The Choquet expected utility model deals with nonadditive probabilities (or capacities). Their dependence on the information the decision-maker has about the possibility of the events is taken into account. Two kinds of information are examined: interval information (for instance, the percentage of white balls in an urn is between 60% and 100%) and comparative information (for instance, the information that there are more white balls than black ones). Some implications are shown with regard to (...) the core of the capacity and to two additive measures which can be derived from capacities: the Shapley value and the nucleolus. Interval information bounds prove to be satisfied by all probabilities in the core, but they are not necessarily satisfied by the nucleolus (when the core is empty) and the Shapley value. We must introduce the constrained nucleolus in order for these bounds to be satisfied, while the Shapley value does not seem to be adjustable. On the contrary, comparative information inequalities prove to be not necessarily satisfied by all probabilities in the core and we must introduce the constrained core in order for these inequalities be satisfied. However, both the nucleolus and the Shapley value satisfy the comparative information inequalities, and the Shapley value does it more strictly than the nucleolus. (shrink)

The original development of the formalism of quantum mechanics involved the study of isolated quantum systems in pure states. Such systems fail to capture important aspects of the warm, wet, and noisy physical world which can better be modelled by quantum statistical mechanics and local quantum field theory using mixed states of continuous systems. In this context, we need to be able to compute quantum probabilities given only partial information. Specifically, suppose that B is a set of (...) operators. This set need not be a von Neumann algebra. Simple axioms are proposed which allow us to identify a function which can be interpreted as the probability, per unit trial of the information specified by B, of observing the (mixed) state of the world restricted to B to be σ when we are given ρ – the restriction to B of a prior state. This probability generalizes the idea of a mixed state (ρ) as being a sum of terms (σ) weighted by probabilities. The unique function satisfying the axioms can be defined in terms of the relative entropy. The analogous inference problem in classical probability would be a situation where we have some information about the prior distribution, but not enough to determine it uniquely. In such a situation in quantum theory, because only what we observe should be taken to be specified, it is not appropriate to assume the existence of a fixed, definite, unknown prior state, beyond the set B about which we have information. The theory was developed for the purposes of a fairly radical attack on the interpretation of quantum theory, involving many-worlds ideas and the abstract characterization of observers as finite information-processing structures, but deals with quantum inference problems of broad generality. (shrink)

If is a random sequence, then the sequence is clearly not random; however, seems to be “about half random”. L. Staiger [Kolmogorov complexity and Hausdorff dimension, Inform. and Comput. 103 159–194 and A tight upper bound on Kolmogorov complexity and uniformly optimal prediction, Theory Comput. Syst. 31 215–229] and K. Tadaki [A generalisation of Chaitin’s halting probability Ω and halting self-similar sets, Hokkaido Math. J. 31 219–253] have studied the degree of randomness of sequences or reals by measuring their “degree (...) of compression”. This line of study leads to various definitions of partial randomness. In this paper we explore some relations between these definitions. Among other results we obtain a characterisation of Σ1-dimension in terms of strong Martin-Löf ε-tests , and we show that ε-randomness for ε is different than the classical 1-randomness. (shrink)

According to a prominent thought, in one’s practical reasoning one should rely only on what one knows. Yet for many choices, the relevant information is uncertain. This has led Schiffer to the following objection: oftentimes, we are fully rational in reasoning from uncertain premises which we do not know. For example, we may decide to take an umbrella based on a 0.4 credence that it will rain. There are various ways proponents of a knowledge norm for practical reasoning can (...) respond. One option is to say that the right way of dealing with uncertain information requires knowledge of probabilities. Another option is to say that credences can be knowledge because they really are beliefs with an unusual kind of content that consists of a set of probability spaces. Mixed accounts are possible as well. On neither of the accounts in the literature, however, can reasoning from uncertain premises be taken as reasoning based on a graded attitude, a credence, toward an ordinary proposition. To make room for this possibility, I argue that reliance comes in degrees. The knowledge norm is only plausible when taken to be concerned with full reliance.Partial reliance, on the other hand, goes hand in hand with credence. (shrink)

