The relation between probabilistic and explanatory reasoning is a classical topic in philosophy of science. Most philosophical analyses are concerned with the compatibility of Inference to the Best Explanation with probabilistic, Bayesian inference, and the impact of explanatory considerations on the assignment of subjective probabilities. This paper reverses the question and asks how causal and explanatory considerations are affected by probabilistic information. We investigate how probabilistic information determines the explanatory value of a hypothesis, and in which (...) sense folk explanatory practice can be said to be rational. Our study identifies three main factors in reasoning about a explanatory hypothesis: cognitive salience, rational acceptability and logical entailment. This corresponds well to the variety of philosophical accounts of explanation. Moreover, we show that these factors are highly sensitive to manipulations of probabilistic information. This finding suggests that probabilistic reasoning is a crucial part of explanatory inferences, and it motivates new avenues of research in the debate about Inference to the Best Explanation and probabilistic measures of explanatory power. (shrink)

The question of how judgments of explanatory value inform probabilistic inference is well studied within psychology and philosophy. Less studied are the questions: How does probabilistic information affect judgments of explanatory value? Does probabilistic information take precedence over causal information in determining explanatory judgments? To answer these questions, we conducted two experimental studies. In Study 1, we found that probabilistic information had a negligible impact on explanatory judgments of event-types with a potentially unlimited number of available, (...) alternative explanations; causal credibility was the main determinant of explanatory value. In Study 2, we found that, for event-token explanations with a definite set of candidate alternatives, probabilistic information strongly affected judgments of explanatory value. In the light of these findings, we reassess under which circumstances explanatory inference is probabilistically sound. (shrink)

We propose a new approach for tracing value change. Value change may lead to a mismatch between current value priorities in society and the values for which technologies were designed in the past, such as energy technologies based on fossil fuels, which were developed when sustainability was not considered a very important value. Better anticipating value change is essential to avoid a lack of social acceptance and moral acceptability of technologies. While value change can be studied historically and qualitatively, (...) we propose a more quantitative approach that uses large text corpora. It uses probabilistic topic models, which allow us to trace values that are latent. We demonstrate the approach for five types of value change in technology. Our approach is useful for testing hypotheses about value change, such as verifying whether value change has occurred and identifying patterns of value change. The approach can be used to trace value change for various technologies and text corpora, including scientific articles, newspaper articles, and policy documents. (shrink)

The value of varied evidence, I propose, lies in the fact that more varied evidence is less coherent on the assumption of the negation of the hypothesis under consideration than less varied evidence. I contrast my own analysis with several other Bayesian analyses of the value of evidential diversity and show how my account explains cases where it seems intuitively that evidential variety is valuable for confirmation.

This paper considers a puzzling conflict between two positions that are each compelling: it is irrational for an agent to pay to avoid `free' evidence before making a decision, and rational agents may have imprecise beliefs and/or desires. Indeed, we show that Good's theorem concerning the invariable choice-worthiness of free evidence does not generalise to the imprecise realm, given the plausible existing decision theories for handling imprecision. A key ingredient in the analysis, and a potential source of controversy, is the (...) general approach taken for resolving sequential decision problems { we make explicit what the key alternatives are and defend our own approach. Furthermore, we endorse a resolution of the aforementioned puzzle { we privilege decision theories that merely permit avoiding free evidence over decision theories for which avoiding free evidence is uniquely admissible. Finally, we situate this particular result about free evidence within the broader `dynamic-coherence' literature. (shrink)

This article considers the popular thesis that a more proportional relationship between a cause and its effect yields a more abstract causal explanation of that effect, which in turn produces a deeper explanation. This thesis is taken to have important implications for choosing the optimal granularity of explanation for a given explanandum. In this article, I argue that this thesis is not generally true of probabilistic causal relationships. In light of this finding, I propose a pragmatic, interest-relative measure of (...) explanatory depth. This measure uses a decision-theoretic model of information pricing to determine the optimal granularity of explanation for a given explanandum, agent, and decision problem. (shrink)

The question of how the probabilistic opinions of different individuals should be aggregated to form a group opinion is controversial. But one assumption seems to be pretty much common ground: for a group of Bayesians, the representation of group opinion should itself be a unique probability distribution, 410–414, [45]; Bordley Management Science, 28, 1137–1148, [5]; Genest et al. The Annals of Statistics, 487–501, [21]; Genest and Zidek Statistical Science, 114–135, [23]; Mongin Journal of Economic Theory, 66, 313–351, [46]; Clemen (...) and Winkler Risk Analysis, 19, 187–203, [7]; Dietrich and List [14]; Herzberg Theory and Decision, 1–19, [28]). We argue that this assumption is not always in order. We show how to extend the canonical mathematical framework for pooling to cover pooling with imprecise probabilities by employing set-valued pooling functions and generalizing common pooling axioms accordingly. As a proof of concept, we then show that one IP construction satisfies a number of central pooling axioms that are not jointly satisfied by any of the standard pooling recipes on pain of triviality. Following Levi, 3–11, [39]), we also argue that IP models admit of a much better philosophical motivation as a model of rational consensus. (shrink)

