In the first edition of LFP, Carnap [2] undertakes a precise probabilistic explication of the concept of confirmation. This is where modern confirmation theory was born (in sin). Carnap was interested mainly in quantitative confirmation (which he took to be fundamental). But, he also gave (derivative) qualitative and comparative explications: • Qualitative. E inductively supports H. • Comparative. E supports H more strongly than E supports H . • Quantitative. E inductively supports H to degree r . Carnap (...) begins by clarifying the explicandum (the informal “inductive support” concept) in various ways, including. (shrink)

In this paper we examine the thesis that the probability of the conditional is the conditional probability. Previous work by a number of authors has shown that in standard numerical probability theories, the addition of the thesis leads to triviality. We introduce very weak, comparative conditional probability structures and discuss some extremely simple constraints. We show that even in such a minimal context, if one adds the thesis that the probability of a conditional is (...) the conditional probability, then one trivializes the theory. Another way of stating the result is that the conditional of conditional probability cannot be represented in the object language on pain of trivializing the theory. (shrink)

This paper studies connections between two alternatives to the standard probability calculus for representing and reasoning about uncertainty: imprecise probability andcomparative probability. The goal is to identify complete logics for reasoning about uncertainty in a comparative probabilistic language whose semantics is given in terms of imprecise probability. Comparative probability operators are interpreted as quantifying over a set of probability measures. Modal and dynamic operators are added for reasoning about epistemic possibility and updating (...) sets of probability measures. (shrink)

The notion of comparative probability defined in Bayesian subjectivist theory stems from an intuitive idea that, for a given pair of events, one event may be considered “more probable” than the other. Yet it is conceivable that there are cases where it is indeterminate as to which event is more probable, due to, e.g., lack of robust statistical information. We take that these cases involve indeterminate comparative probabilities. This paper provides a Savage-style decision-theoretic foundation for indeterminate (...) class='Hi'>comparative probabilities. (shrink)

While pragmatic arguments for numerical probability axioms have received much attention, justifications for axioms of qualitative probability have been less discussed. We offer an argument for the requirement that an agent’s qualitative judgments be probabilistically representable, inspired by, but importantly different from, the Money Pump argument for transitivity of preference and Dutch book arguments for quantitative coherence. The argument is supported by a theorem, to the effect that a subject is systematically susceptible to dominance given her preferred acts, (...) if and only if the subject’s comparative judgments preclude representation by a standard probability measure. (shrink)

Abstract: It seems that from an epistemological point of view comparative probability has many advantages with respect to a probability measure. It is more reasonable as an evaluation of degrees of rational beliefs. It allows the formulation of a comparative indifference principle free from well known paradoxes. Moreover it makes it possible to weaken the principal principle, so that it becomes more reasonable. But the logical systems of comparative probability do not admit an adequate (...) probability updating, which on the contrary is possible for a probability measure. Therefore we are faced with a true epistemological dilemma between comparative and quantitative probability. (shrink)

We prove that the generalized cancellation axiom for incomplete comparative probability relations introduced by Rios Insua and Alon and Lehrer is stronger than the standard cancellation axiom for complete comparative probability relations introduced by Scott, relative to their other axioms for comparative probability in both the finite and infinite cases. This result has been suggested but not proved in the previous literature.

A possible event always seems to be more probable than an impossible event. Although this constraint, usually alluded to as regularity , is prima facie very attractive, it cannot hold for standard probabilities. Moreover, in a recent paper Timothy Williamson has challenged even the idea that regularity can be integrated into a comparative conception of probability by showing that the standard comparative axioms conflict with certain cases if regularity is assumed. In this note, we suggest that there (...) is a natural weakening of the standard comparative axioms. It is shown that these axioms are consistent both with the regularity condition and with the essential feature of Williamson’s example. (shrink)

