The representational epistemic approach to the design of visual displays and notation systems advocates encoding the fundamental conceptual structure of a knowledge domain directly in the structure of a representational system. It is claimed that representations so designed will benefit from greater semantic transparency, which enhances comprehension and ease of learning, and plastic generativity, which makes the meaningful manipulation of the representation easier and less error prone. Epistemic principles for encoding fundamental conceptual structures directly in representational schemes are described. (...) The diagrammatic recodification of probability theory is undertaken to demonstrate how the fundamental conceptual structure of a knowledge domain can be analyzed, how the identified conceptual structure may be encoded in a representational system, and the cognitive benefits that follow. An experiment shows the new probability space diagrams are superior to the conventional approach for learning this conceptually challenging topic. (shrink)

We believe that when assessing the likelihood of uncertain events, statistically unsophisticated people utilize a coarse internal scale that only has a limited number of categories. The success of the nested sets hypothesis may lie in its ability to provide an appropriate set structure of the problem by reducing the computational demands.

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)

This paper is a sequel to the joint publication of Scott and Krauss in which the first aspects of a mathematical theory are developed which might be called "First Order Probability Logic". No attempt will be made to present this additional material in a self-contained form. We will use the same notation and terminology as introduced and explained in Scott and Krauss, and we will frequently refer to the theorems stated and proved in the preceding paper. The main objective (...) of this study is to show that the probability of symmetric probability systems may be represented as a "weighted average" of what might be called "product probabilities". We then discuss some applications of our results to Carnap's "Principle of instantial relevance", which plays an important role in his system of inductive logic. (shrink)

The quantum state of a d-dimensional system can be represented by a probability distribution over the d 2 outcomes of a Symmetric Informationally Complete Positive Operator Valued Measure (SIC-POVM), and then this probability distribution can be represented by a vector of $\mathbb {R}^{d^{2}-1}$ in a (d 2−1)-dimensional simplex, we will call this set of vectors $\mathcal{Q}$ . Other way of represent a d-dimensional system is by the corresponding Bloch vector also in $\mathbb {R}^{d^{2}-1}$ , we will call this (...) set of vectors $\mathcal{B}$ . In this paper it is proved that with the adequate scaling $\mathcal{B}=\mathcal{Q}$ . Also we indicate some features of the shape of $\mathcal{Q}$. (shrink)

The first English translation of Hans Reichenbach's lucid doctoral thesis sheds new light on how Kant’s Critique of Pure Reason was understood in some quarters at the time. The source of several themes in his still influential The Direction of Time, the thesis shows Reichenbach's early focus on the interdependence of physics, probability, and epistemology.

Bayesian, Jeffrey and Field conditionals are compared and it is shown why the last two cannot be reduced to the first. Maximum relative entropy is used in two kinds of justification of the Field conditional and the dispensability of entropy principles in general is discussed.

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)

Nick Shea’s Representation in Cognitive Science commits him to representations in perceptual processing that are about probabilities. This commentary concerns how to adjudicate between this view and an alternative that locates the probabilities rather in the representational states’ associated “attitudes”. As background and motivation, evidence for probabilistic representations in perceptual processing is adduced, and it is shown how, on either conception, one can address a specific challenge Ned Block has raised to this evidence.

In the context of propositional logics, we apply semantics modulo satisfiability—a restricted semantics which comprehends only valuations that satisfy some specific set of formulas—with the aim to efficiently solve some computational tasks. Three possible such applications are developed.We begin by studying the possibility of implicitly representing rational McNaughton functions in Łukasiewicz Infinitely-valued Logic through semantics modulo satisfiability. We theoretically investigate some approaches to such representation concept, called representation modulo satisfiability, and describe a polynomial algorithm that builds representations in (...) the newly introduced system. An implementation of the algorithm, test results and ways to randomly generate rational McNaughton functions for testing are presented. Moreover, we propose an application of such representations to the formal verification of properties of neural networks by means of the reasoning framework of Łukasiewicz Infinitely-valued Logic.Then, we move to the investigation of the satisfiability of joint probabilistic assignments to formulas of Łukasiewicz Infinitely-valued Logic, which is known to be an NP-complete problem. We provide an exact decision algorithm derived from the combination of linear algebraic methods with semantics modulo satisfiability. Also, we provide an implementation for such algorithm for which the phenomenon of phase transition is empirically detected.Lastly, we study the game theory situation of observable games, which are games that are known to reach a Nash equilibrium, however, an external observer does not know what is the exact profile of actions that occur in a specific instance; thus, such observer assigns subjective probabilities to players actions. We study the decision problem of determining if a set of these probabilistic constraints is coherent by reducing it to the problems of satisfiability of probabilistic assignments to logical formulas both in Classical Propositional Logic and Łukasiewicz Infinitely-valued Logic depending on whether only pure equilibria or also mixed equilibria are allowed. Such reductions rely upon the properties of semantics modulo satisfiability. We provide complexity and algorithmic discussion for the coherence problem and, also, for the problem of computing maximal and minimal probabilistic constraints on actions that preserves coherence.prepared by Sandro Márcio da Silva Preto.E-mail: [email protected]:https://doi.org/10.11606/T.45.2021.tde-17062021-163257. (shrink)