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

Over the past two decades, gamblers have begun taking mathematics into account more seriously than ever before. While probability theory is the only rigorous theory modeling the uncertainty, even though in idealized conditions, numerical probabilities are viewed not only as mere mathematical information, but also as a decision-making criterion, especially in gambling. This book presents the mathematics underlying the major games of chance and provides a precise account of the odds associated with all gaming events. It begins by (...) explaining in simple terms the meaning of the concept of probability for the layman and goes on to become an enlightening journey through the mathematics of chance, randomness and risk. It then continues with the basics of discrete probability, combinatorics and counting arguments for those interested in the supporting mathematics. These mathematic sections may be skipped by readers who do not have a minimal background in mathematics; these readers can skip directly to the Guide to Numerical Results to pick the odds and recommendations they need for the desired gaming situation. Doing so is possible due to the organization of that chapter, in which the results are listed at the end of each section, mostly in the form of tables. The chapter titled The Mathematics of Games of Chance presents these games not only as a good application field for probability theory, but also in terms of human actions where probability-based strategies can be tried to achieve favorable results. Through suggestive examples, the reader can see what are the experiments, events and probability fields in games of chance and how probability calculus works there. The main portion of this work is a collection of probability results for each type of game. Each game s section is packed with formulas and tables. Each section also contains a description of the game, a classification of the gaming events and the applicable probability calculations. The primary goal of this work is to allow the reader to quickly find the odds for a specific gaming situation, in order to improve his or her betting/gaming decisions. Every type of gaming event is tabulated in a logical, consistent and comprehensive manner. The complete methodology and complete or partial calculations are shown to teach players how to calculate probability for any situation, for every stage of the game for any game. Here, readers can find the real odds, returned by precise mathematical formulas and not by partial simulations that most software uses. Collections of odds are presented, as well as strategic recommendations based on those odds, where necessary, for each type of gaming situation. The book contains much new and original material that has not been published previously and provides great coverage of probabilities for the following games of chance: Dice, Slots, Roulette, Baccarat, Blackjack, Texas Hold em Poker, Lottery and Sport Bets. Most of games of chance are predisposed to probability-based decisions. This is why the approach is not an exclusively statistical one, but analytical: every gaming event is taken as an individual applied probability problem to solve. A special chapter defines the probability-based strategy and mathematically shows why such strategy is theoretically optimal.". (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)

The logical interpretation of probability, or "objective Bayesianism'' – the theory that (some) probabilities are strictly logical degrees of partial implication – is defended. The main argument against it is that it requires the assignment of prior probabilities, and that any attempt to determine them by symmetry via a "principle of insufficient reason" inevitably leads to paradox. Three replies are advanced: that priors are imprecise or of little weight, so that disagreement about them does not matter, within (...) limits; that it is possible to distinguish reasonable from unreasonable priors on logical grounds; and that in real cases disagreement about priors can usually be explained by differences in the background information. It is argued also that proponents of alternative conceptions of probability, such as frequentists, Bayesians and Popperians, are unable to avoid committing themselves to the basic principles of logical probability. (shrink)

Analyzing situations where information is partial, incomplete or contradictory has created a demand for quantitative belief measures that are weaker than classic probability theory. In this paper, we compare two frameworks that have been proposed for this task, Dempster-Shafer theory and non-standard probability theory based on Belnap-Dunn logic. We show the two frameworks to assume orthogonal perspectives on informational shortcomings, but also provide a partial correspondence result. Lastly, we also compare various dynamical rules of the two frameworks, (...) all seen as generalizations of classic Bayes’ conditioning. (shrink)