1 Choice conjecture In axiomatizing nonclassical extensions of classical sentential logic one tries to make do, if one can, with adding to classical sentential logic a finite number of axiom schemes of the simplest kind and a finite number of inference rules of the simplest kind. The simplest kind of axiom scheme in effect states of a particular formula P that for any substitution of formulas for atoms the result of its application to P is to count as an axiom. (...) The simplest kind of onepremise inference rule in effect states of a particular pair of formulas P and Q that for any substitution of formulas for atoms, if the result of its application to P is a theorem, then the result of its application to Q is to count as a theorem; similarly for many-premise rules. Such are the schemes and rules of all the best-known modal and tense logics, for instance. Sometimes it is difficult to find such simple schemes and rules. In that case one may resort to less simple schemes or less simple rules. There is no generally recognized rigorous definition of "next simplest kind" of scheme. Neither is there any generally recognized definition of "next simplest kind" of rule, and hence there is no fully rigorous enunciation of the choice conjecture, the conjecture that schemes of the next simplest kind can always be avoided in favor of rules of the next simplest kind and vice versa. Nonetheless, there are cases where intuitively one does recognize that the schemes or rules in a given axiomatization are only slightly more complex than the simplest kind, including cases where one does have a choice between adopting slightly-more-complex-than-simplest schemes and adopting slightly-more-complex-than-simplest rules. In tense logic early examples of slightly more complex rules are found in [2] and [3] : there is one example of the embarrassed use of such rules in the former, and many examples of the enthusiastic use of such rules in the latter and its sequels. Accordingly the rules in question have come to be called "Gabbay-style" rules. (shrink)

Evolution would favor organisms that can make recurrent decisions in accordance with classical probability (CP) theory, because such choices would be optimal in the long run. This is illustrated by the base-rate fallacy and probability matching, where nonhumans choose optimally but humans do not. Quantum probability (QP) theory may be able to account for these species differences in terms of orthogonal versus nonorthogonal representations.

Rational agents seem more confident in any possible event than in an impossible event. But if rational credences are real-valued, then there are some possible events that are assigned 0 credence nonetheless. How do we differentiate these events from impossible events then when we order events? de Finetti (1975), Hájek (2012) and Easwaran (2014) suggest that when ordering events, conditional credences and subset relations are as relevant as unconditional credences. I present a counterexample to all their proposals in this paper. (...) While their proposals order possible and impossible events correctly, they deliver the wrong verdict for disjoint possible events assigned equal positive credence. (shrink)

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

This book compares various approaches to the interpretation of quantum mechanics, in particular those which are related to the key words "the Copenhagen interpretation", "the antirealist view", "quantum logic" and "hidden variable theory". Using the concept of "correlation" carefully analyzed in the context of classical probability and in quantum theory, the author provides a framework to compare these approaches. He also develops an extension of probability theory to construct a local hidden variable theory. The book should be of (...) interest for physicists and philosophers of science interested in the foundations of quantum theory. (shrink)

Some propositions add more information to bodies of propositions than do others. We start with intuitive considerations on qualitative comparisons of information added . Central to these are considerations bearing on conjunctions and on negations. We find that we can discern two distinct, incompatible, notions of information added. From the comparative notions we pass to quantitative measurement of information added. In this we borrow heavily from the literature on quantitative representations of qualitative, comparative conditional probability. We look (...) at two ways to obtain a quantitative conception of information added. One, the most direct, mirrors Bernard Koopman’s construction of conditional probability: by making a strong structural assumption, it leads to a measure that is, transparently, some function of a function P which is, formally, an assignment of conditional probability (in fact, a Popper function). P reverses the information added order and mislocates the natural zero of the scale so some transformation of this scale is needed but the derivation of P falls out so readily that no particular transformation suggests itself. The Cox–Good–Aczél method assumes the existence of a quantitative measure matching the qualitative relation, and builds on the structural constraints to obtain a measure of information that can be rescaled as, formally, an assignment of conditional probability. A classical result of Cantor’s, subsequently strengthened by Debreu, goes some way towards justifying the assumption of the existence of a quantitative scale. What the two approaches give us is a pointer towards a novel interpretation of probability as a rescaling of a measure of information added. (shrink)