In their constructive reviews, Frances Egan, Randy Gallistel and Steven Gross have raised some important problems for the account of content advanced by Nicholas Shea in Representation in Cognitive Science. Here the author addresses their main challenges. Egan argues that the account includes an unrecognised pragmatic element; and that it makes contents explanatorily otiose. Gallistel raises questions about homomorphism and correlational information. Gross puts the account to work to resolve a dispute about probabilistic contents in perception, but argues that (...) a question remains about whether probabilities are found in the content or instead in the manner of representation. (shrink)

This chapter is a philosophical survey of some leading approaches in formal epistemology in the so-called ‘Bayesian’ tradition. According to them, a rational agent’s degrees of belief—credences—at a time are representable with probability functions. We also canvas various further putative ‘synchronic’ rationality norms on credences. We then consider ‘diachronic’ norms that are thought to constrain how credences should respond to evidence. We discuss some of the main lines of recent debate, and conclude with some prospects for future research.

Tomographic approach to describing both the states in classical statistical mechanics and the states in quantum mechanics using the fair probability distributions is reviewed. The entropy associated with the probability distribution (tomographic entropy) for classical and quantum systems is studied. The experimental possibility to check the inequalities like the position–momentum uncertainty relations and entropic uncertainty relations are considered.

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 propose a model of uncertain belief. This models coherent beliefs by a filter, ????, on the set of probabilities. That is, it is given by a collection of sets of probabilities which are closed under supersets and finite intersections. This can naturally capture your probabilistic judgements. When you think that it is more likely to be sunny than rainy, we have{????|????(????????????????????)>????(????????????????????)}∈????. When you think that a gamble ???? is desirable, we have {????|Exp????[????]>0}∈????. It naturally extends the model of credal (...) sets; and we will show it captures all the expressive power of the desirable gambles model. It also captures the expressive power of sets of desirable gamble sets (with a mixing axiom, but no Archimadean axiom). (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)

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)

A small probability space representation of quantum mechanical probabilities is defined as a collection of Kolmogorovian probability spaces, each of which is associated with a context of a maximal set of compatible measurements, that portrays quantum probabilities as Kolmogorovian probabilities of classical events. Bell's theorem is stated and analyzed in terms of the small probability space formalism.

As a survey of many technical results in probability theory and probability logic, this monograph by two widely respected scholars offers a valuable compendium of the principal aspects of the formal study of probability. Hugues Leblanc and Peter Roeper explore probability functions appropriate for propositional, quantificational, intuitionistic, and infinitary logic and investigate the connections among probability functions, semantics, and logical consequence. They offer a systematic justification of constraints for various types of probability functions, in (...) particular, an exhaustive account of probability functions adequate for first-order quantificational logic. The relationship between absolute and relative probability functions is fully explored and the book offers a complete account of the representation of relative functions by absolute ones. The volume is designed to review familiar results, to place these results within a broad context, and to extend the discussions in new and interesting ways. Authoritative, articulate, and accessible, it will interest mathematicians and philosophers at both professional and post-graduate levels. (shrink)

de Finetti's representation theorem of exchangeable probabilities as unique mixtures of Bernoullian probabilities is a special case of a result known as the ergodic decomposition theorem. It says that stationary probability measures are unique mixtures of ergodic measures. Stationarity implies convergence of relative frequencies, and ergodicity the uniqueness of limits. Ergodicity therefore captures exactly the idea of objective probability as a limit of relative frequency (up to a set of measure zero), without the unnecessary restriction to probabilistically (...) independent events as in de Finetti's theorem. The ergodic decomposition has in some applications to dynamical systems a physical content, and de Finetti's reductionist interpretation of his result is not adequate in these cases. (shrink)

