Further test of a subjective probability model of social inference

Subjects read paragraphs suggesting that a fictitious person (O) either did or did not possess an attribute X. Then, they reported their beliefs that they would like O (P_L), that O possessed X (P_X), that O was intelligent (P_I) and that O was sarcastic (P_s). Other beliefs necessary to test the predictive accuracy of Wyer's subjective probability model of cognitive organization were also reported. The model accurately predicted P_L on the basis of beliefs about X (the attribute to which information about O directly pertained); this accuracy was greater than that obtained in an earlier study in which real persons were evaluated, and did not depend upon the favorableness of X. However, predictions of P_L based upon beliefs about O's intelligence overestimated obtained values, while predictions based upon beliefs about O's sarcasm underestimated these values; moreover, P_I and P_S were underestimated and overestimated, respectively, by beliefs about X. It was speculated that these discrepancies were due to temporary inconsistencies among beliefs resulting from the information presented which did not have time to be resolved before beliefs were reported. Changes in P_L, P_I, and P_S resulting from additional information about X were accurately predicted in all cases. Implications of the model for other formulations of social evaluation processes are noted.     

Probability weights in rank-dependent utility with binary even-chance independence

We examine the effects of a weak version of expected utility's independence axiom on the probability weighting function in rank-dependent utility. Our weak independence axiom says that a 50-50 lottery between a two-outcome gamble and its certainty equivalent is indifferent to the certainty equivalent. A variety of nonlinear probability weighting functions satisfy this axiom, but most weighting functions proposed by others do not. Nevertheless, the axiom accommodates weighting functions that are quite similar to the inverse S-shaped concave-convex functions of others that overvalue small probabilities and undervalue large probabilities.     

Predictive probability and analogy by similarity in inductive logic

The-continuum of inductive methods was derived from an assumption, called-condition, which says that the probability of finding an individual having propertyx _j depends only on the number of observed individuals having propertyx _j and on the total number of observed individuals. So, according to that assumption, all individuals with properties which are different fromx _j have equal weight with respect to that probability and, in particular, it does not matter whether any individual was observed having some propertysimilar tox _j (the most complete proof of this result is presented in Carnap, 1980).The problem thus remained open to find some general condition, weaker than the-condition, which would allow for thederivation of probability functions which might be sensitive to similarity. Carnap himself suggested a weakening of the-condition which might allow for similarity sensitive probability functions (Carnap, 1980, p. 45) but he did not find the family of probability functions derivable from that principle. The aim of this paper is to present the family of probability functions derivable from Carnap's suggestion and to show how it is derived.In Section 1 the general problem of analogy by similarity in inductive logic is presented, Section 2 outlines the notation and the conceptual background involved in the proof, Section 3 gives the proof, Section 4 discusses Carnap's principle and the result, Section 5 is a brief review of the solutions which have previously been proposed.     

Epistemic closure and commutative, nonassociative residuated structures

K-axiom-based epistemic closure for explicit knowledge is rejected for even the most trivial cases of deductive inferential reasoning on account of the fact that the closure axiom does not extend beyond a raw consequence relation. The recognition that deductive inference concerns interaction as much as it concerns consequence allows for perspectives from logics of multi-agent information flow to be refocused onto mono-agent deductive reasoning. Instead of modeling the information flow between different agents in a communicative or announcement setting, we model the information flow between different states of a single agent as that agent reasons deductively. The resource management of the database of agent states for the deductive reasoning fragment in question is covered by the residuated structure that encodes the nonassociative Lambek Calculus with permutation, bottom, and identity: NLP ₀₁ .     

Tests of product and additive scaling axioms

A product axiom for ratio judgments and an additive axiom for difference judgments were tested for six Ss judging brightness. The product axiom was rejected for all six Ss whereas the additive axiom was accepted for two Ss. Power functions did not fit the data well, although somewhat better for difference judgments than ratio judgments. A two-stage model fitted somewhat better, but failed to satisfy one important implication of that model, previously confirmed for lifted weights. The data gave stronger support for a theory of brightness estimation based on an additive axiom for difference judgments than a product axiom based on ratio judgments.     