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

According to the Lockean thesis, a proposition is believed just in case it is highly probable. While this thesis enjoys strong intuitive support, it is known to conflict with seemingly plausible logical constraints on our beliefs. One way out of this conflict is to make probability 1 a requirement for belief, but most have rejected this option for entailing what they see as an untenable skepticism. Recently, two new solutions to the conflict have been proposed that are alleged to (...) be non-skeptical. We compare these proposals with each other and with the Lockean thesis, in particular with regard to the question of how much we gain by adopting any one of them instead of the probability 1 requirement, that is, of how likely it is that one believes more than the things one is fully certain of. (shrink)

Recent years have witnessed a proliferation of attempts to apply the mathematical theory of probability to the semantics of natural language probability talk. These sorts of “probabilistic” semantics are often motivated by their ability to explain intuitions about inferences involving “likely” and “probably”—intuitions that Angelika Kratzer’s canonical semantics fails to accommodate through a semantics based solely on an ordering of worlds and a qualitative ranking of propositions. However, recent work by Wesley Holliday and Thomas Icard has been widely (...) thought to undercut this motivation: they present a world-ordering semantics that yields essentially the same logic as probabilistic semantics. In this paper, I argue that the challenge remains: defenders of world-ordering semantics have yet to offer a plausible semantics that captures the logic of comparative likelihood. Holliday & Icard’s semantics yields an adequate logic only if models are restricted to Noetherian pre-orders. But I argue that the Noetherian restriction faces problems in cases involving infinitely large domains of epistemic possibilities. As a result, probabilistic semantics remains the better explanation of the data. (shrink)

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

The leading accounts of the nature of causation divide into probability-raising and process-linkage views. On the probability-raising view, causation is rooted in the comparative probability of the effect with the cause versus without. On the process-linkage view, causation is rooted in the existence of a connecting line from cause to effect. I propose a third alternative which synthesizes these views while solving their problem cases. On this alternative, causation is rooted in the comparative probability of (...) the connecting line to the effect with the cause versus without: causes as probability raisers of processes. In the first two sections, I (briefly) outline the probability-raising and process-linkage views and identify a space of problems, and then I focus on developing the alternative probability-raisers-of-processes solution. I identify problems for this view as well, though I conclude that it is at least in important respects a step forward. (shrink)

How do we ascribe subjective probability? In decision theory, this question is often addressed by representation theorems, going back to Ramsey (1926), which tell us how to define or measure subjective probability by observable preferences. However, standard representation theorems make strong rationality assumptions, in particular expected utility maximization. How do we ascribe subjective probability to agents which do not satisfy these strong rationality assumptions? I present a representation theorem with weak rationality assumptions which can be used to (...) define or measure subjective probability for partly irrational agents. (shrink)

We must restrict to mere probability not only statements of comparatively great uncertainty, like predictions about the weather, where we would cautiously ...

It is a consequence of the theory of imprecise credences that there exist situations in which rational agents inevitably become less opinionated toward some propositions as they gather more evidence. The fact that an agent's imprecise credal state can dilate in this way is often treated as a strike against the imprecise approach to inductive inference. Here, we show that dilation is not a mere artifact of this approach by demonstrating that opinion loss is countenanced as rational by a substantially (...) broader class of normative theories than has been previously recognised. Specifically, we show that dilation-like phenomena arise even when one abandons the basic assumption that agents have (precise or imprecise) credences of any kind, and follows directly from bedrock norms for rational comparative confidence judgements of the form `I am at least as confident in p as I am in q'. We then use the comparative confidence framework to develop a novel understanding of what exactly gives rise to dilation-like phenomena. By considering opinion loss in this more general setting, we are able to provide a novel assessment of the prospects for an account of inductive inference that is not saddled with the inevitability of rational opinion loss. (shrink)

This paper addresses the apparent mismatch between the normative and descriptive literatures in the cognitive science of conditional reasoning. Descriptive psychological theories still regard material implication as the normative theory of the conditional. However, over the last 20 years in the philosophy of language and logic the idea that material implication can account for everyday indicative conditionals has been subject to severe criticism. The majority view is now apparently in favour of a subjective conditional probability interpretation. A comparative (...) model fitting exercise is presented that shows that a conditional probability model can explain as much of the data on abstract indicative conditional reasoning tasks as psychological theories that supplement material implication with various rationally unjustified processing assumptions. Consequently, when people are asked to solve laboratory reasoning tasks, they can be seen as simply generalising their everyday probabilistic reasoning strategies to this novel context. (shrink)

