Non-zero probabilities for universal generalizations

This article discusses the classical problem of zero probability of universal generalizations in Rudolf Carnap’s inductive logic. A correction rule for updating the inductive method on the basis of evidence will be presented. It will be shown that this rule has the effect that infinite streams of uniform evidence assume a non-zero limit probability. Since Carnap’s inductive logic is based on finite domains of individuals, the probability of the corresponding universal quantification changes accordingly. This implies that universal generalizations can receive positive prior and posterior probabilities, even for (countably) infinite domains.     

The logical form of universal generalizations

First order logic does not distinguish between different forms of universal generalization; in this paper I argue that lawlike and accidental generalizations (broadly construed) have a different logical form, and that this distinction is syntactically marked in English. I then consider the relevance of this broader conception of lawlikeness to the philosophy of science.     

Grounding and the explanatory role of generalizations

According to Hempel’s (Aspects of scientific explanation and other essays. The Free Press, New York, 1965) influential theory of explanation, explaining why some a is G consists in showing that the truth that a is G follows from a law-like generalization to the effect that all Fs are G together with the initial condition that a is F. While Hempel’s overall account is now widely considered to be deeply flawed, the idea that some generalizations play the explanatory role that the account predicts is still often endorsed by contemporary philosophers of science. This idea, however, conflicts with widely shared views in metaphysics according to which the generalization that all Fs are G is partially explained by the fact that a is G. I discuss two solutions to this conflict that have been proposed recently, argue that they are unsatisfactory, and offer an alternative.     

Note on two generalizations of coefficient alpha

An important relationship is given for two generalizations of coefficient alpha, Rajaratnam, Cronbach and Gleser's generalizability formula for stratified-parallel tests and Raju's coefficient beta.The author gratefully acknowledges the generous assistance given by reviewers and the editor.     

Confirming Mathematical Theories: an Ontologically Agnostic Stance

The Quine/Putnam indispensability approach to the confirmation of mathematical theories in recent times has been the subject of significant criticism. In this paper I explore an alternative to the Quine/Putnam indispensability approach. I begin with a van Fraassen-like distinction between accepting the adequacy of a mathematical theory and believing in the truth of a mathematical theory. Finally, I consider the problem of moving from the adequacy of a mathematical theory to its truth. I argue that the prospects for justifying this move are qualitatively worse in mathematics than they are in science. This revised version was published online in June 2006 with corrections to the Cover Date.     

Confirming Gender Stereotypes: A Social Role Perspective

In this research we examined whether emotional vulnerability leads women and men to confirm gender stereotypes. Emotional vulnerability is a state where one is open to having one's feelings hurt or to experiencing rejection. Drawing on the tenets of social role theory and research related to normative expectations, we propose that emotional vulnerability leads to stereotype confirmation, as normative expectations are less risky and easier to enact than nonnormative behavior. Fifty-nine dating couples were randomly assigned to a high emotional vulnerability or low emotional vulnerability discussion with their partners. When the degree of emotional vulnerability was high men confirmed gender-stereotypes. Women's behavior, on the other hand, was not significantly affected by condition. We discuss these findings in terms of the domain in which gender-typed behaviors occur and the social pressures to act in accordance with gender norms.     

Confirming expectations in asymmetric and symmetric social hypothesis testing

This article examines individuals' expectations in a social hypothesis testing task. Participants selected questions from a list to investigate the presence of personality traits in a target individual. They also identified the responses that they expected to receive and the likelihood of the expected responses. The results of two studies indicated that when people asked questions inquiring about the hypothesized traits that did not entail strong a priori beliefs, they expected to find evidence confirming the hypothesis under investigation. These confirming expectations were more pronounced for symmetric questions, in which the diagnosticity and frequency of the expected evidence did not conflict. When the search for information was asymmetric, confirming expectations were diminished, likely as a consequence of either the rareness or low diagnosticity of the hypothesis-confirming outcome. We also discuss the implications of these findings for confirmation bias.     

A low-cost universal feeder

Construction of a universal feeder for a cost of less than $200 is described. The feeder consists of two plywood panels, the first of which is circular and has grooves cut around its margin to hold reinforcers. This circular panel is secured to the second, rectangular panel so that it is free to rotate. A brush fixed onto the axle of a stepper motor above the circular panel sweeps the reinforcers away from the circular panel. Another stepper motor rotates the circular panel after each delivery to prepare the next reinforcer for delivery.     

Structural generalizations over consonants and vowels in 11-month-old infants

Recent research has suggested consonants and vowels serve different roles during language processing. While statistical computations are preferentially made over consonants but not over vowels, simple structural generalizations are easily made over vowels but not over consonants. Nevertheless, the origins of this asymmetry are unknown. Here we tested if a lifelong experience with language is necessary for vowels to become the preferred target for structural generalizations. We presented 11-month-old infants with a series of CVCVCV nonsense words in which all vowels were arranged according to an AAB rule (first and second vowels were the same, while the third vowel was different). During the test, we presented infants with new words whose vowels either followed or not, the aforementioned rule. We found that infants readily generalized this rule when implemented over the vowels. However, when the same rule was implemented over the consonants, infants could not generalize it to new instances. These results parallel those found with adult participants and demonstrate that several years of experience learning a language are not necessary for functional asymmetries between consonants and vowels to appear.     

Some generalizations of a theoretical distribution of mental test scores

The negative hypergeometric distribution may be generalized to provide distributions involving 3, 4, or more parameters. It is shown that, in the case of the binomial error model, such distributions correspond to quadratic, cubic, etc., regressions of true scores on raw scores. Explicit formulas are given for the bivariate moments required to fit these regressions and so estimate the parameters of the generalized hypergeometric distribution. Equations are also developed for fitting the 3- and 4-parameter distributions directly, i.e., without reference to the model, and the method for generalizing these is indicated. The methods developed are applied to objective test data with satisfactory results.This research was supported in part by Office of Naval Research Contract Nonr2752(00).The author is now at the University of Queensland, St. Lucia, Queensland, Australia.     

Antirealism and universal knowability

Truth’s universal knowability entails its discovery. This threatens antirealism, which is thought to require it. Fortunately, antirealism is not committed to it. Avoiding it requires adoption (and extension) of Dag Prawitz’s position in his long-term disagreement with Michael Dummett on the notion of provability involved in intuitionism’s identification of it with truth. Antirealism (intuitionism generalized) must accommodate a notion of lost-opportunity truth (a kind of recognition-transcendent truth), and even truth consisting in the presence of unperformable verifications. Dummett’s position cannot abide this, while Prawitz’s can. Antirealism’s epistemic notion of truth derives from general features of its meaning theory, not from a universal knowability principle.