We study the synchronization of complex networks by using event-sampling information. The nodes of the network are connected with event-triggered communication via multiple couplings. The couplings are split into several channels. Not all the channels are connected. Only a part of the states of each node can be communicated by the channels. An event detector is designed for each channel to independently determine the sampling moments. The couplings of the network are partial and event-triggered. Both features make that (...) less information can be used for synchronization. The pinning controllers are also designed based on the sampled information. By establishing a time-dependent Lyapunov functional and utilizing an efficient event condition, we derive less conservative criteria for the synchronization of complex networks. Finally, the effectiveness of our main results is verified by an illustrative example, and comparisons are also presented to show how much conservatism can be reduced. (shrink)

In this article, ambiguity attitude is measured through the maximum price a decision maker is willing to pay to know the probability of an event. Two problems are examined in which the decision maker faces an act: in one case, buying information implies playing a lottery, while, in the other case, buying information gives also the option to avoid playing the lottery. In both decision settings, relying on the Choquet expected utility model, we study how the decision maker’s (...) risk and ambiguity attitudes affect the reservation price for ambiguity resolution. These effects are analyzed for different levels of ambiguity of the act. Operating instructions for the elicitation of the reservation price for ambiguity resolution in an experimental setting are provided at the end of the article. (shrink)

The arrow of time is a familiar phenomenon we all know from our experience: we remember the past but not the future and control the future but not the past. However, it takes an effort to keep records of the past, and to affect the future. For example, it would take an immense effort to unmix coffee and milk, although we easily mix them. Such time directed phenomena are sub- sumed under the Second Law of Thermodynamics. This law characterizes our (...) experience of the arrow of time in terms of an increase of a theoretical magnitude called entropy. Statistical mechan- ics tries to explain the Second Law as an effect of the behavior of the microscopic particles that make up the universe. Since our senses are too coarse to see this microstructure, statistical mechanics describes our experience in terms of probability, or in terms of the partial information we have about the particles. In this paper we explain the workings of statistical mechanics; how it accounts for probability as an objective feature of the world based on the underlying dynamics; how it accounts for our memories of the past; how the statistical mechanical probability under- writes the Second Law; and how, at the same time, it leads to a violation of the Second Law by the so-called Maxwell’s Demon. (shrink)

A predicate of personal taste occurring in a sentence in which the perspectival information is not linguistically articulated by an experiencer phrase may have two different readings. In case the speaker of a bare sentence formed with a predicate of personal taste uses the subjective predicate encoding perspectival information in one way and the hearer interprets it in another way, the agents’ acts are not coordinated. In this paper I offer an answer to the question of how a (...) hearer can strategically interact with a speaker on the intended perspectival information so that both agents can optimally solve their coordination problem. In this sense, I offer a game-theoretical account of the strategic communication with expressions referring to agents’ perspectives, communication which involves the interaction between a speaker who intends to convey some perspectival information and who chooses to utter a bare sentence formed with a predicate of personal taste, instead of a sentence in which the perspectival information is linguistically articulated by an experiencer phrase, and a hearer who has to choose between interpreting the uttered sentence in conformity with the speaker’s autocentric use of the predicate of personal taste or in conformity with the speaker’s exocentric use. (shrink)

This paper suggests an epistemic interpretation of Belnap’s branching space-times theory based on Everett’s relative state formulation of the measurement operation in quantum mechanics. The informational branching models of the universe are evolving structures defined from a partial ordering relation on the set of memory states of the impersonal observer. The totally ordered set of their information contents defines a linear “time” scale to which the decoherent alternative histories of the informational universe can be referred—which is quite necessary (...) for assigning them a probability distribution. The “historical” state of a physical system is represented in an appropriate extended Hilbert space and an algebra of multi-branch operators is developed. An age operator computes the informational depth of historical states and its standard deviation can be used to provide a universal information/energy uncertainty relation. An information operator computes the encoding complexity of historical states, the rate of change of its average value accounting for the process of correlation destruction inherent to the branching dynamics. In the informational branching models of the universe, the asymmetry of phenomena in nature appears as a mere consequence of the subject’s activity of measuring, which defines the flow of time-information. (shrink)