We construct, for any finite dimension n, a new hidden measurement model for quantum mechanics based on representing quantum transition probabilities by the volume of regions in projective Hilbert space. For n=2 our model is equivalent to the Aerts sphere model and serves as a generalization of it for dimensions n .≥ 3 We also show how to construct a hidden variables scheme based on hidden measurements and we discuss how joint distributions arise in our hidden variables scheme and their (...) relationship with the results of Fine [J. Math. Phys. 23 1306 (1982)]. (shrink)

The representation theorems of expected utility theory show that having certain types of preferences is both necessary and sufficient for being representable as having subjective probabilities. However, unless the expected utility framework is simply assumed, such preferences are also consistent with being representable as having degrees of belief that do not obey the laws of probability. This fact shows that being representable as having subjective probabilities is not necessarily the same as having subjective probabilities. Probabilism can be defended (...) on the basis of the representation theorems only if attributions of degrees of belief are understood either antirealistically or purely qualitatively, or if the representation theorems are supplemented by arguments based on other considerations (simplicity, consilience, and so on) that single out the representation of a person as having subjective probabilities as the only true representation of the mental state of any person whose preferences conform to the axioms of expected utility theory. (shrink)

This paper investigates the role of conditionals in hypothetical reasoning and rational decision making. Its main result is a proof of a representation theorem for preferences defined on sets of sentences (and, in particular, conditional sentences), where an agent’s preference for one sentence over another is understood to be a preference for receiving the news conveyed by the former. The theorem shows that a rational preference ordering of conditional sentences determines probability and desirability representations of the agent’s degrees (...) of belief and desire that satisfy, in the case of non-conditional sentences, the axioms of Jeffrey’s decision theory and, in the case of conditional sentences, Adams’ expression for the probabilities of conditionals. Furthermore, the probability representation is shown to be unique and the desirability representation unique up to positive linear transformation. (shrink)

This paper presents an effort to induce a Bayesian belief network (BBN) from crime data, namely the national crime victimization survey (NCVS). This BBN defines a joint probability distribution over a set of variables that were employed to record a set of crime incidents, with particular focus on characteristics of the victim. The goals are to generate a BBN to capture how characteristics of crime incidents are related to one another, and to make this information available to domain specialists. (...) The novelty associated with the study reported in this paper lies in the use of a Bayesian network to represent a complex data set to non-experts in a way that facilitates automated analysis. Validation of the BBN’s ability to approximate the joint probability distribution over the set of variables entailed in the NCVS data set is accomplished through a variety of sources including mathematical techniques and human experts for appropriate triangulation. Validation results indicate that the BBN induced from the NCVS data set is a good joint probability model for the set of attributes in the domain, and accordingly can serve as an effective query tool. (shrink)

Philosophers investigating the interpretation and use of conditional sentences have long been intrigued by the intuitive correspondence between the probability of a conditional `if A, then C' and the conditional probability of C, given A. Attempts to account for this intuition within a general probabilistic theory of belief, meaning and use have been plagued by a danger of trivialization, which has proven to be remarkably recalcitrant and absorbed much of the creative effort in the area. But there is (...) a strategy for avoiding triviality that has been known for almost as long as the triviality results themselves. What is lacking is a straightforward integration of this approach in a larger framework of belief representation and dynamics. This paper discusses some of the issues involved and proposes an account of belief update by conditionalization. (shrink)