The concept of time emerges as an ordering structure in a classical statistical ensemble. Probability distributions p τ (t) at a given time t obtain by integrating out the past and future. We discuss all-time probability distributions that realize a unitary time evolution as described by rotations of the real wave function $q_{\tau}(t)=\pm \sqrt{p_{\tau}(t)}$ . We establish a map to quantum physics and the Schrödinger equation. Suitable classical observables are mapped to quantum operators. The non-commutativity of the operator product is (...) traced back to the incomplete statistics of the local-time subsystem. Our investigation of classical statistics is based on two-level observables that take the values one or zero. Then the wave functions can be mapped to elements of a Grassmann algebra. Quantum field theories for fermions arise naturally from our formulation of probabilistic time. (shrink)

Scoring rules measure the accuracy or epistemic utility of a credence assignment. A significant literature uses plausible conditions on scoring rules on finite sample spaces to argue for both probabilism—the doctrine that credences ought to satisfy the axioms of probabilism—and for the optimality of Bayesian update as a response to evidence. I prove a number of formal results regarding scoring rules on infinite sample spaces that impact the extension of these arguments to infinite sample spaces. A common condition in the (...) arguments for probabilism and Bayesian update is strict propriety: that according to each probabilistic credence, the expected accuracy of any other credence is worse. Much of the discussion needs to divide depending on whether we require finite or countable additivity of our probabilities. I show that in a number of natural infinite finitely additive cases, there simply do not exist strictly proper scoring rules, and the prospects for arguments for probabilism and Bayesian update are limited. In many natural infinite countably additive cases, on the other hand, there do exist strictly proper scoring rules that are continuous on the probabilities, and which support arguments for Bayesian update, but which do not support arguments for probabilism. There may be more hope for accuracy-based arguments if we drop the assumption that scores are extended-real-valued. I sketch a framework for scoring rules whose values are nets of extended reals, and show the existence of a strictly proper net-valued scoring rules in all infinite cases, both for f.a. and c.a. probabilities. These can be used in an argument for Bayesian update, but it is not at present known what is to be said about probabilism in this case. (shrink)

It is shown that the probabilistic theories of coherence proposed up to now produce a number of counter-intuitive results. The last section provides some reasons for believing that no probabilistic measure will ever be able to adequately capture coherence. First, there can be no function whose arguments are nothing but tuples of probabilities, and which assigns different values to pairs of propositions {A, B} and {A, C} if A implies both B and C, or their negations, and (...) if P(B)=P(C). But such sets may indeed differ in their degree of coherence. Second, coherence is sensitive to explanatory relations between the propositions in question. Explanation, however, can hardly be captured solely in terms of probability. (shrink)

I defend an analog of probabilism that characterizes rationally coherent estimates for chances. Specifically, I demonstrate the following accuracy-dominance result for stochastic theories in the C*-algebraic framework: supposing an assignment of chance values is possible if and only if it is given by a pure state on a given algebra, your estimates for chances avoid accuracy-dominance if and only if they are given by a state on that algebra. When your estimates avoid accuracy-dominance (roughly: when you cannot guarantee that (...) other estimates would be more accurate), I say that they are sufficiently coherent. In formal epistemology and quantum foundations, the notion of rational coherence that gets more attention requires that you never allow for a sure loss (or “Dutch book”) in a given sort of betting game; I call this notion full coherence. I characterize when these two notions of rational coherence align, and I show that there is a quantum state giving estimates that are sufficiently coherent, but not fully coherent. (shrink)

In this article, I introduce the term “cognitivism” as a name for the thesis that degrees of belief are equivalent to full beliefs about truth-valued propositions. The thesis (of cognitivism) that degrees of belief are equivalent to full beliefs is equivocal, inasmuch as different sorts of equivalence may be postulated between degrees of belief and full beliefs. The simplest sort of equivalence (and the sort of equivalence that I discuss here) identifies having a given degree of belief with having a (...) full belief with a specific content. This sort of view was proposed in [C. Howson and P. Urbach, Scientific reasoning: the Bayesian approach. Chicago: Open Court (1996)].In addition to embracing a form of cognitivism about degrees of belief, Howson and Urbach argued for a brand of probabilism. I call a view, such as Howson and Urbach’s, which combines probabilism with cognitivism about degrees of belief “cognitivist probabilism”. In order to address some problems with Howson and Urbach’s view, I propose a view that incorperates several of modifications of Howson and Urbach’s version of cognitivist probabilism. The view that I finally propose upholds cognitivism about degrees of belief, but deviates from the letter of probabilism, in allowing that a rational agent’s degrees of belief need not conform to the axioms of probability, in the case where the agent’s cognitive resources are limited. (shrink)

In ethics, ‘probabilism’ refers to a position defended by a number of Catholic theologians, mainly in the sixteenth and seventeenth centuries. They held that, when one is uncertain which of a range of actions is the right one to perform, it is permissible to perform any which has a good chance of being the right one—even if there is another which has a better chance. This paper considers the value of this position from the viewpoint of modern ethical philosophy. The (...) unusual nature of probabilism as a theory focusing upon permissibility, rather than right-making properties, is explored and related to some modern attempts to set out ‘satisficing’ and ‘hybrid’ ethical theories. Such theories try to distinguish between what is best and what is permissible, and probabilism can be understood as an alternative way of supplementing a theory of right-making properties by adding to it a theory of permissibility. But a more radical version is also possible, where one abandons any attempt to identify right actions or right-making properties, and instead considers permissibility alone. Accordingly, a ‘multi-account theory’ of permissibility is proposed and defended as a model of how many people actually make moral decisions. (shrink)