This paper presents a new language for isomorphic representations of legalknowledge in feature structures. The language includes predefinedstructures based on situation theory for common-sense categories, andpredefined structures based on Van Kralingens frame-based conceptualmodelling language for legal rules. It is shown that the flexibility of thefeature-structure formalism can exploited to allow for structure-preservingrepresentations of non-primitive concepts, and to enable various types ofinteraction and cross- reference between language elements. A fragment of theDutch Opium Act is used to illustrate how modelling and reasoning (...) proceed in practice. (shrink)

Close connections between probability theory and the theory of belief change emerge if the codomain of probability functions is extended from the real-valued interval [0, 1] to a hyperreal interval with the same limits. Full beliefs are identified as propositions with a probability at most infinitesimally smaller than 1. Full beliefs can then be given up, and changes in the set of full beliefs follow a pattern very close to that of AGM revision. In this contribution, iterated revision is investigated. (...) The iterated changes in the set of full beliefs generated by repeated revisions of a hyperreal probability function can, semantically, be modelled with the same basic structure as the sphere models of belief change theory. The changes on the set of full beliefs induced by probability revision satisfy the Darwiche–Pearl postulates for iterated belief change. (shrink)

The arrow of time is a familiar phenomenon we all know from our experience: we remember the past but not the future and control the future but not the past. However, it takes an effort to keep records of the past, and to affect the future. For example, it would take an immense effort to unmix coffee and milk, although we easily mix them. Such time directed phenomena are subsumed under the Second Law of Thermodynamics. This law characterizes our experience (...) of the arrow of time in terms of an increase of a theoretical magnitude called entropy. Statistical mechanics tries to explain the Second Law as an effect of the behavior of the microscopic particles that make up the universe. Since our senses are too coarse to see this microstructure, statistical mechanics describes our experience in terms of probability, or in terms of the partial information we have about the particles. In this paper we explain the workings of statistical mechanics; how it accounts for probability as an objective feature of the world based on the underlying dynamics; how it accounts for our memories of the past; how the statistical mechanical probability underwrites the Second Law; and how, at the same time, it leads to a violation of the Second Law by the so‐called Maxwell’s Demon. (shrink)

Combining testimonial reports from independent and partially reliable information sources is an important epistemological problem of uncertain reasoning. Within the framework of Dempster–Shafer theory, we propose a general model of partially reliable sources, which includes several previously known results as special cases. The paper reproduces these results on the basis of a comprehensive model taxonomy. This gives a number of new insights and thereby contributes to a better understanding of this important application of reasoning with uncertain and incomplete (...) class='Hi'>information. (shrink)

In everyday life, as well as in science, we have to deal with and act on the basis of partial (i.e. incomplete, uncertain, or even inconsistent) information. This observation is the source of a broad research activity from which a number of competing approaches have arisen. There is some disagreement concerning the way in which partial or full ignorance is and should be handled. The most successful approaches include both quantitative aspects (by means of probability theory) and (...) qualitative aspect (by means of graphical or causal models or logic). Some of these approaches have important impacts on various disciplines including philosophy, computer science, statistics, mathematics, physics, and social science. Most notably, the relation between causal and probabilistic information is a topic of interest in all of these fields. (shrink)