Reichenbach's metaphysical picture

Conclusion To recapitulate, then, for Reichenbach probability is the foundation of both metaphysics and epistemology. Metaphysically, probability is fundamental because it is the probability relations among the sequences of events in the world that gives rise to causality, time, and space. Epistemologically, probability is fundamental because empirical knowledge is simply knowledge of probabilities. Even knowledge of observation sentences is considered to be probabilistic knowledge by Reichenbach (EP, pp. 183–188), because Reichenbach's fallibilism leads him to hold that no observation sentence is absolutely incorrigible, and with the advance of scientific knowledge we need to inquire into the probability that our singular observation judgments may be in error.My aim here has not been to argue that Reichenbach succeeded in his magnificent attempt any more than Carnap succeeded in his. But I hope to have convinced you that is was one of the most magnificent attempts by any empiricist philosopher of this or of any other century, and I believe that the effort to understand it and to master its details will as richly repay us as the much greater effort which has been devoted to the study of Carnap's work has already repayed us.     

Choice probabilities and choice functions

Choice probabilities are basic to much of the theory of individual choice behavior in mathematical psychology. On the other hand, consumer economics has relied primarily on preference relations and choice functions for its theories of individual choice. Although there are sizable literatures on the connections between choice probabilities and preference relations, and between preference relations and choice functions, little has been done—apart from their common ties to preference relations—to connect choice probabilities and choice functions. The latter connection is studied in this paper. A family of choice functions that depends on a threshold parameter is defined from a choice probability function. It is then shown what must be true of the choice probability function so that the choice functions satisfy three traditional rationality conditions. Conversely, it is shown what must be true of the choice functions so that the choice probability function satisfies a version of Luce's axiom for individual choice probabilities.     

Martin's Axiom,Omitting Types,and Complete Representations in Algebraic Logic

We give a new characterization of the class of completely representable cylindric algebras of dimension 2 #lt; n w via special neat embeddings. We prove an independence result connecting cylindric algebra to Martin's axiom. Finally we apply our results to finite-variable first order logic showing that Henkin and Orey's omitting types theorem fails for L _n, the first order logic restricted to the first n variables when 2 #lt; n#lt;w. L _n has been recently (and quite extensively) studied as a many-dimensional modal logic.     

A General Setting for Dedekind's Axiomatization of the Positive Integers

A Dedekind algebra is an ordered pair (B, h), where B is a non-empty set and h is a similarity transformation on B. Among the Dedekind algebras is the sequence of the positive integers. From a contemporary perspective, Dedekind established that the second-order theory of the sequence of the positive integers is categorical and finitely axiomatizable. The purpose here is to show that this seemingly isolated result is a consequence of more general results in the model theory of second-order languages. Each Dedekind algebra can be decomposed into a family of disjoint, countable subalgebras called the configurations of the algebra. There are ?₀ isomorphism types of configurations. Each Dedekind algebra is associated with a cardinal-valued function on ω called its configuration signature. The configuration signature counts the number of configurations in each isomorphism type that occurs in the decomposition of the algebra. Two Dedekind algebras are isomorphic iff their configuration signatures are identical. The second-order theory of any countably infinite Dedekind algebra is categorical, and there are countably infinite Dedekind algebras whose second-order theories are not finitely axiomatizable. It is shown that there is a condition on configuration signatures necessary and sufficient for the second-order theory of a Dedekind algebra to be finitely axiomatizable. It follows that the second-order theory of the sequence of the positive integers is categorical and finitely axiomatizable.     

A New Analysis of Quasianalysis

We investigate the conditions under which quasianalysis, i.e., Carnap's method of abstraction in his Aufbau, yields adequate results. In particular, we state both necessary and sufficient conditions for the so-called faithfulness and fullness of quasianalysis, and analyze adequacy as the conjunction of faithfulness and fullness. It is shown that there is no method of (re-)constructing properties from similarity that delivers adequate results in all possible cases, if the same set of individuals is presupposed for properties and for similarity, and if similarity is a relation of finite arity. The theory is applied to various examples, including Russell's construction of temporal instants and Carnap's constitution of the phenomenal counterparts to quality spheres. Our results explain why the former is adequate while the latter is bound to fail.     

