In finite probability theory, events are subsets S⊆U of the outcome set. Subsets can be represented by 1-dimensional column vectors. By extending the representation of events to two dimensional matrices, we can introduce "superposition events." Probabilities are introduced for classical events, superposition events, and their mixtures by using density matrices. Then probabilities for experiments or `measurements' of all these events can be determined in a manner exactly like in quantum mechanics (QM) using density matrices. Moreover the transformation of the density (...) matrices induced by the experiments or `measurements' is the Lüders mixture operation as in QM. And finally by moving the machinery into the n-dimensional vector space over ℤ₂, different basis sets become different outcome sets. That `non-commutative' extension of finite probability theory yields the pedagogical model of quantum mechanics over ℤ₂ that can model many characteristic non-classical results of QM. (shrink)

We describe a dual-process theory of how individuals estimate the probabilities of unique events, such as Hillary Clinton becoming U.S. President. It postulates that uncertainty is a guide to improbability. In its computer implementation, an intuitive system 1 simulates evidence in mental models and forms analog non-numerical representations of the magnitude of degrees of belief. This system has minimal computational power and combines evidence using a small repertoire of primitive operations. It resolves the uncertainty of divergent evidence for single events, (...) for conjunctions of events, and for inclusive disjunctions of events, by taking a primitive average of non-numerical probabilities. It computes conditional probabilities in a tractable way, treating the given event as evidence that may be relevant to the probability of the dependent event. A deliberative system 2 maps the resulting representations into numerical probabilities. With access to working memory, it carries out arithmetical operations in combining numerical estimates. Experiments corroborated the theory's predictions. Participants concurred in estimates of real possibilities. They violated the complete joint probability distribution in the predicted ways, when they made estimates about conjunctions: P, P, P, disjunctions: P, P, P, and conditional probabilities P, P, P. They were faster to estimate the probabilities of compound propositions when they had already estimated the probabilities of each of their components. We discuss the implications of these results for theories of probabilistic reasoning. (shrink)

This paper reports an experiment which investigates a possible cognitive antecedent of event-splitting effects (ESEs) experimentally observed by Starmer and Sugden (1993) and Humphrey (1995) – the learning of absolute frequency of event category impacting on the learning of probability of event category – and reveals some evidence that it is responsible for observed ESEs. It is also suggested and empirically substantiated that stripped-down prospect theory will accurately predict ESEs in some decision making tasks, but will not (...) perform well in others. This contention, it is argued, is indicative of fundamental descriptive shortcomings in the economic conception of choice under uncertainty and may entail implications beyond the direct concerns of this paper. (shrink)

We introduce the concept of partial event as a pair of disjoint sets, respectively the favorable and the unfavorable cases. Partial events can be seen as a De Morgan algebra with a single fixed point for the complement. We introduce the concept of a measure of partial probability, based on a set of axioms resembling Kolmogoroff’s. Finally we define a concept of conditional probability for partial events and apply this concept to the analysis of the two-slit experiment in quantum (...) mechanics. (shrink)

A logic for classical conditional events was investigated by Dubois and Prade. In their approach, the truth value of a conditional event may be undetermined. In this paper we extend the treatment to many-valued events. Then we support the thesis that probability over partially undetermined events is a conditional probability, and we interpret it in terms of bets in the style of de Finetti. Finally, we show that the whole investigation can be carried out in a logical and algebraic (...) setting, and we find a logical characterization of coherence for assessments of partially undetermined events. (shrink)

Subjects were requested to choose between gambles, where the outcome of one gamble depended on a single elementary event, and the other depended on an event compounded of a series of such elementary events. The data supported the hypothesis that the subjective probability of a compound event is systematically biased in the direction of the probability of its components resulting in overestimation of conjunctive events and underestimation of disjunctive events. Studies pertaining to this topic are discussed.