The essays that make up my dissertation share a methodological approach that aims to explore the philosophical implications of linguists' accounts of ordinary language use. In particular, all of them focus on epistemic natural language and the implications that linguists' accounts of such language has for epistemology. The first essay focuses on the debate about the norms that govern assertion and shows the ways in which research on natural language evidentiality has direct bearing. This essay uses existing cross-linguistic data about (...) assertions in Quechua and Cheyenne to argue that assertions and the norms that govern them are more complex than allowed for in extant views. What makes Quechua and Cheyenne important is that they allow speakers to assert sentences the content of which they do not believe, or even believe to be false, as long as the sentence contains the right evidential marker. This is a problem for the current theories as they all take belief in the proposition being expressed as a minimal requirement for a speaker to felicitously assert. Given the data, I argue that we should see evidential markers as modifying the norm that is in place governing felicity of the current content. I go on to present three implementations of a context-sensitive norm and argue that only the ones that are evidence-based or completely contextual can properly capture the entire set of linguistic data. In the second essay I argue that epistemic uses of 'should' can be modelling using the standard Kratzerian modal canon. In Kratzer's system, modals induce quantification over some partially ordered, restricted class of worlds. The relevant partially ordered, restricted class of worlds is generally fixed by two ingredients: a modal base and an ordering source. Modelling epistemic uses of 'should' requires us to rethink and expand what has traditionally been thought of as making up an epistemic modal base. Traditionally it has been thought that epistemic modal bases just include information about probabilities but this thought needs to be updated in order to bring epistemic 'should' into the fold. I argue that there are good theoretical reasons for having as uniform a semantics as possible then show that the context-sensitive semantic model that I develop meets all empirical demands. I end the essay by arguing that this updated way of thinking about epistemic modals bases has implications for epistemology. The final essay outlines the broad type of methodological approach mentioned above that guides the research throughout the dissertation. In it, I argue that the results of linguistics' theorizing about semantics, especially within the epistemic domain, ought to be seen as an accurate guide to reality. Looking to the way that language evolves over time and what it takes for information to become encoded into language is what drives this result. Buttressing the evolutionary explanation with results from the Condorcet Jury Theorem gives us more reason to believe that language is an accurate, albeit defeasible, guide to reality. If these results hold, they not only have broad implications for how we ought to be conducting our epistemological theorizing but our philosophical theorizing in general. (shrink)

This note is a postscript to Ramsey's 'Truth and Probability'. It replaces that article's psychological reading of subjective probability with a reading of it as a consistency condition on the theory that we act to maximise expected utility.

In lieu of an abstract, here is a brief excerpt of the content:110 DAVID HUME'S INVISIBLE HAND IN THE WEALTH OF NATIONS THE PUBLIC CHOICE OF MORAL INFORMATION Introduction The thesis I shall defend is that there are systematic aspects of Adam Smith's economics which make little sense when read in isolation from a literature in which David Hume provides the signal contributions. Consequently, parts of Hume's own work are stripped of meaning, isolated as they are from later developments. (...) The puzzles about Smith's analysis which I find illuminated by Hume's are the following: 1. Why does Smith both defend the importance of economic growth for the material of well-being of the working class and claim that property originates as spoil, impoverishing the working class? After all, in the classical system the existence of property is one requisite for economic growth. 2. Why does Smith exalt the assumption of self-interested behavior in private activity but claim that irrationality dominates the political process? 3. Why does Smith deny the importance of benevolence in Wealth of Nations and insist on the importance of morality Sentiment morality in Theory of Moral its?3 The three questions have a common feature: they are related to the question of how public goods are provided. The particular public good in question is the institution of private property. As with all true I am obligated to the Center for Study of Public Choice for generous support of my research. Suzanne Levy gets the credit for the extent to which this approaches English. I have benefited from acute comments from Charles Rowley and Charles Griswold. The errors and obscurities which remain are solely my responsibility. Ill public goods, society considered as a whole benefits from having such an institution, but each individual considered separately can gain by anti-social behavior.4 Public goods provide a considerable difficulty for any social institution; neither competition in the market nor government requests that individuals reveal how much they wish to be taxed are certain to be efficient. There are aspects of the classical economic analysis which stress the importance of moral information as a partial solution to the public goods problem. By "moral information" I only require that the theory is employed as a guide and that if an individual acts contrary to the theory, this choice is not cited as "falsifying" the theory. Rather, something is said to be "wrong" or "wicked" about the choice.6 Much of modern economic analysis assumes that individuals have perfect information; this postulate rules out any significant role for moral information in the provision of the public goods. To avoid assuming away such a role for morality, we will drop the perfect information assumption and then examine if we can make sense of the classics' attack on the public goods problem. Of course, the specification that individuals see their own interests very poorly is consistent with Smith's general approach. The technique employed below is to look at the issue which Hume and Smith addressed as a problem in modern economics. The key difficulty is to determine under what conditions moral information could guide individuals with very imperfect perceptions to maximize their utility. In particular, suppose we find that moral guidance tells an individual how efficiently to produce. Is this morality self-enforcing or are resources required to persuade people to act in accord with these teachings? If resources are required, we will not pay much attention to just how this persuasion 112 is provided, e.g., whether we tell people that theft is "evil" or whether we hang a few thieves. General production theory tells us we probably will do both, but this is a separate issue. The argument to be advanced is that moral information has self-enforcement properties in private activities but these properties are missing in public activities. In this account, then, real resources will be required for implementing the morality. This consideration in turn raises an interesting issue. If the government expends these resources, what reason is there to believe that they will be provided to serve the public interest and not the interest of the government? Is it possible that the government can prosper by keeping people ignorant or by giving them advice... (shrink)