Life distinguishes itself from non-life in taking advantage of the cohesion of temporal origin which non-life cannot afford. The temporal cohesion letting the local participants adhere to each other in a contemporaneous manner refers to an instance of the precedent product being pulled into the subsequent production. Setting the precedent is equivalent to preparing the conditions for the subsequent to follow. A concrete implementation of the cohesion of temporal origin, compared with the spatial cohesion common in physics, is found in (...) the natural construction of a reaction cycle with use of the temporal affinity exerted from the immediate local environment. That construction is a temporally local phenomenon in the experiential domain, rather than in the theoretical. The cohesion originating in the local environment is due to the local act of measurement by the environment. A major component of the local environment to each reactant in the reaction cycle is the cycle itself. The cohesion specific to the reaction cycle rests upon the fact that every reaction product from the upstream production in the cycle comes to be fed upon by the immediate downstream production. Every production constituting the reaction cycle is temporally adjacent to and contemporaneous with the similar others residing in the whole cycle, in sharp contrast to the case of the open-ended linear chain of reaction. One externalist scheme of appreciating the internalist enterprise of constructing a durable reaction cycle in a contemporaneous manner may become possible as referring to the Bayesian probability. The durable reaction cycle may be made actual with probability unity under the condition that the products from the preceding production come with the protocol for the similar production to come subsequently. (shrink)

Many groups make decisions over multiple interconnected propositions. The “doctrinal paradox” or “discursive dilemma” shows that propositionwise majority voting can generate inconsistent collective sets of judgments, even when individual sets of judgments are all consistent. I develop a simple model for determining the probability of the paradox, given various assumptions about the probability distribution of individual sets of judgments, including impartial culture and impartial anonymous culture assumptions. I prove several convergence results, identifying when the probability of the (...) paradox converges to 1, and when it converges to 0, as the number of individuals increases. Drawing on the Condorcet jury theorem and work by Bovens and Rabinowicz , I use the model to assess the “truth-tracking” performance of two decision procedures, the premise- and conclusion-based procedures. I compare the present results with existing results on the probability of Condorcet’s paradox. I suggest that the doctrinal paradox is likely to occur under plausible conditions. (shrink)

Iterated conditionals of the form If p, then if q, r are an important topic in philosophical logic. In recent years, psychologists have gained much knowledge about how people understand simple conditionals, but there are virtually no published psychological studies of iterated conditionals. This paper presents experimental evidence from a study comparing the iterated form, If p, then if q, r with the “imported,” noniterated form, If p and q, then r, using a probability evaluation task and a truth-table (...) task, and taking into account qualitative individual differences. This allows us to critically contrast philosophical and psychological approaches that make diverging predictions regarding the interpretation of these forms. The results strongly support the probabilistic Adams conditional and the “new paradigm” that takes this conditional as a starting point. (shrink)

This paper assesses the comparative reliability of two belief-revision rules relevant to the epistemology of disagreement, the Equal Weight and Stay the Course rules. I use two measures of reliability for probabilistic belief-revision rules, calibration and Brier Scoring, to give a precise account of epistemic peerhood and epistemic reliability. On the calibration measure of reliability, epistemic peerhood is easy to come by, and employing the Equal Weight rule generally renders you less reliable than Staying the Course. On the Brier-Score (...) measure of reliability, epistemic peerhood is much more difficult to come by, but employing the Equal Weight rule always renders you more reliable than Staying the Course. I conclude with some normative lessons we can draw from these formal results. (shrink)

PROBEX (PROBabilities from EXemplars), a model of probabilistic inference and probability judgment based on generic knowledge is presented. Its properties are that: (a) it provides an exemplar model satisfying bounded rationality; (b) it is a “lazy” algorithm that presumes no pre‐computed abstractions; (c) it implements a hybrid‐representation, similarity‐graded probability. We investigate the ecological rationality of PROBEX and find that it compares favorably with Take‐The‐Best and multiple regression (Gigerenzer, Todd, & the ABC Research Group, 1999). PROBEX is fitted to (...) the point estimates, decisions, and probability assessments by human participants. The best fit is obtained for a version that weights frequency heavily and retrieves only two exemplars. It is proposed that PROBEX implements speed and frugality in a psychologically plausible way. (shrink)

The paper investigates exchangeability in the context of probability logic. We study generalizations of basic inference rules and inferences involving cardinalities. We compare the results with those obtained in the case in which only identical probabilities are assumed.