The paper investigates what are the proper procedures for calculating the probability on certain evidence of a particular object e having a property, Q, e.g. of Eclipse winning the Derby. Let `α ' denote the conjunction of properties known to be possessed by e, and P(Q)/α the probability of an object which is α being Q. One view is that the probability of e being Q is given by the best confirmed value of P(Q)/α . This view is shown not (...) to be generally true, but to provide a useful approximation in many cases. Then given that we have information about the observed frequencies of Q among objects having one or more of the properties whose conjunction forms α , the paper shows how to establish which value of P(Q)/α is the best confirmed one. (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)

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)

Quantum mechanics and probability theory share one peculiarity. Both have well established mathematical formalisms, yet both are subject to controversy about the meaning and interpretation of their basic concepts. Since probability plays a fundamental role in QM, the conceptual problems of one theory can affect the other. We first classify the interpretations of probability into three major classes: inferential probability, ensemble probability, and propensity. Class is the basis of inductive logic; deals with the frequencies of events in repeatable experiments; describes (...) a form of causality that is weaker than determinism. An important, but neglected, paper by P. Humphreys demonstrated that propensity must differ mathematically, as well as conceptually, from probability, but he did not develop a theory of propensity. Such a theory is developed in this paper. Propensity theory shares many, but not all, of the axioms of probability theory. As a consequence, propensity supports the Law of Large Numbers from probability theory, but does not support Bayes theorem. Although there are particular problems within QM to which any of the classes of probability may be applied, it is argued that the intrinsic quantum probabilities are most naturally interpreted as quantum propensities. This does not alter the familiar statistical interpretation of QM. But the interpretation of quantum states as representing knowledge is untenable. Examples show that a density matrix fails to represent knowledge. (shrink)

The term probability can be used in two main senses. In the frequency interpretation it is a limiting ratio in a sequence of repeatable events. In the Bayesian view, probability is a mental construct representing uncertainty. This 2002 book is about these two types of probability and investigates how, despite being adopted by scientists and statisticians in the eighteenth and nineteenth centuries, Bayesianism was discredited as a theory of scientific inference during the 1920s and 1930s. Through the examination of a (...) dispute between two British scientists, the author argues that a choice between the two interpretations is not forced by pure logic or the mathematics of the situation, but depends on the experiences and aims of the individuals involved. The book should be of interest to students and scientists interested in statistics and probability theories and to general readers with an interest in the history, sociology and philosophy of science. (shrink)

The external worlds do not objectively exist for living systems because these worlds are unknown from within systems. How can they escape solipsism to survive and reproduce as open systems? Living systems must construct their hypothetical models of external entities in the form of their internal structures to determine how to change states (i.e., sense and act) appropriately to achieve a favorable probability distribution of the events they experience. The model construction involves the generation of symbols referring to external entities. (...) This paper attempts to provide a new view that living systems are an inverse-causality operator. Inverse causality (IC) is an algorithmic process that generates symbols referring to external reality states based on a given data sequence. For applying this logical model involving if–then entailments to living systems involving material interactions, the cognizers-system model was employed to represent the IC process; here, living systems were modeled as a subject of cognition and action. A focal subject system is described as a cognizer composed of sub-cognizers, such as a sensor, a signal transducer, and an effector. Analysis using this model proposes that living systems invented the “measurers” for conducting IC operations through their evolution. (shrink)

The first clear and precise statement of the axioms of qualitative probability was given by de Finetti ([1], Section 13). A more detailed treatment, based however on more complex axioms for conditional qualitative probability, was given later by Koopman [5]. De Finetti and Koopman derived a probability measure from a qualitative probability under the assumption that, for any integer n, there are n mutually exclusive, equally probable events. L. J. Savage [6] has shown that this strong assumption is unnecessary. (...) More precisely, he proves that if a qualitative probability is only fine and tight, then there is one and only one probability measure compatible with it. No property equivalent to countable additivity has been used as yet in the development of qualitative probability theory. However, since the concept of countable additivity is of such fundamental importance in measure theory, it is to be expected that an equivalent property would be of interest in qualitative probability theory, and that in particular it would simplify the proof of the existence of compatible probability measures. Such a property is introduced in this paper, under the name of monotone continuity. It is shown that, if a qualitative probability is atomless and monotonely continuous, then there is one and only one probability measure compatible with it, and this probability measure is countably additive. It is also proved that any fine and tight qualitative probability can be extended to a monotonely continuous qualitative probability, and therefore, contrary to what might be expected, there is no loss in generality if we consider only qualitative probabilities which are monotonely continuous. At the present time there is still a controversy over the interpretation which should be given to the word probability in the scientific and technical literature. Although the present writer subscribes to the opinion that this interpretation may be different in different contexts, in this paper we do not enter into this controversy. We simply remark that a qualitative probability, as a numerical one, may be interpreted either as an objective or as a subjective probability, and therefore the following axiomatic theory is compatible with both interpretations of probability. (shrink)

