The significance of the ergodic decomposition of stationary measures for the interpretation of probability

De Finetti's representation theorem is a special case of the ergodic decomposition of stationary probability measures. The problems of the interpretation of probabilities centred around de Finetti's theorem are extended to this more general situation. The ergodic decomposition theorem has a physical background in the ergodic theory of dynamical systems. Thereby the interpretations of probabilities in the cases of de Finetti's theorem and its generalization and in ergodic theory are systematically connected to each other.This paper is an extended version of footnote 5 of von Plato (1981).     

De Finetti's earliest works on the foundations of probability

Bruno de Finetti's earliest works on the foundations of probability are reviewed. These include the notion of exchangeability and the theory of random processes with independent increments. The latter theory relates to de Finetti's ideas for a probabilistic science more generally. Different aspects of his work are united by his foundational programme for a theory of subjective probabilities.This work would not have been possible in its present form without the generous help of Prof. Maria Carla Galavotti. For several years, she has unfailingly answered all my requests for papers difficult of access, and discussed the details of interpretation as well as the philosophy of probability in general. To her I express my gratitude for everything.     

Algebras of Intervals and a Logic of Conditional Assertions

Intervals in boolean algebras enter into the study of conditional assertions (or events) in two ways: directly, either from intuitive arguments or from Goodman, Nguyen and Walker's representation theorem, as suitable mathematical entities to bear conditional probabilities, or indirectly, via a representation theorem for the family of algebras associated with de Finetti's three-valued logic of conditional assertions/events. Further representation theorems forge a connection with rough sets. The representation theorems and an equivalent of the boolean prime ideal theorem yield an algebraic completeness theorem for the three-valued logic. This in turn leads to a Henkin-style completeness theorem. Adequacy with respect to a family of Kripke models for de Finetti's logic, ukasiewicz's three-valued logic and Priest's Logic of Paradox is demonstrated. The extension to first-order yields a short proof of adequacy for Körner's logic of inexact predicates.     

Possibility and probability

Conclusion De Finetti was a strong proponent of allowing 0 credal probabilities to be assigned to serious possibilities. I have sought to show that (pace Shimony) strict coherence can be obeyed provided that its scope of applicability is restricted to partitions into states generated by finitely many ultimate payoffs. When countable additivity is obeyed, a restricted version of ISC can be applied to partitions generated by countably many ultimate payoffs. Once this is appreciated, perhaps the compelling character of the Shimony argument will be less overwhelming and the attractiveness of de Finetti's more permissive attitude will become more apparent.I want to push the permissive tendency in de Finetti still further. It seems doubtful that RUIWC should be required as de Finetti apparently suggested. It is also excessively dogmatic and restrictive to require that the credal states of ideally situated rational agents be numerically definite (Levi 1974, 1980). And de Finetti's rejection of objectivism in statistics overreached itself when he dismissed objective probabilities as meaningless metaphysical artefacts (Levi 1986). In this respect, the philosophically most important lessons de Finetti has to teach us are to be found not in his celebrated representation theorem but in his discussions of the relations between 0-probability and possibility, conditional probability and countable additivity. Perhaps, the technical issues involved are remote and pedantic. But the attitude de Finetti sought to inculcate is of profound importance.Thanks are due to Teddy Seidenfeld whose comments have improved this paper substantially. He is not to blame for its lingering defects.     

Finite forms of de Finetti's theorem on exchangeability

A geometrical interpretation of independence and exchangeability leads to understanding the failure of de Finetti's theorem for a finite exchangeable sequence. In particular an exchangeable sequence of length r which can be extended to an exchangeable sequence of length k is almost a mixture of independent experiments, the error going to zero like 1/k.     

Unknown probabilities

From a point of view like de Finetti's, what is the judgmental reality underlying the objectivistic claim that a physical magnitude X determines the objective probability that a hypothesis H is true? When you have definite conditional judgmental probabilities for H given the various unknown values of X, a plausible answer is sufficiency, i.e., invariance of those conditional probabilities as your probability distribution over the values of X varies. A different answer, in terms of conditional exchangeability, is offered for use when such definite conditional probabilities are absent.     

Laws, Ceteris Paribus Conditions, and the Dynamics of Belief

The characteristic difference between laws and accidental generalizations lies in our epistemic or inductive attitude towards them. This idea has taken various forms and dominated the discussion about lawlikeness in the last decades. Likewise, the issue about ceteris paribus conditions is essentially about how we epistemically deal with exceptions. Hence, ranking theory with its resources of defeasible reasoning seems ideally suited to explicate these points in a formal way. This is what the paper attempts to do. Thus it will turn out that a law is simply the deterministic analogue of a sequence of independent, identically distributed random variables. This entails that de Finetti's representation theorems can be directly transformed into an account of confirmation of laws thus conceived. This revised version was published online in July 2006 with corrections to the Cover Date.     