The paper starts with the distinction between conjunction-of-parts accounts and disjunction-of-possibilities accounts to truthlikeness. In Sect. 3, three distinctions between kinds of truthlikeness measures are introduced: comparative versus numeric t-measures, t-measures for qualitative versus quantitative theories, and t-measures for deterministic versus probabilistic truth. These three kinds of truthlikeness are explicated and developed within a version of conjunctive part accounts based on content elements. The focus lies on measures of probabilistic truthlikeness, that are divided into t-measures for statistical probabilities (...) and single case probabilities. The logical notion of probabilistic truthlikeness can be treated as a subcase of deterministic truthlikeness for quantitative theories. In contrast, the epistemic notion of probabilistic truthlikeness creates genuinely new problems, especially for hypotheses about single case probabilities that are evaluated not by comparison to observed frequencies, but by comparison to the truth values of single event statements. By the method of meta-induction, competing theories about single case probabilities can be aggregated into a combined theory with optimal predictive success and epistemic truthlikeness. (shrink)

We present new probabilistic generalizations of Pearl’s entailment in System Z and Lehmann’s lexicographic entailment, called Zλ- and lexλ-entailment, which are parameterized through a value λ ∈ [0,1] that describes the strength of the inheritance of purely probabilistic knowledge. In the special cases of λ = 0 and λ = 1, the notions of Zλ- and lexλ-entailment coincide with probabilistic generalizations of Pearl’s entailment in System Z and Lehmann’s lexicographic entailment that have been recently introduced by the (...) author. We show that the notions of Zλ- and lexλ-entailment have similar properties as their classical counterparts. In particular, they both satisfy the rationality postulates of System P and the property of Rational Monotonicity. Moreover, Zλ-entailment is weaker than lexλ-entailment, and both Zλ- and lexλ-entailment are proper generalizations of their classical counterparts. (shrink)

Using an asymptotic characterization of probabilistic finite state languages over a one-letter alphabet we construct a probabilistic language with regular support that cannot be generated by probabilistic CFGs. Since all probability values used in the example are rational, our work is immune to the criticism leveled by Suppes (Synthese 22:95–116, 1970 ) against the work of Ellis ( 1969 ) who first constructed probabilistic FSLs that admit no probabilistic FSGs. Some implications for probabilistic (...) language modeling by HMMs are discussed. (shrink)

I PSYCHOLOGY 5 The Probabilistic Revolution in Psychology--an Overview Gerd Gigerenzer 7 1 Probabilistic Thinking and the Fight against Subjectivity Gerd Gigerenzer 11 2 Statistical Method and the Historical Development of Research Practice in American Psychology Kurt Danziger 35 3 Survival of the Fittest Probabilist: Brunswik, Thurstone, and the Two Disciplines of Psychology Gerd Gigerenzer 49 4 A Perspective for Viewing the Integration of Probability Theory in Psychology David J. Murray 73 II SOCIOLOGY 101 5 The Two Empirical (...) Roots of Social Theory and the Probability Revolution Anthony Oberschall 103 III ECONOMICS 133 The Probabilistic Revolution in Economics--an Overview Mary S. Morgan 135 6 Why Was There No Probabilistic Revolution in Economic Thought? Claude Mènard 139 7 The Rise of Macroeconomic Calculations in Economic Statistics Robert A. Horváth 147 8 Statistics without Probability and Haavelmo’s Revolution in Econometrics Mary S. Morgan 171 IV PHYSIOLOGY 199 9 Experimental Physiology and Statistical Inference: The Therapeutic Trial in Nineteenth-Century Germany William Coleman 201 V EVOLUTIONARY BIOLOGY 227 The Probabilistic Revolution in Evolutionary Biology--an Overview John H. Beatty 229 10 Natural Selection as a Causal, Empirical, and Probabilistic Theory M. J. S. Hodge 233 11 Dobzhansky and Drift: Facts, Values, and Chance in Evolutionary Biology John H. Beatty 271 12 Random Genetic Drift, R. A. Fisher, and the Oxford School of Ecological Genetics John R. G. Turner 313 13 On the Prior Probability of the Existence of Life Bernd-Olaf Küppers 355 VI PHYSICS 371 The Probabilistic Revolution in Physics--an Overview Lorenz Krüger 373 14 Probabilistic Physics the Classical Way Jan von Plato 379 15 Max Born and the Reality of Quantum Probabilities Nancy Cartwright 409 16 Philosophical Problems of Quantum Theory: The Response of American Physicists Nancy Cartwright. (shrink)