Probability theory promises to deliver an exact and unified foundation for inquiry in epistemology and philosophy of science. But philosophy of religion is also fertile ground for the application of probabilistic thinking. This volume presents original contributions from twelve contemporary researchers, both established and emerging, to offer a representative sample of the work currently being carried out in this potentially rich field of inquiry. Grouped into five parts, the chapters span a broad range of traditional issues in religious epistemology. The (...) first three parts discuss the evidential impact of various considerations that have been brought to bear on the question of the existence of God. These include witness reports of the occurrence of miraculous events, the existence of complex biological adaptations, the apparent 'fine-tuning' for life of various physical constants and the existence of seemingly unnecessary evil. The fourth part addresses a number of issues raised by Pascal's famous pragmatic argument for theistic belief. A final part offers probabilistic perspectives on the rationality of faith and the epistemic significance of religious disagreement. (shrink)

It is possible that a fair coin tossed infinitely many times will always land heads. So the probability of such a sequence of outcomes should, intuitively, be positive, albeit miniscule: 0 probability ought to be reserved for impossible events. And, furthermore, since the tosses are independent and the probability of heads (and tails) on a single toss is half, all sequences are equiprobable. But Williamson has adduced an argument that purports to show that our intuitions notwithstanding, the probability of an (...) infinite sequence is 0. In this paper, I rebut his argument.No Abstract. (shrink)

A probability distribution is regular if no possible event is assigned probability zero. While some hold that probabilities should always be regular, three counter-arguments have been posed based on examples where, if regularity holds, then perfectly similar events must have different probabilities. Howson (2017) and Benci et al. (2016) have raised technical objections to these symmetry arguments, but we see here that their objections fail. Howson says that Williamson’s (2007) “isomorphic” events are not in fact isomorphic, but Howson is (...) speaking of set-theoretic representations of events in a probability model. While those sets are not isomorphic, Williamson’s physical events are, in the relevant sense. Benci et al. claim that all three arguments rest on a conflation of different models, but they do not. They are founded on the premise that similar events should have the same probability in the same model, or in one case, on the assumption that a single rotation-invariant distribution is possible. Having failed to refute the symmetry arguments on such technical grounds, one could deny their implicit premises, which is a heavy cost, or adopt varying degrees of instrumentalism or pluralism about regularity, but that would not serve the project of accurately modelling chances. (shrink)

Probabilistic inference forms lead from point probabilities of the premises to interval probabilities of the conclusion. The probabilistic version of Modus Ponens, for example, licenses the inference from \({P(A) = \alpha}\) and \({P(B|A) = \beta}\) to \({P(B)\in [\alpha\beta, \alpha\beta + 1 - \alpha]}\) . We study generalized inference forms with three or more premises. The generalized Modus Ponens, for example, leads from \({P(A_{1}) = \alpha_{1}, \ldots, P(A_{n})= \alpha_{n}}\) and \({P(B|A_{1} \wedge \cdots \wedge A_{n}) = \beta}\) to an according interval for (...) P(B). We present the probability intervals for the conclusions of the generalized versions of Cut, Cautious Monotonicity, Modus Tollens, Bayes’ Theorem, and some SYSTEM O rules. Recently, Gilio has shown that generalized inference forms “degrade”—more premises lead to less precise conclusions, i.e., to wider probability intervals of the conclusion. We also study Adam’s probability preservation properties in generalized inference forms. Special attention is devoted to zero probabilities of the conditioning events. These zero probabilities often lead to different intervals in the coherence and the Kolmogorov approach. (shrink)