In this paper, we illustrate some serious difficulties involved in conveying information about uncertain risks and securing informed consent for risky interventions in a clinical setting. We argue that in order to secure informed consent for a medical intervention, physicians often need to do more than report a bare, numerical probability value. When probabilities are given, securing informed consent generally requires communicating how probability expressions are to be interpreted and communicating something about the quality and quantity of the (...) evidence for the probabilities reported. Patients may also require guidance on how probability claims may or may not be relevant to their decisions, and physicians should be ready to help patients understand these issues. (shrink)

As they are conventionally formulated, Boolean games assume that players make their choices in ignorance of the choices being made by other players – they are games of simultaneous moves. For many settings, this is clearly unrealistic. In this paper, we show how Boolean games can be enriched by dependency graphs which explicitly represent the informational dependencies between variables in a game. More precisely, dependency graphs play two roles. First, when we say that variable x depends on variable y, then (...) we mean that when a strategy assigns a value to variable x, it can be informed by the value that has been assigned to y. Second, and as a consequence of the first property, they capture a richer and more plausible model of concurrency than the simultaneous-action model implicit in conventional Boolean games. Dependency graphs implicitly define a partial ordering of the run-time events in a game: if x is dependent on y, then the assignment of a value to y must precede the assignment of a value to x; if x and y are independent, however, then we can say nothing about the ordering of assignments to these variables—the assignments may occur concurrently. We refer to Boolean games with dependency graphs as partial-order Boolean games. After motivating and presenting the partial-order Boolean games model, we explore its properties. We show that while some problems associated with our new games have the same complexity as in conventional Boolean games, for others the complexity blows up dramatically. We also show that the concurrency in partial-order Boolean games can be modelled using a closure-operator semantics, and conclude by considering the relationship of our model to Independence-Friendly logic. (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)

Wittgenstein did not write very much on the topic of probability. The little we have comes from a few short pages of the Tractatus, some 'remarks' from the 1930s, and the informal conversations which went on during that decade with the Vienna Circle. Nevertheless, Wittgenstein's views were highly influential in the later development of the logical theory of probability. This paper will attempt to clarify and defend Wittgenstein's conception of probability against some oft-cited criticisms that stem from a misunderstanding of (...) his views. Max Black, for instance, criticises Wittgenstein for formulating a theory of probability that is capable of being used only against the backdrop of the ideal language of the Tractatus. I argue that on the contrary, by appealing to the 'hypothetical laws of nature', Wittgenstein is able to make sense of probability statements involving propositions that have not been completely analysed. G.H. von Wright criticises Wittgenstein's characterisation of these very hypothetical laws. He argues that by introducing them Wittgenstein makes what is distinctive about his theory superfluous, for the hypothetical laws are directly inspired by statistical observations and hence these observations indirectly determine the mechanism by which the logical theory of probability operates. I argue that this is not the case at all, and that while statistical observations play a part in the formation of the hypothetical laws, these observations are only necessary, but not sufficient conditions for the introduction of these hypotheses. (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 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)