Various semantics for studying the square of opposition and the hexagon of opposition have been proposed recently. We interpret sentences by imprecise (set-valued) probability assessments on a finite sequence of conditional events. We introduce the acceptability of a sentence within coherence-based probability theory. We analyze the relations of the square and of the hexagon in terms of acceptability. Then, we show how to construct probabilistic versions of the square and of the hexagon of opposition by forming suitable tripartitions of (...) the set of all coherent assessments. Finally, as an application, we present new versions of the square and of the hexagon involving generalized quantifiers. (shrink)

This work is in two parts. The main aim of part 1 is a systematic examination of deductive, probabilistic, inductive and purely inductive dependence relations within the framework of Kolmogorov probability semantics. The main aim of part 2 is a systematic comparison of (in all) 20 different relations of probabilistic (in)dependence within the framework of Popper probability semantics (for Kolmogorov probability semantics does not allow such a comparison). Added to this comparison is an examination of (in all) 15 (...) purely inductive dependence relations. ————Part 1 leads in an axiomatic step-by-step development from the elementary classical truth value semantics of a sentential-logical language, called ‘L’, (chapter 1) to the elementary Kolmogorov probability semantics of L (chapter 2), which is then extended to four axiomatic semantical theories of dependence relations between the formulae of L. First the elementary Kolmogorov probability semantics of L is extended to a theory, called ‘Kdd’, of the relations of deductive dependence and deductive independence between formulae of L (chapter 3). Then Kdd is extended to a theory, called ‘Kpd1’, of the degree to which formulae of L probabilistically depend on each other in regard to a given probability distribution on the set of all formulae of L (chapter 4). Kpd1, in its turn, gets extended to a theory, called ‘Kpd2’, of the relations of probabilistic dependence and independence, relativized to unary Kolmogorov probability functions defined on L (chapter 5). Then Kpd2 is extended to a theory, called ‘Kid’, of the relations of inductive dependence and inductive independence, again relativized to unary Kolmogorov probability functions defined on L (chapter 6). Finally, Kid is extended to a theory, called ‘Kpid’, of the relations of purely inductive positive and negative dependence, relativized to unary Kolmogorov probability functions defined on L (chapter 7). ——Chapter 1, which deals with the familiar notions of truth value functions, tautologies, consequence relations and relations of logical opposition, is naturally the shortest chapter of part 1.——In chapter 2, the elementary classical semantics of L is extended to the elementary Kolmogorov probability semantics of L, i.e. to an axiomatic theory of unary and of binary Kolmogorov probability functions defined on the set of formulae of L. Because of the elementary character of this theory, chapter 2 is also rather short.——Chapter 3 introduces the first theory on dependence relations, to wit: Kdd, the theory of deductive (in)dependence between formulae of L. I follow here the well-known idea of Popper and Miller, who have used it in a famous discussion on the nature of probabilistic support for their arguments that probabilistic support is deductive, not inductive. I develop Kdd in the form of about 100 theorems, making ample use of the fact that deductive independence is nothing but subcontrary opposition, and close with a remark on the fundamental difference between deductive and logical dependence—two relations the ideas of which are all too easily mixed up.——Chapters 4 and 5 deal extensively with the traditional ideas of probabilistic (in)dependence, applied to formulae rather than to events. As always, I proceed axiomatically in a step-by-step process under systematic viewpoints and obtain about 300 theorems in this way. In the formulation of the theorems, I took special care to state clearly and expressly so-called tacit assumptions, especially those concerning the probability values of the formulae said to be dependent on each other. These assumptions are usually missing in the literature, due either to economy of writing or to sloppiness of thinking. Presumably, both chapters contain little that is new, their value lying more in the systematic grouping and organic development of the theorems than in the newness of these.——In chapter 6, I extend the axiomatic theory about probabilistic (in)dependence which has been elaborated in chapter 5, to an axiomatic theory of inductive (in)dependence by requiring of the relation of inductive (in)dependence that it be probabilistic (in)dependence, but not also logical implication or logical opposition. I point out the differences between probabilistic and inductive (in)dependence by means of some 60 theorems and close my examination of inductive (in)dependence by considering its relationship to the notion of support in the philosophy of science.——Finally, in chapter 7, the last of part 1, I take the step from inductive dependence to what I call ‘purely inductive dependence’ by combining the idea of inductive dependence with that of deductive independence in a way which is suggested by writings of Popper and Miller. I arrive at two noteworthy theorems. Firstly, there is indeed no purely inductive support. But secondly, and perhaps amazingly, countersupport is purely inductive.————Whereas the probabilistic framework of part 1 of the present work is Kolmogorov probability semantics, the framework of part 2 is Popper probability semantics, which is not only worth examining as a fascinating alternative to orthodox Kolmogorov probability semantics, but also allows us to examine dependence relations more deeply, than Kolmogorov probability semantics does. Part 2 leads—again in an axiomatic step-by-step development—from the basic Popper probability semantics of L, called ‘Pb’, (chapter 8) via a probabilistic theory of logical attributes, called ‘Ps’, (chapter 9) to four axiomatic semantical theories of dependence relations between the formulae of L. First, Ps is extended to a theory, called ‘Pdd’, of the relations of deductive dependence and deductive independence between formulae of L (chapter 10). Then Pdd is extended to a theory, called ‘Ppd’, of (in all) 20 relations of probabilistic (in)dependence, relativized to binary Popper probability functions defined on L (chapter 11). Ppd, in its turn, is extended to a theory, called ‘Pid’, of (in all) 10 relations of inductive dependence, again relativized to binary Popper probability functions defined on L (first part of chapter 12). Finally, Pid is extended to a theory, called ‘Ppid’, of (in all) 15 relations of purely inductive positive and negative dependence, relativized to binary Popper probability functions defined on L (second part of chapter 12).——Chapter 8, the first chapter of part 2 of the present work, is entirely preparatory. It introduces the axioms and about 180 theorems (150 of them together with their proofs) of basic Popper probability semantics in order to set this kind of semantics under way.——Then, in chapter 9, basic Popper probability semantics is extended to a probabilistic theory of logical properties of and relations between the formulae of L. Although I think that the way I did this extension is of some interest in itself, the main task of chapter 9 is again a preparatory one: to yield the indispensable lemmata (about 90 in number) for the theorems concerning probabilistic dependence relations in chapter 11 and concerning inductive dependence relations in chapter 12.——Chapter 10 brings the extension of Ps to the theory Pdd of deductive (in)dependence. Only half a dozen theorems are noted here for later use in the Pdd-extensions Ppd and Ppid. In view of the over 100 theorems already gained on this topic in the Kolmogorovian framework (cf. chapter 3), a similar extensive elaboration of Pdd would have been superfluous.——Chapter 11 is the most important one of part 2. It consists of a systematic comparison of 20 probabilistic (in)dependence concepts by means of about 230 theorems, obtained within the axiomatic theory Ppd, which is built up as an extension of Pdd. The main points of comparison were: differences in logical strength; reflexivity and symmetry; behaviour under the condition that the probability values of the formulae in question are extreme. It turned out that each of the examined concepts violates a strong and straightforward version of the intuitive requirement that probabilistic dependence should go with logical dependence. Whereas the corresponding chapter 5 in part 1 of the present work may not have led to new theorems, chapter 11 yields dozens of them in the process of comparison of concepts of dependence and independence which had—as far as I know—never before been treated in a single theoretical framework. With Popper probability semantics, this framework has become available, and here I have simply made full use of it.——In chapter 12, I extend the theory Ppd of probabilistic (in)dependence to the theories Pid and Ppid of inductive and purely inductive dependence, in a way very similar to that in which I have extended the theory Kpd2 to the theories Kid and Kpid in chapters 6 and 7. The first main result of Kpid (roughly: there is no purely inductive support) could be repeated for four of the five purely inductive positive dependence relations considered in chapter 12, whereas the second main result of Kpid (roughly: there is purely inductive countersupport ) could be repeated for each of the five examined purely inductive negative dependence relations. Chapter 12 closes with a brief recapitulation and critical discussion of the main results. (shrink)