A proposal for an objective interpretation of probability is introduced and discussed: probabilities as deriving from ranges in suitably structured initial-state spaces. Roughly, the probability of an event on a chance trial is the proportion of initial states that lead to the event in question within the space of all possible initial states associated with this type of experiment, provided that the proportion is approximately the same in any not too small subregion of the space. This I would (...) like to call the “natural-range conception” of probability. Providing a substantial alternative to frequency or propensity accounts of probability in a deterministic setting, it is closely related to the so-called “method of arbitrary functions”. It is explicated, confronted with certain problems, and some ideas how these might be overcome are sketched and discussed. (shrink)

The probability ‘measure’ for measurements at two consecutive mo- ments of time is non-additive. These probabilities, on the other hand, may be determined by the limit of relative frequency of measured events, which are by nature additive. We demonstrate that there are only two ways to resolve this problem. The first solution places emphasis on the precise use of the concept of conditional probability for successive mea- surements. The physically correct conditional probabilities define additive probabilities for two-time measurements. These probabilities (...) depend ex- plicitly on the resolution of the physical device and do not, therefore, correspond to a function of the associated projection operators. It follows that quantum theory distinguishes between physical events and proposi- tions about events, the latter are not represented by projection operators and that the outcomes of two-time experiments cannot be described by quantum logic. The alternative explanation is rather radical: it is conceivable that the relative frequencies for two-time measurements do not converge, unless a particular consistency condition is satisfied. If this is true, a strong revision of the quantum mechanical formalism may prove necessary. We stress that it is possible to perform experiments that will distinguish the two alternatives. (shrink)

Probability measures can be constructed using the measure-theoretic techniques of Caratheodory and Hausdorff. Under these constructions one obtains first an outer measure over "events" or "propositions." Then, if one restricts this outer measure to the measurable propositions, one finally obtains a classical probability theory. What I argue is that outer measures can also be used to yield the structures of probability theories in quantum mechanics, provided we permit them to range over at least some unmeasurable propositions. I thereby show that (...) nonclassical probability theories can be seen to arise naturally within the framework of possible worlds semantics. (shrink)

According to principles of probability coordination, such as Miller's Principle or Lewis's Principal Principle, you ought to set your subjective probability for an event equal to what you take to be the objective probability of the event. For example, you should expect events with a very high probability to occur and those with a very low probability not to occur. This paper examines the grounds of such principles. It is argued that any attempt to justify a principle of (...) probability coordination encounters the same difficulties as attempts to justify induction. As a result, no justification can be found. (shrink)

This paper continues our work on a coherence-based probability semantics for Aristotelian syllogisms (Gilio, Pfeifer, and Sanfilippo, 2016; Pfeifer and Sanfilippo, 2018) by studying Figure III under coherence. We interpret the syllogistic sentence types by suitable conditional probability assessments. Since the probabilistic inference of P|S from the premise set {.

Consider the regularity thesis that each possible event has non-zero probability. Hájek challenges this in two ways: there can be nonmeasurable events that have no probability at all and on a large enough sample space, some probabilities will have to be zero. But arguments for the existence of nonmeasurable events depend on the axiom of choice. We shall show that the existence of anything like regular probabilities is by itself enough to imply a weak version of AC sufficient to (...) prove the Banach–Tarski Paradox on the decomposition of a ball into two equally sized balls, and hence to show the existence of nonmeasurable events. This provides a powerful argument against unrestricted orthodox Bayesianism that works even without AC. A corollary of our formal result is that if every partial order extends to a total preorder while maintaining strict comparisons, then the Banach–Tarski Paradox holds. This yields an argument that incommensurability cannot be avoided in ambitious versions of decision theory. (shrink)