Everettian quantum mechanics (EQM) results in ‘multiple, emergent, branching quasi-classical realities’ (Wallace [2012]). The possible outcomes of measurement as per ‘orthodox’ quantum mechanics are, in EQM, all instantiated. Given this metaphysics, Everettians face the ‘probability problem’—how to make sense of probabilities and recover the Born rule. To solve the probability problem, Wallace, following Deutsch ([1999]), has derived a quantum representation theorem. I argue that Wallace’s solution to the probability problem is unsuccessful, as follows. First, I examine (...) one of the axioms of rationality used to derive the theorem, ‘branching indifference’ (BI). I argue that Wallace is not successful in showing that BI is rational. While I think it is correct to put the burden of proof on Wallace to motivate BI as an axiom of rationality, it does not follow from his failing to do so that BI is not rational. Thus, second, I show that there is an alternative strategy for setting one’s credences in the face of branching which is rational and which violates BI. This is ‘branch counting’ (BC). Wallace is aware of BC and has proffered various arguments against it. However, third, I argue that Wallace’s arguments against BC are unpersuasive. I conclude that the probability problem in EQM persists. 1 Introduction2 Branching Indifference 2.1 The positive argument for branching indifference 2.2 The negative argument for branching indifference 2.3 Branching indifference section summary3 Branch Counting 3.1 Branch counting is rational under the subjective-uncertainty viewpoint 3.2 Branch counting is rational under the objective-determinism viewpoint 3.3 Branch counting section summary4 Number of Branches 4.1 Veracity of the framework 4.2 No such thing as the number of branches 4.2.1 The number of branches is indeterminate 4.2.2 The number of branches is indeterminable 4.3 Rationality and weight5 Conclusion. (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)

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)

I derive a sufficient condition for a belief set to be representable by a probability function: if at least one comparative confidence ordering of a certain type satisfies Scott’s axiom, then the belief set used to induce that ordering is representable. This provides support for Kenny Easwaran’s project of analyzing doxastic states in terms of belief sets rather than credences.

A probability distribution is regular if it does not assign probability zero to any possible event. 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 and Benci et al. have raised technical objections to these symmetry arguments, but we see here that their objections fail. Howson says that Williamson’s “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)

A diagnostic judgment of a teacher can be seen as an inference from manifest observable evidence on a student’s behavior to his or her latent traits. This can be described by a Bayesian model of in-ference: The teacher starts from a set of assumptions on the student (hypotheses), with subjective probabilities for each hypothesis (priors). Subsequently, he or she uses observed evidence (stu-dents’ responses to tasks) and knowledge on conditional probabilities of this evidence (likelihoods) to revise these assumptions. Many systematic (...) deviations from this model (biases, e.g. base-rate neglect, inverse fallacy) are reported in literature on Bayesian reasoning. In a teacher’s situation the information (hypotheses, priors, likelihoods) is usually not explicitly represented numerically (as in most research on Bayesian reasoning), but only by qualitative esti-mations in the mind of the teacher. In our study, we ask to which extent individuals (approximately) apply a rational Bayesian strategy or resort on other biased strategies of processing information for their diagnostic judgments. We explicitly pose this question with respect to non-numerical settings. To investigate this question, we developed a scenario that visually displays all relevant information (hypotheses, priors, likelihoods) in a graphically displayed hypothesis space (called “hypothegon”) – without recurring to numerical representations or mathematical procedures. 42 pre-service teach-ers were asked to judge the plausibility of different misconceptions of six students based on their responses to decimal comparison tasks (e.g. 3.39 > 3.4). Applying a Bayesian classification proce-dure, we identified three updating strategies: A Bayesian update strategy (BUS, processing all probabilities), a combined evidence strategy (CES, ignoring the prior probabilities but including all likelihoods), and a single evidence strategy (SES, only using the likelihood of the most probable hypothesis). In study 1, an instruction on the relevance of using all probabilities (priors and likelihoods) only weakly increased the processing of more information. In study 2, we found strong evidence that a visual explication of the prior-likelihood interaction led to an increase of processing the interaction of all relevant information. These results show that the phenomena found in general research on Bayesian reasoning in numerical settings extend to diagnostic judgments in non-numerical settings. (shrink)

The probability that a fair coin tossed yesterday landed heads is either 0 or 1, but the probability that it would land heads was 0.5. In order to account for the latter type of probabilities, past probabilities, a temporal restriction operator is introduced and axiomatically characterized. It is used to construct a representation of conditional past probabilities. The logic of past probabilities turns out to be strictly weaker than the logic of standard probabilities.