By and large, the conditional connective in three-valued logic has two different functions. First, by means of a deduction theorem, it can express a specific relation of logical consequence in the logical language itself. Second, it can represent natural language structures such as "if/then'' or "implies''. This chapter surveys both approaches, shows why none of them will typically end up with a three-valued material conditional, and elaborates on connections to probabilistic reasoning.

A coherence-based probability semantics for categorical syllogisms of Figure I, which have transitive structures, has been proposed recently (Gilio, Pfeifer, & Sanfilippo [15]). We extend this work by studying Figure II under coherence. Camestres is an example of a Figure II syllogism: from Every P is M and No S is M infer No S is P. We interpret these sentences by suitable conditional probability assessments. Since the probabilistic inference of ~????|???? from the premise set {????|????, ~????|????} is not (...) informative, we add ????(????|(????∨????))>0 as a probabilistic constraint (i.e., an “existential import assumption”) to obtain probabilistic informativeness. We show how to propagate the assigned (precise or interval-valued) probabilities to the sequence of conditional events (????|????,~????|????,????|(????∨????)) to the conclusion ~????|???? . Thereby, we give a probabilistic meaning to the other syllogisms of Figure II. Moreover, our semantics also allows for generalizing the traditional syllogisms to new ones involving generalized quantifiers (like Most S are P) and syllogisms in terms of defaults and negated defaults. (shrink)

Safety is a concern in almost all branches of engineering. Whereas safety was traditionally introduced by applying safety factors or margins to the calculated maximum load, this approach is increasingly replaced with probabilistic risk assessment (PRA) as a tool for dimensioning safety measures. In this paper, the two approaches are compared in terms of what they aim at and what they can, in fact, achieve. The outcome of this comparison suggests that the two approaches should be seen as complementary (...) rather than mutually exclusive. PRA is particularly useful for priority setting and for the effect evaluation of safety measures; however, in most applications, uncertainties prevent PRA from providing an objective probability of failure or value of damage. Safety factors are indispensible for dealing with dangers that cannot be assigned meaningful probabilities. (shrink)

By and large, the conditional connective in three-valued logic has two different functions. First, by means of a deduction theorem, it can express a specific relation of logical consequence in the logical language itself. Second, it can represent natural language structures such as "if/then'" or "implies''. This chapter surveys both approaches, shows why none of them will typically end up with a three-valued material conditional, and elaborates on connections to probabilistic reasoning.