Non-monotonic Probability Theory and Photon Polarization

A non-monotonic theory of probability is put forward and shown to have applicability in the quantum domain. It is obtained simply by replacing Kolmogorov’s positivity axiom, which places the lower bound for probabilities at zero, with an axiom that reduces that lower bound to minus one. Kolmogorov’s theory of probability is monotonic, meaning that the probability of A is less then or equal to that of B whenever A entails B. The new theory violates monotonicity, as its name suggests; yet, many standard theorems are also theorems of the new theory since Kolmogorov’s other axioms are retained. What is of particular interest is that the new theory can accommodate quantum phenomena (photon polarization experiments) while preserving Boolean operations, unlike Kolmogorov’s theory. Although non-standard notions of probability have been discussed extensively in the physics literature, they have received very little attention in the philosophical literature. One likely explanation for that difference is that their applicability is typically demonstrated in esoteric settings that involve technical complications. That barrier is effectively removed for non-monotonic probability theory by providing it with a homely setting in the quantum domain. Although the initial steps taken in this paper are quite substantial, there is much else to be done, such as demonstrating the applicability of non-monotonic probability theory to other quantum systems and elaborating the interpretive framework that is provisionally put forward here. Such matters will be developed in other works.     

Binary choice probabilities: on the varieties of stochastic transitivity

Let {P_λ} denote the family of decisiveness relations {$\text{P}{}_{\mathrm{\lambda }}\text{:}\text{1}\text{2}\text{≤ λ < 1}$} with aP_λb if and only if P(a,b) > λ, where P is a binary choice probability function. Families in which all decisiveness relations are of the same type, such as all strict partial orders or all semiorders, are characterized by stochastic transitivity conditions. The conditions used for this purpose differ in various ways from the traditional forms of strong, moderate, and weak stochastic transitivity. The family {P_λ} is then examined from the viewpoint of interval representation models, the most general of which is aP_λb if and only if I(a, λ) > I(b, λ), where the I's are real intervals with I(a, λ) > I(b, λ) if and only if the first interval is completely to the right of the second. With I(a, λ) = [f(a, λ), f(a, λ) + σ(a, λ)], the specializations of the interval model that are discussed include those where the location function f (for left end-points) depends only on the set A of alternatives or stimuli and where the length function σ depends only on A or on λ or neither.     

The variance error of the P50-discriminant

The P₅₀-discriminant has been reported elsewhere in connection with its use in predicting whether selective service registrants if inducted would become normal operative soldiers or would commit offenses causing their imprisonment. The standard error of the P₅₀-discriminant is a good measure to use in determining how far to the side of this statistic a particular case falls. The standard error formula itself has also been published elsewhere; but its derivation, as the variance error, is given here.The author gratefully acknowledges the very extensive assistance kindly given to him by Dr. Truman L. Kelley and Dr. Frederick Mosteller. This assistance was given without reference to the utility of the P₅₀-discriminant, upon which matter the author reports elsewhere and for which he takes full responsibility.     

The Insufficiency of the Dutch Book Argument

It is a common view that the axioms of probability can be derived from the following assumptions: (a) probabilities reflect (rational) degrees of belief, (b) degrees of belief can be measured as betting quotients; and (c) a rational agent must select betting quotients that are coherent. In this paper, I argue that a consideration of reasonable betting behaviour, with respect to the alleged derivation of the first axiom of probability, suggests that (b) and (c) are incorrect. In particular, I show how a rational agent might assign a ‘probability’ of zero to an event which she is sure will occur.     

The Simplest Axiom System for Plane Hyperbolic Geometry Revisited

Using the axiom system provided by Carsten Augat in [1], it is shown that the only 6-variable statement among the axioms of the axiom system for plane hyperbolic geometry (in Tarski’s language L _B≡), we had provided in [3], is superfluous. The resulting axiom system is the simplest possible one, in the sense that each axiom is a statement in prenex form about at most 5 points, and there is no axiom system consisting entirely of at most 4-variable statements.     

Archimedean qualitative probabilities

Necessary and sufficient conditions for the existence of a probability measure agreeing with a weak order on an algebra of events are given. In the case of a countable algebra they consist of an extension of Kraft, Pratt, and Seidenberg's (1959. Annals of Mathematical Statistics, 38, 780–786) additivity condition through the requirement of an Archimedean property. In the case of a σ-algebra and a σ-additive agreeing probability, Villegas' (1964. Annals of Mathematical Statistics, 35, 1787–1796) monotone continuity condition, which becomes necessary, is merely added to them.