Conditional probability is one of the central concepts in probability theory. Some notion of conditional probability is part of every interpretation of probability. The basic mathematical fact about conditional probability is that p(A |B) = p(A ∧B)/p(B) where this is defined. However, while it has been typical to take this as a definition or analysis of conditional probability, some (perhaps most prominently Hájek, 2003) have argued that conditional probability should instead be taken as the primitive notion, so that this formula (...) is at best coextensive, and at worst sometimes gets it wrong. -/- Section 1.1 considers the concept of conditional probability in each of the major families of interpretation of probability. Section 1.2 considers a conceptual argument for the claim that conditional probability is prior to unconditional probability, while Section 1.3 considers a family of mathematical arguments for this claim, leading to consideration specifically of the question of how to understand probability conditional on events of probability 0. Section 1.4 discusses several mathematical principles that have been alleged to be important for understanding how probability 0 behaves, and raises a dilemma for probability conditional on events of probability 0. Section 2 and Section 3 take the two horns of this dilemma and describe the two main competing families of mathematical accounts of conditional probability for events of probability 0. Section 4 summarizes the results, and their significance for the two arguments that conditional probability is prior to unconditional probability. (shrink)

The article shows, using elementary examples, that risk assessment depends both on the probability of future events and on their results. Various approaches to the integration of probabilities and numerical estimates of possible outcomes are shown and analyzed. The subjective nature of risk is substantiated. An interdisciplinary definition of risk is proposed, not associated with any science or group of sciences. The definition is based on the idea that risk is a consequence of decisions made by the subject, who evaluates (...) their consequences. The question is raised about responsibility for decisions under risk conditions. (shrink)

Today philosophical discussion on indicative conditionals is dominated by the so called Lewis Triviality Results, according to which, tehere is no binary connective '-->' (let alone truth-functional) such that the probability of p --> q equals the probability of q conditionally on p, so that P(p --> q)= P(q|p). This tenet, that suggests that conditonals lack truth-values, has been challenged in 1991 by Goodman et al. who show that using a suitable three-valued logic the above equation may be restored. In (...) this paper it is first analysed a long neglected paper by Bruno de Finetti, written in 1935, where the essentials of Goodman's theory was clearly outlined. It is also stressed that de Finetti anticipated Kleene's as well as Bochvar and Blamey ideas. In the second part of the paper it is argued that the de Finetti-Goodman's original theory is defective and leads to absurd results. However, a new semantics, called semantics of hypervaluations, is here defined, that avoids the defects of the original theory. This appears to be a powerful challenge to Lewis Triviality results and to the thesis by which conditionals lack truth-values as well. (shrink)

Various proposals for generalizing event spaces for probability functions have been put forth in the mathematical, scientific, and philosophic literatures. In cognitive psychology such generalizations are used for explaining puzzling results in decision theory and for modeling the influence of context effects. This commentary discusses proposals for generalizing probability theory to event spaces that are not necessarily boolean algebras. Two prominent examples are quantum probability theory, which is based on the set of closed subspaces of a Hilbert space, (...) and topological probability theory, which is based on the set of open sets of a topology. Both have been applied to a variety of cognitive situations. This commentary focuses on how event space properties can influence probability concepts and impact cognitive modeling. (shrink)

It is argued that probability should be defined implicitly by the distributions of possible measurement values characteristic of a theory. These distributions are tested by, but not defined in terms of, relative frequencies of occurrences of events of a specified kind. The adoption of an a priori probability in an empirical investigation constitutes part of the formulation of a theory. In particular, an assumption of equiprobability in a given situation is merely one hypothesis inter alia, which can be tested, like (...) any other assumption. Probability in relation to some theories – for example quantum mechanics – need not satisfy the Kolmogorov axioms. To illustrate how two theories about the same system can generate quite different probability concepts, and not just different probabilistic predictions, a team game for three players is described. If only classical methods are allowed, a 75% success rate at best can be achieved. Nevertheless, a quantum strategy exists that gives a 100% probability of winning. (shrink)