In this paper we exploit the notions of conjoined and iterated conditionals, which are defined in the setting of coherence by means of suitable conditional random quantities with values in the interval [0,1]. We examine the iterated conditional (B|K)|(A|H), by showing that A|H p-entails B|K if and only if (B|K)|(A|H) = 1. Then, we show that a p-consistent family F={E1|H1, E2|H2} p-entails a conditional event E3|H3 if and only if E3|H3= 1, or (E3|H3)|QC(S) = 1 for some nonempty subset (...) S of F, where QC(S) is the quasi conjunction of the conditional events in S. Then, we examine the inference rules And, Cut, Cautious Monotonicity, and Or of System P and other well known inference rules (Modus Ponens, Modus Tollens, Bayes). We also show that QC(F)|C(F) = 1, where C(F) is the conjunction of the conditional events in F. We characterize p-entailment by showing that F p-entails E3|H3 if and only if (E3|H3)|C(F) = 1. Finally, we examine Denial of the antecedent and Affirmation of the consequent, where the p-entailment of (E3|H3) from F does not hold, by showing that (E3|H3)|C(F) is not equal to 1. (shrink)

This paper introduces the logic of evidence and truth \ as an extension of the Belnap–Dunn four-valued logic \. \ is a slightly modified version of the logic \, presented in Carnielli and Rodrigues. While \ is equipped only with a classicality operator \, \ is equipped with a non-classicality operator \ as well, dual to \. Both \ and \ are logics of formal inconsistency and undeterminedness in which the operator \ recovers classical logic for propositions in its scope. (...) Evidence is a notion weaker than truth in the sense that there may be evidence for a proposition \ even if \ is not true. As well as \, \ is able to express preservation of evidence and preservation of truth. The primary aim of this paper is to propose a probabilistic semantics for \ where statements \\) and \\) express, respectively, the amount of evidence available for \ and the degree to which the evidence for \ is expected to behave classically—or non-classically for \ \). A probabilistic scenario is paracomplete when \ + P 1\), and in both cases, \ < 1\). If \ = 1\), or \ = 0\), classical probability is recovered for \. The proposition \, a theorem of \, partitions what we call the information space, and thus allows us to obtain some new versions of known results of standard probability theory. (shrink)

The goal of this paper is to review and critically discuss the philosophical aspects of probabilistic seismic hazard analysis (PSHA). Given that estimates of seismic hazard are typically riddled with uncertainty, diferent epistemic values (related to the pursuit of scientifc knowledge) compete in the selection of seismic hazard models, in a context infuenced by non-epistemic values (related to practical goals and aims) as well. We frst distinguish between the diferent types of uncertainty in PSHA. We claim that (...) epistemic and nonepistemic considerations are closely related in the selection of the appropriate estimate of seismic hazard by the experts. Finally, we argue that the division of scientifc responsibility among the experts can lead to responsibility gaps. This raises a problem for the ownership of the results (“no one’s model” problem) similar to the “problem of many hands” in the ethics of technology. We conclude with a plea for a close collaboration between philosophy and engineering. (shrink)

We present a probabilistic extension to active path analyses of token causation (Halpern & Pearl 2001, forthcoming; Hitchcock 2001). The extension uses the generalized notion of intervention presented in (Korb et al. 2004): we allow an intervention to set any probability distribution over the intervention variables, not just a single value. The resulting account can handle a wide range of examples. We do not claim the account is complete --- only that it fills an obvious gap in previous active-path (...) approaches. It still succumbs to recent counterexamples by Hiddleston (2005), because it does not explicitly consider causal processes. We claim three benefits: a detailed comparison of three active-path approaches, a probabilistic extension for each, and an algorithmic formulation. (shrink)

Does the philosophy of Radical Probabilism have enough structure to enable it to address fundamental epistemological questions? The requirement of dynamic coherence provides the structure for radical probabilist epistemology. This structure is sufficient to establish (i) the value of knowledge and (ii) long run convergence of degrees of belief.

Stochastic forecasts in complex environments can benefit from combining the estimates of large groups of forecasters (“judges”). But aggregating multiple opinions faces several challenges. First, human judges are notoriously incoherent when their forecasts involve logically complex events. Second, individual judges may have specialized knowledge, so different judges may produce forecasts for different events. Third, the credibility of individual judges might vary, and one would like to pay greater attention to more trustworthy forecasts. These considerations limit the value of simple aggregation (...) methods like linear averaging. In this paper, a new algorithm is proposed for combining probabilistic assessments from a large pool of judges. Two measures of a judge’s likely credibility are introduced and used in the algorithm to determine the judge’s weight in aggregation. The algorithm was tested on a data set of nearly half a million probability estimates of events related to the 2008 U.S. presidential election (∼ 16000 judges). (shrink)

We call attention to an underappreciated way in which non-epistemic values influence evidence evaluation in science. Our argument draws upon some well-known features of scientific modeling. We show that, when scientific models stand in for background knowledge in Bayesian and other probabilistic methods for evidence evaluation, conclusions can be influenced by the non-epistemic values that shaped the setting of priorities in model development. Moreover, it is often infeasible to correct for this influence. We further suggest that, while (...) this value influence is not particularly prone to the problem of wishful thinking, it could have problematic non-epistemic consequences in some cases. (shrink)