I introduce a mathematical account of expectation based on a qualitative criterion of coherence for qualitative comparisons between gambles (or random quantities). The qualitative comparisons may be interpreted as an agent’s comparative preference judgments over options or more directly as an agent’s comparative expectation judgments over random quantities. The criterion of coherence is reminiscent of de Finetti’s quantitative criterion of coherence for betting, yet it does not impose an Archimedean condition on an agent’s comparative judgments, it does (...) not require the binary relation reflecting an agent’s comparative judgments to be reflexive, complete or even transitive, and it applies to an absolutely arbitrary collection of gambles, free of structural conditions (e.g., closure, measurability, etc.). Moreover, unlike de Finetti’s criterion of coherence, the qualitative criterion respects the principle of weak dominance, a standard of rational decision making that obliges an agent to reject a gamble that is possibly worse and certainly no better than another gamble available for choice. Despite these weak assumptions, I establish a qualitative analogue of de Finetti’s Fundamental Theorem of Prevision, from which it follows that any coherent system of comparative expectations can be extended to a weakly ordered coherent system of comparative expectations over any collection of gambles containing the initial set of gambles of interest. The extended weakly ordered coherent system of comparative expectations satisfies familiar additivity and scale invariance postulates (i.e., independence) when the extended collection forms a linear space. In the course of these developments, I recast de Finetti’s quantitative account of coherent prevision in the qualitative framework adopted in this article. I show that comparative previsions satisfy qualitative analogues of de Finetti’s famous bookmaking theorem and his Fundamental Theorem of Prevision.The results of this article complement those of another article (Pedersen, Strictly coherent preferences, no holds barred, Manuscript, 2013). I explain how those results entail that any coherent weakly ordered system of comparative expectations over a unital linear space can be represented by an expectation function taking values in a (possibly non-Archimedean) totally ordered field extension of the system of real numbers. The ordered field extension consists of formal power series in a single infinitesimal, a natural and economical representation that provides a relief map tracing numerical non-Archimedean features to qualitative non-Archimedean features. (shrink)

In this article, I aim at showing how powers may ground different types of probability in the universe. In Section 1 I single out several dimensions along which the probability of something can be determined. Each of such dimensions can be further specified at the type-level or at the token-level. In Section 2 I introduce some metaphysical assumptions about powers. In Section 3 I show how powers can ground single-case probabilities and frequency-probabilities in a deterministic setting. Later on, (...) in Section 4, I move to a theoretical framework where the falsity of determinism is assumed. Within such a framework, I first argue that some probabilities are grounded on basic powers. Moreover, in Section 5, I introduce tendencies and suggest that they are endowed with specific degrees of activation that may change over time. Such degrees explain why tendencies are more likely to be activated than non-activated, or vice versa. In Section 6 I compare my account of tendencies with other accounts. Finally, in Section 7, I anticipate some general objections against my account – objections that it shares with propensity-accounts of probability – and against degrees of activation of tendencies. (shrink)

Iterated conditionals of the form If p, then if q, r are an important topic in philosophical logic. In recent years, psychologists have gained much knowledge about how people understand simple conditionals, but there are virtually no published psychological studies of iterated conditionals. This paper presents experimental evidence from a study comparing the iterated form, If p, then if q, r with the “imported,” noniterated form, If p and q, then r, using a probability evaluation task and a truth‐table (...) task, and taking into account qualitative individual differences. This allows us to critically contrast philosophical and psychological approaches that make diverging predictions regarding the interpretation of these forms. The results strongly support the probabilistic Adams conditional and the “new paradigm” that takes this conditional as a starting point. (shrink)