There is a plethora of confirmation measures in the literature. Zalabardo considers four such measures: PD, PR, LD, and LR. He argues for LR and against each of PD, PR, and LD. First, he argues that PR is the better of the two probability measures. Next, he argues that LR is the better of the two likelihood measures. Finally, he argues that LR is superior to PR. I set aside LD and focus on the trio of PD, PR, and LR. (...) The question I address is whether Zalabardo succeeds in showing that LR is superior to each of PD and PR. I argue that the answer is negative. I also argue, though, that measures such as PD and PR, on one hand, and measures such as LR, on the other hand, are naturally understood as explications of distinct senses of confirmation. (shrink)

When do probability distribution functions (PDFs) about future climate misrepresent uncertainty? How can we recognise when such misrepresentation occurs and thus avoid it in reasoning about or communicating our uncertainty? And when we should not use a PDF, what should we do instead? In this paper we address these three questions. We start by providing a classification of types of uncertainty and using this classification to illustrate when PDFs misrepresent our uncertainty in a way that may adversely affect decisions. We (...) then discuss when it is reasonable and appropriate to use a PDF to reason about or communicate uncertainty about climate. We consider two perspectives on this issue. On one, which we argue is preferable, available theory and evidence in climate science basically excludes using PDFs to represent our uncertainty. On the other, PDFs can legitimately be provided when resting on appropriate expert judgement and recognition of associated risks. Once we have specified the border between appropriate and inappropriate uses of PDFs, we explore alternatives to their use. We briefly describe two formal alternatives, namely imprecise probabilities and possibilistic distribution functions, as well as informal possibilistic alternatives. We suggest that the possibilistic alternatives are preferable. -/- . (shrink)

In many real-life situations, we are interested in the value of a physical quantity y that is difficult or impossible to measure directly. To estimate y, we find some easier-to-measure quantities x1, … , xn which are related to y by a known relation y = f. Measurements are never 100% accurate; hence, the measured values equation image are different from xi, and the resulting estimate equation image is different from the desired value y = f. How different can it (...) be? Traditional engineering approach to error estimation in data processing assumes that we know the probabilities of different measurement errors equation image. In many practical situations, we only know the upper bound Δi for this error; hence, after the measurement, the only information that we have about xi is that it belongs to the interval equation image. In this case, it is important to find the range y of all possible values of y = f when xi ∈ xi. We start the paper with a brief overview of the computational complexity of the corresponding interval computation problems: Most of the related problems turn out to be, in general, at least NP-hard. In this paper, we show how the use of quantum computing can speed up some computations related to interval and probabilistic uncertainty. We end the paper with speculations on whether “hypothetic” physical devices can compute NP-hard problems faster than in exponential time. Most of the paper's results were first presented at NAFIPS'2003 [30]. (shrink)

Previous studies have emphasized a person’s biological sex as a factor which influences online search behavior. This study aims to investigate how people search online for political information and if gendered online search exists. We examined online search behavior via eye tracking while the participants searched for information about political party positions on the Internet. A content analysis of the eye tracking data followed and was evaluated with a special focus on the role of biological sex and social (...) gender, and the relationship of both factors with other variables, such as self-reported prior political knowledge, political interest, and Internet skills. The results accord with previous studies in that the sex influences the online search process. However, this relationship was partially mediated by self-reported political interest and prior knowledge. This outcome calls for a more critical use of the sex variable in reference to political online search behavior, and the inclusion of sex and gender related variables. (shrink)