In recent years, some authors have proposed quantitative measures of the coherence of sets of propositions. Such probabilistic measures of coherence (PMCs) are, in general terms, functions that take as their argument a set of propositions (along with some probability distribution) and yield as their value a number that is supposed to represent the degree of coherence of the set. In this paper, I introduce a minimal constraint on PMC theories, the weak stability principle, and show that any correct, (...) coherent, and complete PMC cannot satisfy it. As a matter of fact, the argument offered in this paper can be applied to any coherence theory that uses a priori procedures. I briefly explore some consequences of this fact. (shrink)

Inductive probabilistic reasoning is understood as the application of inference patterns that use statistical background information to assign (subjective) probabilities to single events. The simplest such inference pattern is direct inference: from “70% of As are Bs” and “a is an A” infer that a is a B with probability 0.7. Direct inference is generalized by Jeffrey’s rule and the principle of cross-entropy minimization. To adequately formalize inductive probabilistic reasoning is an interesting topic for artificial intelligence, as an (...) autonomous system acting in a complex environment may have to base its actions on a probabilistic model of its environment, and the probabilities needed to form this model can often be obtained by combining statistical background information with particular observations made, i.e., by inductive probabilistic reasoning. In this paper a formal framework for inductive probabilistic reasoning is developed: syntactically it consists of an extension of the language of first-order predicate logic that allows to express statements about both statistical and subjective probabilities. Semantics for this representation language are developed that give rise to two distinct entailment relations: a relation ⊨ that models strict, probabilistically valid, inferences, and a relation that models inductive probabilistic inferences. The inductive entailment relation is obtained by implementing cross-entropy minimization in a preferred model semantics. A main objective of our approach is to ensure that for both entailment relations complete proof systems exist. This is achieved by allowing probability distributions in our semantic models that use non-standard probability values. A number of results are presented that show that in several important aspects the resulting logic behaves just like a logic based on real-valued probabilities alone. (shrink)

We present a probabilistic extension to active path analyses of token causation. The extension uses the generalized notion of intervention presented in : we allow an intervention to set any probability distribution over the intervention variables, not just a single value. The resulting account can handle a wide range of examples. We do not claim the account is complete --- only that it fills an obvious gap in previous active-path approaches. It still succumbs to recent counterexamples by Hiddleston, because (...) it does not explicitly consider causal processes. We claim three benefits: a detailed comparison of three active-path approaches, a probabilistic extension for each, and an algorithmic formulation. (shrink)

A central problem in the cognitive sciences is identifying the link between consciousness and neural computation. The key features of consciousness—including the emergence of representative information content and the initiation of volitional action—are correlated with neural activity in the cerebral cortex, but not computational processes in spinal reflex circuits or classical computing architecture. To take a new approach toward considering the problem of consciousness, it may be worth re‐examining some outstanding puzzles in neuroscience, focusing on differences between the cerebral cortex (...) and spinal reflex circuits. First, the mammalian cerebral cortex exhibits exascale computational power, a feature that is not strictly correlated with the number of binary computational units; second, individual computational units engage in noisy coding, allowing random electrical events to gate signaling outcomes; third, this noisy coding results in the synchronous firing of statistically random populations of cells across the neural network, at a range of nested frequencies; fourth, the system grows into a more ordered state over time, as it encodes the predictive value gained through observation; and finally, the cerebral cortex is extraordinarily energy efficient, with very little free energy lost to entropy during the work of information processing. Here, I argue that each of these five key features suggest the mammalian brain engages in probabilistic computation. Indeed, by modeling the physical mechanisms of probabilistic computation, we may find a better way to explain the unique emergent features arising from cortical neural networks. (shrink)

In the literature there are at least two models for probabilistic belief revision: Bayesian updating and imaging [Lewis, D. K. (1973), Counterfactuals, Blackwell, Oxford; Gärdenfors, P. (1988), Knowledge in flux: modeling the dynamics of epistemic states, MIT Press, Cambridge, MA]. In this paper we focus on imaging rules that can be described by the following procedure: (1) Identify every state with some real valued vector of characteristics, and accordingly identify every probabilistic belief with an expected vector of characteristics; (...) (2) For every initial belief and every piece of information, choose the revised belief which is compatible with this information and for which the expected vector of characteristics has minimal Euclidean distance to the expected vector of characteristics of the initial belief. This class of rules thus satisfies an intuitive notion of minimal belief revision. The main result in this paper is to provide an axiomatic characterization of this class of imaging rules. (shrink)