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)

An early, very preliminary edition of this book was circulated in 1962 under the title Set-theoretical Structures in Science. There are many reasons for maintaining that such structures play a role in the philosophy of science. Perhaps the best is that they provide the right setting for investigating problems of representation and invariance in any systematic part of science, past or present. Examples are easy to cite. Sophisticated analysis of the nature of representation in perception is to be (...) found already in Plato and Aristotle. One of the great intellectual triumphs of the nineteenth century was the mechanical explanation of such familiar concepts as temperature and pressure by their representation in terms of the motion of particles. A more disturbing change of viewpoint was the realization at the beginning of the twentieth century that the separate invariant properties of space and time must be replaced by the space-time invariants of Einstein's special relativity. Another example, the focus of the longest chapter in this book, is controversy extending over several centuries on the proper representation of probability. The six major positions on this question are critically examined. Topics covered in other chapters include an unusually detailed treatment of theoretical and experimental work on visual space, the two senses of invariance represented by weak and strong reversibility of causal processes, and the representation of hidden variables in quantum mechanics. The final chapter concentrates on different kinds of representations of language, concluding with some empirical results on brain-wave representations of words and sentences. (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)

This dissertation interprets the lack of uniqueness in probability representations of agents' degrees of belief in the decision theory of Richard Jeffrey as a formal statement of an important epistemological problem: the underdetermination of our attributions of belief and desire to agents by the evidence of their observed behaviour. A solution is pursued through investigation of agents' attitudes to information of a conditional nature. ;As a first step, Jeffrey's theory is extended to agents' conditional attitudes of belief and desire (...) by introduction of a definition of conditional desirability. The second step is an analysis of agents' attitudes to conditionals. Following Adams it is argued that the probability of a conditional is the conditional probability of its consequent given the antecedent. Since numerous 'trivialization' results show that no proposition could be such that this were the case, it is proposed that conditionals express a variable proposition as a function of the context in which they are asserted. The consequent technical treatment of conditionals as proposition-valued functions of sets of possible worlds delivers Adams' thesis as a natural consequence without fear of trivialization. ;Conditionals are the underlying objects of the decision theory that is presented in the final chapter. A representation theorem is proven for preferences defined over sets of such conditionals satisfying certain closure conditions, where an agent is understood to prefer one conditional over another just in case she prefers that the former by true rather than the latter. The theorem establishes that a rational preference ranking of conditionals implies the existence of a unique probability function and a desirability function unique up to choice of scale, that agree with the preference ranking, and that these functions satisfy both the axioms of Jeffrey's decision calculus and Adams' thesis. The theorem is interpreted as saying that elicitation of an agent's choices amongst conditional options suffices to determine our attributions of belief and desire to them. (shrink)

Hans Reichenbach has been not only one of the founding fathers of logical empiricism but also one of the most prominent figures in the philosophy of science of the past century. While some of his ideas continue to be of interest in current philosophical programs, an important part of his early work has been neglected, and some of it has been unavailable to English readers. Among Reichenbach’s overlooked (and untranslated) early works, his doctoral thesis of 1915, The Concept of (...) class='Hi'>Probability in the Mathematical Representation of Reality, deserves special attention, both for the topics covered and for its significance for a proper understanding of his intellectual trajectory. This volume anticipates most of the fundamental themes of his later philosophy. In particular, it addresses the issue of the application of probability statements to reality, as well as the relationship between probability and causality—questions that have been at the core of his research throughout his life. (shrink)

Probability kinematics is studied in detail within the framework of elementary probability theory. The merits and demerits of Jeffrey's and Field's models are discussed. In particular, the principle of maximum relative entropy and other principles are used in an epistemic justification of generalized conditionals. A representation of conditionals in terms of Bayesian conditionals is worked out in the framework of external kinematics.

A series of recent studies have explored the impact of people's judgments regarding physical law, morality, and probability. Surprisingly, such studies indicate that these three apparently unrelated types of judgments often have precisely the same impact. We argue that these findings provide evidence for a more general hypothesis about the kind of cognition people use to think about possibilities. Specifically, we suggest that this aspect of people's cognition is best understood using an idea developed within work in the formal (...) semantics tradition, namely the notion of modality. On the view we propose, people may have separate representations for physical, moral and probabilistic considerations, but they also integrate these various considerations into a unified representation of modality. (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)