The conjunction fallacy has been a key topic in debates on the rationality of human reasoning and its limitations. Despite extensive inquiry, however, the attempt to provide a satisfactory account of the phenomenon has proved challenging. Here we elaborate the suggestion (first discussed by Sides, Osherson, Bonini, & Viale, 2002) that in standard conjunction problems the fallacious probability judgements observed experimentally are typically guided by sound assessments of _confirmation_ relations, meant in terms of contemporary Bayesian confirmation theory. Our main (...) formal result is a confirmation-theoretic account of the conjunction fallacy, which is proven _robust_ (i.e., not depending on various alternative ways of measuring degrees of confirmation). The proposed analysis is shown distinct from contentions that the conjunction effect is in fact not a fallacy, and is compared with major competing explanations of the phenomenon, including earlier references to a confirmation-theoretic account. (shrink)

Following the pioneer work of Bruno De Finetti [12], conditional probability spaces (allowing for conditioning with events of measure zero) have been studied since (at least) the 1950's. Perhaps the most salient axiomatizations are Karl Popper's in [31], and Alfred Renyi's in [33]. Nonstandard probability spaces [34] are a well know alternative to this approach. Vann McGee proposed in [30] a result relating both approaches by showing that the standard values of infinitesimal probability functions are representable as (...) Popper functions, and that every Popper function is representable in terms of the standard real values of some infinitesimal measure. Our main goal in this article is to study the constraints on (qualitative and probabilistic) change imposed by an extended version of McGee's result. We focus on an extension capable of allowing for iterated changes of view. Such extension, we argue, seems to be needed in almost all considered applications. Since most of the available axiomatizations stipulate (definitionally) important constraints on iterated change, we propose a non-questionbegging framework, Iterative Probability Systems (IPS) and we show that every Popper function can be regarded as a Bayesian IPS. A generalized version of McGee's result is then proved and several of its consequences considered. In particular we note that our proof requires the imposition of Cumulativity, i.e. the principle that a proposition that is accepted at any stage of an iterative process of acceptance will continue to be accepted at any later stage. The plausibility and range of applicability of Cumulativity is then studied. In particular we appeal to a method for defining belief from conditional probability (first proposed in [42] and then slightly modified in [6] and [3]) in order to characterize the notion of qualitative change induced by Cumulative models of probability kinematics. The resulting cumulative notion is then compared with existing axiomatizations of belief change and probabilistic supposition. We also consider applications in the probabilistic accounts of conditionals [1] and [30]. (shrink)

I argue that the propensity interpretation of fitness, properly understood, not only solves the explanatory circularity problem and the mismatch problem, but can also withstand the Pandora’s box full of problems that have been thrown at it. Fitness is the propensity (i.e., probabilistic ability, based on heritable physical traits) for organisms or types of organisms to survive and reproduce in particular environments and in particular populations for a specified number of generations; if greater than one generation, “reproduction” includes descendants of (...) descendants. Fitness values can be described in terms of distributions of propensities to produce varying number of offspring and can be modeled for any number of generations using computer simulations, thus providing both predictive power and a means for comparing the fitness of different phenotypes. Fitness is a causal concept, most notably at the population level, where fitness differences are causally responsible for differences in reproductive success. Relative fitness is ultimately what matters for natural selection. (shrink)

Inductive reasoning requires exploiting links between evidence and hypotheses. This can be done focusing either on the posterior probability of the hypothesis when updated on the new evidence or on the impact of the new evidence on the credibility of the hypothesis. But are these two cognitive representations equally reliable? This study investigates this question by comparing probability and impact judgments on the same experimental materials. The results indicate that impact judgments are more consistent in time and more (...) accurate than probability judgments. Impact judgments also predict the direction of errors in probability judgments. These findings suggest that human inductive reasoning relies more on estimating evidential impact than on posterior probability. (shrink)