The question of whether humans represent grammatical knowledge as a binary condition on membership in a set of well-formed sentences, or as a probabilistic property has been the subject of debate among linguists, psychologists, and cognitive scientists for many decades. Acceptability judgments present a serious problem for both classical binary and probabilistic theories of grammaticality. These judgements are gradient in nature, and so cannot be directly accommodated in a binary formal grammar. However, it is also not possible to (...) simply reduce acceptability to probability. The acceptability of a sentence is not the same as the likelihood of its occurrence, which is, in part, determined by factors like sentence length and lexical frequency. In this paper, we present the results of a set of large-scale experiments using crowd-sourced acceptability judgments that demonstrate gradience to be a pervasive feature in acceptability judgments. We then show how one can predict acceptability judgments on the basis of probability by augmenting probabilistic language models with an acceptability measure. This is a function that normalizes probability values to eliminate the confounding factors of length and lexical frequency. We describe a sequence of modeling experiments with unsupervised language models drawn from state-of-the-art machine learning methods in natural language processing. Several of these models achieve very encouraging levels of accuracy in the acceptability prediction task, as measured by the correlation between the acceptability measure scores and mean human acceptability values. We consider the relevance of these results to the debate on the nature of grammatical competence, and we argue that they support the view that linguistic knowledge can be intrinsically probabilistic. (shrink)

One of the main assumptions of mathematical tools in science is represented by the idea of measurability and additivity of reality. For discovering the physical universe additive measures such as mass, force, energy, temperature, etc. are used. Economics and conventional business intelligence try to continue this empiricist tradition and in statistical and econometric tools they appeal only to the measurable aspects of reality. However, a lot of important variables of economic systems cannot be observable and additive in principle. These variables (...) can be called symbolic values or symbolic meanings and studied within symbolic interactionism, the theory developed since George Herbert Mead and Herbert Blumer. In statistical and econometric tools of business intelligence we accept only phenomena with causal connections measured by additive measures. In the paper we show that in the social world we deal with symbolic interactions which can be studied by non-additive labels. For accepting the variety of such phenomena we should avoid additivity of basic labels and construct a new probabilistic method in business intelligence based on non-Archimedean probabilities. (shrink)

TODIM is a well-known multiple-criteria decision-making which considers the bounded rationality of decision makers based on prospect theory. However, in the classical TODIM, the perceived probability weighting function and the difference of the risk attitudes for gains and losses are not consistent with the original idea of PT. Moreover, probabilistic hesitant fuzzy information shows its superiority in handling the situation that the DMs hesitate among several possible values with different possibilities. Hence, a novel TODIM with probabilistic hesitant (...) fuzzy information is proposed in this paper to simulate the perceptions of the DMs in PT. To show the advantages of the proposed method, a novel TODIM is combined with hesitant fuzzy information. Finally, a case study is carried out to demonstrate the feasibility of the proposed method, and a series of comparative analyses and the sensitivity analyses are used to show the stability of the proposed method. (shrink)

In this paper incomplete data sets, or data sets with missing attribute values, have three interpretations, lost values, attribute-concept values and ‘do not care’ conditions. Additionally, the process of data mining is based on two types of probabilistic approximations, global and saturated. We present results of experiments on mining incomplete data sets using six approaches, combining three interpretations of missing attribute values with two types of probabilistic approximations. We compare our six approaches, using the (...) error rate computed as a result of ten-fold cross validation as a criterion of quality. We show that for some data sets the error rate is significantly smaller (5% level of significance) for lost values, for some data sets the smaller error rate is associated with attribute-concept values, and sometimes with ‘do not care’ conditions. Again, for some approaches the error rate is significantly smaller for saturated probabilistic approximations than for global probabilistic approximations, while for some approaches it is the other way around. Thus, for an incomplete data set, the best approach to data mining should be chosen by trying all six approaches. (shrink)

Evidence-based practice in education involves basing decisions about what to do on evidence about the relative effectiveness of available interventions (e.g. programmes, products, practices). This article considers two influential critiques of evidence-based education (EBE) pertaining to its treatment of values. The ‘general critique’ condemns EBE for excluding values from decisions about what to do in education. The ‘specific critique’ condemns EBE for relying on a deterministic view of causality in education which disregards the complex, value-laden nature of educational (...) contexts. I argue that virtually all versions of EBE escape the general critique, including the dominant intervention-centred approach that relies on experimental research to discover ‘what works’, because the predictions EBE aims to support are only one premise in a broader normative argument. Further, intervention-centred EBE can avoid much of the specific values-based critique because it is consistent with a probabilistic, rather than deterministic, understanding of causality. However, I argue that only a context-centred approach to EBE that relies on evidence about the specific target setting from local sources in addition to evidence from theory and mixed methods research can fully address the specific critique by accommodating critics’ descriptive claims about the nature of educational contexts. (shrink)

Generalized probabilistic learning takes place in a black-box where present probabilities lead to future probabilities by way of a hidden learning process. The idea that generalized learning can be partially characterized by saying that it doesn’t foreseeably lead to harmful decisions is explored. It is shown that a martingale principle follows for finite probability spaces.

To the extent that we have reasons to avoid these “bad B -properties”, these arguments provide reasons not to have an incoherent credence function b — and perhaps even reasons to have a coherent one. But, note that these two traditional arguments for probabilism involve what might be called “pragmatic” reasons (not) to be (in)coherent. In the case of the Dutch Book argument, the “bad” property is pragmatically bad (to the extent that one values money). But, it is not (...) clear whether the DBA pinpoints any epistemic defect of incoherent agents. The same can be said for Representation Theorem arguments, since they involve the structure of an agent’s preferences. (shrink)