The exchangeable multinomial model as an approach to testing deterministic axioms of choice and measurement

The multinomial (Dirichlet) model, derived from de Finetti's concept of exchangeability, is proposed as a general Bayesian framework to test axioms on data, in particular, deterministic axioms characterizing theories of choice or measurement. For testing, the proposed framework does not require a deterministic axiom to be cast in a probabilistic form (e.g., casting deterministic transitivity as weak stochastic transitivity). The generality of this framework is demonstrated through empirical tests of 16 different axioms, including transitivity, consequence monotonicity, segregation, additivity of joint receipt, stochastic dominance, coalescing, restricted branch independence, double cancellation, triple cancellation, and the Thomsen condition. The model generalizes many previously proposed methods of axiom testing under measurement error, is analytically tractable, and provides a Bayesian framework for the random relation approach to probabilistic measurement (J. Math. Psychol. 40 (1996) 219). A hierarchical and nonparametric generalization of the model is discussed.     

The Idea of a Proof-Theoretic Semantics and the Meaning of the Logical Operations

This is a purely conceptual paper. It aims at presenting and putting into perspective the idea of a proof-theoretic semantics of the logical operations. The first section briefly surveys various semantic paradigms, and Section 2 focuses on one particular paradigm, namely the proof-theoretic semantics of the logical operations. This revised version was published online in June 2006 with corrections to the Cover Date.     

Careful What You Wish

Dilip Ninan has raised a puzzle for centered world accounts of de re attitude reports extended to accommodate what he calls “counterfactual attitudes.” As a solution, Ninan introduces multiple centers to the standard centered world framework, resulting in a more robust semantics for de re attitude reports. However, while the so-called multi-centered world proposal solves Ninan’s counterfactual puzzle, this additional machinery is not without problems. In Section 1, I present the centered world account of attitude reports, followed by the extension to counterfactual attitudes which Ninan targets with his puzzle. In Section 2, I pose the counterfactual puzzle and present Ninan’s multi-centered world solution, emphasizing similarities and differences between multi-centered and centered world accounts of attitude reports. In Section 3, I argue the counterfactual attitude wishing falls under the purview of the multi-centered proposal, but that the proposal generates false predictions for wish reports concerning unsatisfiable content. I canvass responses, ultimately concluding Ninan’s proposal requires substantial revisions.     

A Luminous and Splendid Truth: On the Mystery of Predestination in Matthias Scheeben

Matthias Joseph Scheeben has been described as one of the greatest and least read theologians of the modern era. This article provides an overview of his theology of predestination, which remains a significant but little-studied aspect of his thought. Section I offers a general sketch of Scheeben's theology of predestination, employing the chapter on this topic in The Mysteries of Christianity as a primary source. Section II takes a deeper look at Scheeben's theology of predestination through an engagement with relevant sections of his untranslated masterwork, the Dogmatik. This section will highlight Scheeben's relationship to both the de auxiliis controversy and the thought of Thomas Aquinas. Connections between Scheeben's theology of predestination and his other theological positions are noted throughout.     

A Feminist Multicultural Perspective on Supervision

Sixteen members of the Section for the Advancement of Women conference work group identified themes and issues relevant to a feminist multicultural perspective on supervision. Issues raised included feminist silence and White privilege, the importance of self‐examination, the courage to be anxious, personal commitment to uncertainty, and isolation versus support. Specific risks for supervisors and supervisees participating in feminist multicultural supervision are identified. Dieciséis miembros de una Sección para el Adelantamiento de Mujeres conferencia de grupo del trabajo identificaron los temas y ausuntos pertinente de una perspectiva feminista multicultural en la supervisión. Los asuntos elevados incluyeron el silencio feminista y el privilegio Blanco, la importancia del autoexamen, el valor para estar ansioso, compromiso personal para la incertidumbre, y el aislamiento en oposicion de apoyo. Los riesgos específicos para supervisores y supervisa tomando parte en la supervisión feminista multicultural se identifica.     

For and Against de Régnon: Trinitarianism East and West

Modern systematic theologians frequently claim that Eastern trinitarianism begins with the persons while Western trinitarianism begins with the divine essence, and fails to truly differentiate the persons. This article explores the origins of this claim in the writing of Theodore de Régnon, noting recent criticism of his paradigm's applicability to the patristic era. In response to these criticisms, I argue first for the limited validity of the de Régnon paradigm by demonstrating its applicability in the Carolingian–Photian dispute surrounding the filioque. I then argue against the deployment of the paradigm as a controlling schema in systematic theology by pointing to problems this narrative has created in the theology of Robert Jenson and John Zizioulas.     