Order of information plays a crucial role in the process of updating beliefs across time. In fact, the presence of order effects makes a classical or Bayesian approach to inference difficult. As a result, the existing models of inference, such as the belief-adjustment model, merely provide an ad hoc explanation for these effects. We postulate a quantum inference model for order effects based on the axiomatic principles of quantum probability theory. The quantum inference model explains order effects by transforming (...) a state vector with different sequences of operators for different orderings of information. We demonstrate this process by fitting the quantum model to data collected in a medical diagnostic task and a jury decision-making task. To further test the quantum inference model, a new jury decision-making experiment is developed. Using the results of this experiment, we compare the quantum inference model with two versions of the belief-adjustment model, the adding model and the averaging model. We show that both the quantum model and the adding model provide good fits to the data. To distinguish the quantum model from the adding model, we develop a new experiment involving extreme evidence. The results from this new experiment suggest that the adding model faces limitations when accounting for tasks involving extreme evidence, whereas the quantum inference model does not. Ultimately, we argue that the quantum model provides a more coherent account for order effects that was not possible before. (shrink)

This paper addresses the apparent mismatch between the normative and descriptive literatures in the cognitive science of conditional reasoning. Descriptive psychological theories still regard material implication as the normative theory of the conditional. However, over the last 20 years in the philosophy of language and logic the idea that material implication can account for everyday indicative conditionals has been subject to severe criticism. The majority view is now apparently in favour of a subjective conditional probability interpretation. A comparative (...) model fitting exercise is presented that shows that a conditional probability model can explain as much of the data on abstract indicative conditional reasoning tasks as psychological theories that supplement material implication with various rationally unjustified processing assumptions. Consequently, when people are asked to solve laboratory reasoning tasks, they can be seen as simply generalising their everyday probabilistic reasoning strategies to this novel context. (shrink)

It is increasingly common in cognitive science and philosophy of perception to regard perceptual processing as a probabilistic engine, taking into account uncertainty in computing representations of the distal environment. Models of this kind often postulate probabilistic representations, or what we will call probabilistic states,. These are states that in some sense mark or represent information about the probabilities of distal conditions. It has also been argued that perceptual experience itself in some sense represents uncertainty (Morrison _Analytic Philosophy_ 57 (1): (...) 15 48, 2016 ). In this article, we will first consider three models of sensory activity from perceptual neuroscience, namely signal detection theory (SDT), probabilistic population codes (PPC), and sampling. We will then reflect on the sense in which the probabilistic states introduced in these models are probabilistic representations. To sharpen this discussion, we will compare and contrast these probabilistic states to credences as they are understood in epistemology. We will suggest that probabilistic representation, in an appropriately robust sense, can be understood as a form of analog representation. In the last part of the paper, we apply this to the issue of whether conscious experience represents uncertainty—we will interpret this as the claim that there are phenomenal features of experience that serve as analog probabilistic representations. (shrink)

This volume defends a novel approach to the philosophy of physics: it is the first book devoted to a comparative study of probability, causality, and propensity, and their various interrelations, within the context of contemporary physics -- particularly quantum and statistical physics. The philosophical debates and distinctions are firmly grounded upon examples from actual physics, thus exemplifying a robustly empiricist approach. The essays, by both prominent scholars in the field and promising young researchers, constitute a pioneer effort in (...) bringing out the connections between probabilistic, causal and dispositional aspects of the quantum domain. The book will appeal to specialists in philosophy and foundations of physics, philosophy of science in general, metaphysics, ontology of physics theories, and philosophy of probability. (shrink)

