assumes that if Papias did not know John he ordered the disciples randomly, and conflates the probability of A given B with the probability of B given A. In discussing these errors, I aim to inform both Johannine scholarship and the use of probabilistic methods in historical reasoning. (shrink)

A definition of causation as probability-raising is threatened by two kinds of counterexample: first, when a cause lowers the probability of its effect; and second, when the probability of an effect is raised by a non-cause. In this paper, I present an account that deals successfully with problem cases of both these kinds. In doing so, I also explore some novel implications of incorporating into the metaphysical investigation considerations of causal psychology.

Nonmonotonic reasoning is often claimed to mimic human common sense reasoning. Only a few studies, though, investigated this claim empirically. In the present paper four psychological experiments are reported, that investigate three rules of system p, namely the and, the left logical equivalence, and the or rule. The actual inferences of the subjects are compared with the coherent normative upper and lower probability bounds derived from a non-infinitesimal probability semantics of system p. We found a relatively good agreement of (...) human reasoning and principles of nonmonotonic reasoning according to the coherence interpretation of system p. Contrary to the results reported in the “heuristics and biases” tradition, the subjects committed relatively few upper bound violations (conjunction fallacies). More lower than upper bound violations were observed. When the premises were presented in terms of intervals higher mean lower bounds were observed as when the premises were presented in terms of point percentages. (shrink)

In standard Bayesian probability revision, the adoption of full beliefs (propositions with probability 1) is irreversible. Once an agent has full belief in a proposition, no subsequent revision can remove that belief. This is an unrealistic feature, and it also makes probability revision incompatible with belief change theory, which focuses on how the set of full beliefs is modified through both additions and retractions. This problem in probability theory can be solved in a model that (i) lets the codomain of (...) the probability function be a hyperreal-valued rather than the real-valued closed interval [0, 1], and (ii) identifies the full beliefs as the propositions whose probability is either 1 or infinitesimally smaller than 1. In this model, changes in the probability function will result in changes in the set of full beliefs (belief set), which constitutes a submodel that can be conceived as the “tip of the iceberg” within the larger model that also contains beliefs on lower levels of probability. The patterns of change in the set of full beliefs in this modified Bayesian model coincides with the corresponding pattern in a slightly modified version of AGM revision, which is commonly conceived as the gold standard of (dichotomous) belief change. The modification only concerns the marginal case of revision by an inconsistent input sentence. These results show that probability revision and dichotomous belief change can be unified in one and the same framework, or – if we so wish – that belief change theory can be subsumed under a modified version of probability revision that allows for iterated change and for the removal of full beliefs. (shrink)

The status of risk factors and disease remains a disputed question in the theory and practice of medicine and healthcare, and so does the related question of delineating disease boundaries. I present a framework based on Bernard Lonergan’s account of emergent probability for differentiating (1) generically distinct levels of systematic function within organisms and between organisms and their environments and (2) the methods of functional, genetic, and statistical investigation. I then argue on this basis that it is possible to understand (...) disease in terms of biological or higher intra-level dysfunction, risk factors—including genetic risk factors—in terms of statistical inter-level conditioning of a given stage or developmental sequence of systematic functioning, and the empirical boundaries of disease in terms of the limits of both functional categorization (from an epistemic standpoint) and upper-level integration of lower-level processes and events (from an ontological standpoint). (shrink)

ABSTRACTThe new paradigm in the psychology of reasoning redirects the investigation of deduction conceptually and methodologically because the premises and the conclusion of the inferences are assumed to be uncertain. A probabilistic counterpart of the concept of logical validity and a method to assess whether individuals comply with it must be defined. Conceptually, we used de Finetti's coherence as a normative framework to assess individuals' performance. Methodologically, we presented inference schemas whose premises had various levels of probability that contained non-numerical (...) expressions and, as a control, sure levels. Depending on the inference schemas, from 60% to 80% of the participants produced coherent conclusions when the premises were uncertain. The data also show that except for schemas involving conjunction, performance was consistently lower with certain than uncertain premises, the rate of conjunction fallacy was consistently low (not exceeding 20%,.. (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)

I’ll describe a range of systems for nonmonotonic conditionals that behave like conditional probabilities above a threshold. The rules that govern each system are probabilistically sound in that each rule holds when the conditionals are interpreted as conditional probabilities above a threshold level specific to that system. The well-known preferential and rational consequence relations turn out to be special cases in which the threshold level is 1. I’ll describe systems that employ weaker rules appropriate to thresholds lower (...) than 1, and compare them to these two standard systems. (shrink)

Adolescence has been linked to an enhanced tolerance of uncertainty and risky behavior and is possibly connected to an increased response toward rewards. However, previous research has produced inconsistent findings. To investigate whether these findings are due to different reward probabilities used in the experimental design, we extended a monetary incentive delay task by including three different reward probabilities. Using functional magnetic resonance imaging, 25 healthy adolescents and 22 adults were studied during anticipation of rewards in the VS. (...) Differently colored cue stimuli indicated either a monetary or verbal trial and symbolized different reward probabilities, to which the participants were blinded. Results demonstrated faster reaction times for lower reward probabilities in both age groups. Adolescents were slower through all conditions and had less activation on a neural level. Imaging results showed a three-way interaction between age group x condition x reward probability with differences in percent signal change between adolescents and adults for the high reward probabilities while adolescents demonstrated differences for the lowest. Therefore, previous inconsistent findings could be due to different reward probabilities, which makes examining these crucial for a better understanding of adolescent and adult behavior. (shrink)