Construct Validity of the DSM–5 Section III Personality Trait Profile for Borderline Personality Disorder

This study evaluated the nomological network of the borderline personality disorder (BPD) trait profile in the Diagnostic and Statistical Manual of Mental Disorders (5th ed. [DSM–5]) Section III. BPD symptoms include a variety of maladaptive thoughts and behaviors, and it is important to determine if the Section III trait operationalization for BPD captures these behavioral symptoms, as well as shows similar associations as the traditional Section II version with external criteria. For this purpose, we used a sample of 285 undergraduate students and conducted correlation and regression analyses to delineate the associations between Section III BPD traits and conceptually relevant external criteria. A Section III Total score was meaningfully associated with all criteria. Moreover, externalizing psychopathology tended to be most highly associated with disinhibitory Section III BPD traits, whereas internalizing psychopathology tended to have its strongest unique associations with traits reflective of negative affectivity. These results provide support for the construct validity of the trait profile for BPD in DSM–5 Section III.     

The Ontology of Epistemic Reasons

Epistemic reasons are mental states. They are not propositions or non-mental facts. The discussion proceeds as follows. Section 1 introduces the topic. Section 2 gives two concrete examples of how our topic directly affects the internalism/externalism debate in normative epistemology. Section 3 responds to an argument against the view that reasons are mental states. Section 4 presents two problems for the view that reasons are propositions. Section 5 presents two problems for the view that reasons are non-mental facts. Section 6 argues that reasons are mental states. Section 7 responds to objections.     

Some unifying fixed point principles

This article is written for both the general mathematican and the specialist in mathematical logic. No prior knowledge of metamathematics, recursion theory or combinatory logic is presupposed, although this paper deals with quite general abstractions of standard results in those three areas. Our purpose is to show how some apparently diverse results in these areas can be derived from a common construction. In Section 1 we consider five classical fixed point arguments (or rather, generalizations of them) which we present as problems that the reader might enjoy trying to solve. Solutions are given at the end of the section. In Section 2 we show how all these solutions can be obtained as special cases of a single fixed point theorem. In Section 3 we consider another generalization of the five fixed point results of Section 1 and show that this is of the same strength as that of Section 2. In Section 4 we show some curious strengthenings of results of Section 3 which we believe to be of some interest on their own accounts.     

The MMPI–2 Restructured Form Personality Psychopathology Five Scales: Bridging DSM–5 Section 2 Personality Disorders and DSM–5 Section 3 Personality Trait Dimensions

This study examined in a college sample and a sample of non-treatment-seeking, trauma-exposed veterans the association between the MMPI–2 Restructured Form (MMPI–2–RF) Personality Psychopathology Five (PSY–5) Scales and DSM–5 Section 2 personality disorder (PD) criteria, the same system used in DSM–IV–TR, and the proposed broad personality trait dimensions contained in Section 3 of DSM–5. DSM–5 Section 2 PD symptoms were assessed using the SCID–II–PQ, and applying a replicated rational selection procedure to the SCID–II–PQ item pool, proxies for the DSM–5 Section 3 dimensions and select facets were constructed. The MMPI–2–RF PSY–5 scales demonstrated appropriate convergent and discriminant associations with both Section 2 PDs and Section 3 dimensions in both samples. These findings suggest the MMPI–2–RF PSY–5 scales can serve both conceptually and practically as a bridge between the DSM–5 Section 2 PD criteria and the DSM–5 Section 3 personality features.     

Age Neutrality of Categorically and Dimensionally Measured DSM–5 Section II Personality Disorder Symptoms

Studies on the face validity of DSM–5 Section II categorical personality disorder (PD) symptoms indicate a bias against older adults. To extend these results, this article explores whether categorically and dimensionally scored PD symptoms of DSM–5 Section II, as measured in the Assessment of DSM–IV Personality Disorders (ADP–IV; Schotte & de Doncker, 1994), corroborate potential age bias across younger (aged 18–34), middle-aged (35–59 years), and older adults (aged 60–75). Differential item functioning (DIF) analyses, following a classical test theory approach, showed that 2 of the 79 symptoms were measured differently across 3 age groups when categorically assessed, and 4 when dimensionally measured. Nevertheless, subsequent differential test functioning analyses supported a low aggregated impact of DIF on the dimensional scales, justifying mean-level comparisons across age groups. Generalizability of the results is discussed in light of methodological issues concerning the research of age neutrality of PD symptoms, including the employed measurement instrument, PD symptom measurement approach, and sample and age range used to describe older adults.     

A sideways look at configural encoding: two different effects of face rotation

Inversion has a disproportionate disruptive effect on the recognition of faces. This may be due to the disruption of holistic or configural encoding employed to recognise upright faces. The paradigm developed by Tanaka and Farah (1993 Quarterly Journal of Experimental Psychology, Section A 46 225-246) was used to investigate the effect of 90 degrees (or orthogonal) rotation on configural encoding. Faces learnt in the orthogonal condition were not recognised as well as upright faces, but a whole-face advantage was found in both cases. This whole-face advantage did not occur for inverted faces. It appears that 90 degrees rotation affects recognition but not specifically configural encoding. It is concluded that rotating a face can have at least two different effects on face processing depending on the range of rotation. Implications for the nature of facial dimensions and the expertise account of the inversion effect are considered.