Probability Semantics for Quantifier Logic

By supplying propositional calculus with a probability semantics we showed, in our 1996, that finite stochastic problems can be treated by logic-theoretic means equally as well as by the usual set-theoretic ones. In the present paper we continue the investigation to further the use of logical notions in probability theory. It is shown that quantifier logic, when supplied with a probability semantics, is capable of treating stochastic problems involving countably many trials.     

On the provenance of judgments of conditional probability

In standard treatments of probability, is defined as the ratio of to , provided that . This account of conditional probability suggests a psychological question, namely, whether estimates of arise in the mind via implicit calculation of . We tested this hypothesis (Experiment 1) by presenting brief visual scenes composed of forms, and collecting estimates of relevant probabilities. Direct estimates of conditional probability were not well predicted by . Direct estimates were also closer to the objective probabilities defined by the stimuli, compared to estimates computed from the foregoing ratio. The hypothesis that arises from the ratio fared better (Experiment 2). In a third experiment, the same hypotheses were evaluated in the context of subjective estimates of the chance of future events.     

Quantum mechanics and violations of the sure-thing principle: The use of probability interference and other concepts

The use of quantum mechanical concepts in social science is a fairly new phenomenon. This paper uses one of quantum mechanics’ most basic concepts, probability interference, to explain the violation of an important decision theory principle (the ‘sure-thing principle’). We also attempt to introduce other quantum mechanical concepts in relation to the sure-thing principle violation.     

Empirical comparison of Markov and quantum models of decision making

There are at least two general theories for building probabilistic-dynamical systems: one is Markov theory and another is quantum theory. These two mathematical frameworks share many fundamental ideas, but they also differ in some key properties. On the one hand, Markov theory obeys the law of total probability, but quantum theory does not; on the other hand, quantum theory obeys the doubly stochastic law, but Markov theory does not. Therefore, the decision about whether to use a Markov or a quantum system depends on which of these laws are empirically obeyed in an application. This article derives two general methods for testing these theories that are parameter free, and presents a new experimental test. The article concludes with a review of experimental findings from cognitive psychology that evaluate these two properties.     

The Probability of Small Schedule Values and Preference for Random-Interval Schedules

Preference for working on variable schedules and temporal discrimination were simultaneously examined in two experiments using a discrete-trial, concurrent-chains arrangement with fixed interval (FI) and random interval (RI) terminal links. The random schedule was generated by first sampling a probability distribution after the programmed delay to reinforcement on the FI schedule had elapsed, and thus the RI never produced a component schedule value shorter than the FI and maintained a rate of reinforcement half that of the FI. Despite these features, the FI was not strongly preferred. The probability of obtaining the smallest programmed delay to reinforcement on the RI schedule was manipulated in Experiment 1, and the interaction of this probability and initial link length was examined in Experiment 2. As the probability of obtaining small values in the RI increased, preference for the schedule increased while the discriminated time of reinforcer availability in the terminal link decreased. Both of these effects were attenuated by lengthening the initial links. The results support the view that in addition to the delay to reinforcement, the probability of obtaining a short delay is an important choice-affecting variable that likely contributes to the robust preferences for variable, as opposed to fixed, schedules of reinforcement.     

Perceived certainty based on verbal probability phrases: Effect of directionality and its dependence on method

Verbal probability phrases (e.g. "possible" or "doubtful") have a feature called "directionality" ( Teigen & Brun, 1995 ), which focuses listeners on event occurrence or nonoccurrence. We conducted an experiment about certainty estimations based on verbal probabilities in order to examine the effect of directionality on perceived certainty. In measuring perceived certainty, we used scale-based method involving responses with a scale (e.g. 101 points' scale, 0 = unlikely to 100 = likely) and numerical method involving responses such as "50%." We found that, although the effect of directionality on perceived certainty was observed in using the scale-based method, the effect disappeared when the numerical method was used. We discuss these results from two types of information processing (intuitive, associative processing and deliberate, rule-based processing).     

Complexity as a guide to understanding decision bias: A contribution to the favorite‐longshot bias debate

This paper investigates the origins of a widespread decision bias in betting markets, the favorite‐longshot bias (FLB); in particular, whether it is caused by cognitive errors on the part of bettors or by the pricing policies of bookmakers. The methodology is based on previous literature, which has suggested that: (i) races, as decision tasks for bettors, can be distinguished by their degree of complexity and their attractiveness to those with access to privileged information (insiders), (ii) cognitive errors increase as complexity increases, and (iii) bookmakers set odds in a manner to protect themselves from insiders. The degree of FLB was examined in races of varying complexity and attractiveness to insiders using a dataset of 8545 races drawn from the parallel bookmaker and pari‐mutuel markets operating in the UK in 2004. The results, interpreted in the light of the cognitive error and complexity literature, suggest that neither bettors' nor bookmakers' cognitive errors are the main cause of the bias. Rather, bettors' preferences for risk and the deliberate pricing policies of bookmakers play key roles in influencing the bias in markets where bookmakers and pari‐mutuel operators coexist. Copyright © 2008 John Wiley & Sons, Ltd.     

Proximity bias: Interactive effect of spatial distance and outcome valence on probability judgments

Across a range of decision contexts, we provide evidence of a novel proximity bias in probability judgments, whereby spatial distance and outcome valence systematically interact in determining probability judgments. Six hypothetical and incentive-compatible experiments (combined N = 4007) show that a positive outcome is estimated as more likely to occur when near than distant, whereas a negative outcome is estimated as less likely to occur when near than distant (studies 1–6). The proximity bias is explained by wishful thinking and thus perceptions of outcome desirability (study 3), and it does not manifest when an outcome is less relevant for the self, such as the case of outcomes with little consequence for the self (studies 4 and 5) or when estimating outcomes for others who are irrelevant to the self (study 6). Overall, the proximity bias we document deepens our understanding of the antecedents of probability judgments.     

Relevance differently affects the truth,acceptability, and probability evaluations of “and”, “but”, “therefore”, and “if–then”

In this study, we investigate the influence of reason-relation readings of indicative conditionals and “and”/“but”/“therefore” sentences on various cognitive assessments. According to the Frege–Grice tradition, a dissociation is expected. Specifically, differences in the reason-relation reading of these sentences should affect participants’ evaluations of their acceptability but not of their truth value. In two experiments we tested this assumption by introducing a relevance manipulation into the truth-table task as well as in other tasks assessing the participants’ acceptability and probability evaluations. Across the two experiments, a strong dissociation was found. The reason-relation reading of all four sentences strongly affected their probability and acceptability evaluations, but hardly affected their respective truth evaluations. Implications of this result for recent work on indicative conditionals are discussed.     

The Role of Lexical Frequency in the Acceptability of Syntactic Variants: Evidence From that‐Clauses in Polish

A number of studies report that frequency is a poor predictor of acceptability, in particular at the lower end of the frequency spectrum. Because acceptability judgments provide a substantial part of the empirical foundation of dominant linguistic traditions, understanding how acceptability relates to frequency, one of the most robust predictors of human performance, is crucial. The relation between low frequency and acceptability is investigated using corpus‐ and behavioral data on the distribution of infinitival and finite that‐complements in Polish. Polish verbs exhibit substantial subordination variation and for the majority of verbs taking an infinitival complement, the that‐complement occurs with low frequency (<0.66 ipm). These low‐frequency that‐clauses, in turn, exhibit large differences in how acceptable they are to native speakers. It is argued that acceptability judgments are based on configurations of internally structured exemplars, the acceptability of which cannot reliably be assessed until sufficient evidence about the core component has accumulated.