Thresholds for disease extinction provide essential information for the prevention and control of diseases. In this paper, a stochastic epidemic model, a continuous-time Markov chain, for the transmission dynamics of West Nile virus in birds is developed based on the assumptions of its analogous deterministic model. The branching process is applied to derive the extinction threshold for the stochastic model and conditions for disease extinction or persistence. The probability of disease extinction computed from the branching process is shown to (...) be in good agreement with the probability approximated from numerical simulations. The disease dynamics of both models are compared to ascertain the effect of demographic stochasticity on West Nile virus dynamics. Analytical and numerical results show differences in model predictions and asymptotic dynamics between stochastic and deterministic models that are crucial for the prevention of disease outbreaks. It is found that there is a high probability of disease extinction if the disease emerges from exposed mosquitoes unlike if it emerges from infectious mosquitoes and birds. Finite-time to disease extinction is estimated using sample paths and it is shown that the epidemic duration is shortest if the disease is introduced by exposed mosquitoes. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy and Rhetoric 36.3 (2003) 223-248 [Access article in PDF] Probabilities, Signs, Necessary Signs, Idia, and Topoi:The Confusing Discussion of Materials for Enthymemes in the Rhetoric Brad McAdon This essay examines three groups of "sources" or "materials" of enthymemes in Aristotle's Rhetoric. According to the text of the Rhetoric, enthymemes are derived from, among other things, probabilities, signs, and necessary signs, and/or from the topics, and/or from idia as (...) the following table indicates.1The Rhetoric does not explain any relationships that may exist between these different sources (or materials) from which enthymemes are said. This problem, as I am framing it, is two-fold: First, these different conceptions of materials from which enthymemes are said are never reconciled into a coherent account (or theory) in the Rhetoric. Rather, each conception of materials for the enthymeme is presented in a way that obscures, rather than facilitates, our understanding of the materials of enthymemes in the Rhetoric. Moreover, when Rhetoric 2.22-23 and its textual contexts are compared to 1.2 and its textual contexts, several differences will emerge [End Page 223] in respect to the nature of the materials of enthymemes and the distinctions in enthymemes that further cloud our understanding of the text. Second, many contemporary commentators place more emphasis on the dialectical aspects of rhetoric than the demonstrative. For while rhetoric is said to be closely related to dialectic in important ways in the early portions of the Rhetoric (especially 1.1-2), there are, arguably, more similarities between the materials of enthymemes in the Prior and Posterior Analytics (and thus demonstrative argument) than in the Topics (and dialectical argument). This suggests that the discussion of the materials of enthymemes in the Rhetoric is more demonstrative in nature (or at least in emphasis) than dialectical, and I hope to demonstrate that this is the case.As far as I can tell, Grimaldi's discussions of the "Eikota and semeia" (1972, 104-15) and "The topoi" (115-35) are the only contemporary attempts to explicitly address the relationship between probabilities and signs and the topoi in the Rhetoric. 2 Grimaldi's account is confusing in many respects. For example, in addition to understanding the idia as topoi, he claims that "[w]hen Aristotle has presented us with the sources of the enthymeme, eikota and semeia, and told us that they are the premises of enthymematic reasoning (A3, 59a7-10) the question which immediately arises is where does the rhetor turn for the material which will provide him with such eikota and semeia. The answer is: the topics" (115). Yet, a little further down the same page, he claims that "the topics are sources, or loci, both particular and general, to which one must have recourse in constructing probable argumentation by enthymeme in an effort to effect pistis" (115). So, in effect, Grimaldi claims that probabilities and signs are the sources of enthymemes and then that the topics are the sources of probabilities and signs, 3 but, as demonstrated above, the text of the Rhetoric clearly claims that the topoi are the sources of enthymemes just as probabilities, signs, and necessary signs are sources of enthymemes. Grimaldi fails to address or even acknowledge this. Rather, in his attempt to maintain the unity of the text of the Rhetoric, he has advanced this working relationship between eikota and se-meiaand the topoi in what he claims is a consistent theory in the Rhetoric concerning the relationship of eikota and se-meia (and tekme-ria)to the topoi that, I will argue, is not supported by the text as Grimaldi suggests. More recently, Robin Smith (1989 and 1997) provides carefully informed commentary on some of these terms—topos, idia, eikos, se-meia, and tekme-ria—but, and this is not his purpose, he does not provide a discussion that explicitly examines how (or if) these different terms relate to one another as sources of enthymemes. Thus, this essay is an attempt to explore and succinctly articulate the problems with the... (shrink)

We present four new change detection methods that create an automated change map from a probability map. In this case, the probability map was derived from a 3D model. The primary application of interest is aerial photographic applications, where the appearance, disappearance or change in position of small objects of a selectable class (e.g., cars) must be detected at a high success rate in spite of variations in magnification, lighting and background across the image. The methods rely on (...) an earlier derivation of a probability map. We describe the theory of the four methods, namely Bernoulli variables, Markov Random Fields, connected change, and relaxation-based segmentation, evaluate and compare their performance experimentally on a set probability maps derived from aerial photographs. (shrink)