Christoph Jäger (2004) argues that Dretske's information theory of knowledge raises a serious problem for his denial of closure of knowledge under known entailment: Information is closed under known entailment (even under entailment simpliciter); given that Dretske explains the concept of knowledge in terms of "information", it is hard to stick with his denial of closure for knowledge. Thus, one of the two basic claims of Dretske would have to go. Since giving up the denial of closure (...) would commit Dretske to skepticism, it would most probably be better to rather give up the information-theoretic account of knowledge. But that means that one of the best externalist views of knowledge has to be given up. I argue here that Jäger is mistaken and that there is no problem for Dretske. There is a rather easy way out of Jäger’s problem. (shrink)

The relevance of smart speakers is steadily increasing, allowing users perform several daily tasks. From a commercial perspective, smart speakers also provide recommendations of products and services that may influence the consumer decision-making process. However, previous studies have mainly focused on the adoption of smart speakers, but there is a lack of proper guidelines that help design the way these devices should offer their consumption recommendations. Based on a stimulus-organism-response approach, we analyze how two features of smart speakers' recommendations influence (...) on the effectiveness of such recommendations through its impact on user engagement and attitude. Data was collected from a sample of undergrad students in Spain using an experiment design that focused on a restaurant recommendation, and analyzed using partial least squares. On the one hand, our results suggests that gender congruence generates user engagement with the smart speaker. On the other hand, message length is positively related to attitudes towards the restaurant, at a declining rate. In addition, while better attitudes lead to higher visiting intentions, the influence of engagement on visiting intentions is partially mediated via attitudes. Thus, our findings contribute to understand the antecedents of users' engagement with smart speakers, as well as its impact on the customers' willingness to follow smart speakers' recommendations, constituting a base to analyze the impact of artificial intelligence solutions aimed to smooth the transitions of a customer through the stages of purchase process. (shrink)

In everyday life, as well as in science, we have to deal with and act on the basis of partial (i.e. incomplete, uncertain, or even inconsistent) information. This observation is the source of a broad research activity from which a number of competing approaches have arisen. There is some disagreement concerning the way in which partial or full ignorance is and should be handled. The most successful approaches include both quantitative aspects (by means of probability theory) and (...) qualitative aspect (by means of graphical or causal models or logic). Some of these approaches have important impacts on various disciplines including philosophy, computer science, statistics, mathematics, physics, and social science. Most notably, the relation between causal and probabilistic information is a topic of interest in all of these fields. The general question of how uncertainty is related to ignorance, and the connection thereof to causality, have been the key topics of the international summer school at the University of Konstanz, Germany, in August 2004. This was the third event in a series of summer schools organized by the interdisciplinary research group “Philosophy, Probability, and Modeling”. The group was in operation from September 1st, 2001, until August 31, 2005. The first summer school, held in September 2002, focused on the application of probability theory to topics of philosophical interest. Problems and questions that arise when probabilistic models and techniques are being applied in the special sciences were then discussed during the second summer school, held in July 2003. The fourth and final event in the series, in August 2005, was devoted to philosophical questions related to the application and interpretation of probabilities in physics. Each of the four summer schools attracted 40–50 people from many different countries. It was an inspiring forum for an interdisciplinary discussion among researchers and students from many different areas. This special issue collects a selection of the invited and contributed papers presented at the summer school in 2004. We acknowledge the generous support of the Center for Junior Research Fellows at the University of Konstanz, the Gesellschaft für analytische Philosophie e.V., the Centre for Philosophy of Natural and Social Science at the London School of Economics, the Alexander von Humboldt Foundation, the Federal Ministry of Education and Research and the Program for the Investment in the Future (ZIP) of the German Government through a Sofja Kovalevskaja Award. We thank the referees of the papers for their work, and James Moor for his interest in dedicating a special issue of this journal to Causality, Uncertainty and Ignorance. (shrink)