Reasoning as we read: Establishing the probability of causal conditionals

Indicative conditionals of the form if p then q (e.g., if student tuition fees rise, then applications for university places will fall) invite consideration of a hypothetical event (e.g., tuition fees rising) and of one of its possible consequences (e.g., applications falling). Since a rise in tuition fees is an uncertain event with equally uncertain consequences, a reader may believe the statement to a greater or lesser extent. As a conditional is read, the earliest point at which this probabilistic evaluation can take place is as the consequent clause is wrapped up (e.g., as the critical word fall is read in the example above). Wrap-up processing occurs at the end of the clause, as it is evaluated and integrated into the evolving discourse representation. Five sources of probability may plausibly influence the evaluation of a conditional as it is wrapped up; these are P(p), P(q), P(pq), P(q|p), and P(not-p or q). A total of 128 conditionals were constructed, with these probabilities calculated for each item in a pretest. The conditionals were then embedded in vignettes and read by 36 participants on a word-by-word basis. Using linear mixed-effects modeling, we found that wrap-up reading times were predicted by pretest ratings of P(p) and P(q|p). There was no influence of P(q), P(pq), or P(not-p or q) on wrap-up reading times. Our findings are consistent with the suppositional theory of conditionals proposed by Evans and Over (2004) but do not support the mental-models theory advanced by Johnson-Laird and Byrne (2002).     

The Bayesian explanation of transmission failure

Even if our justified beliefs are closed under known entailment, there may still be instances of transmission failure. Transmission failure occurs when P entails Q, but a subject cannot acquire a justified belief that Q by deducing it from P. Paradigm cases of transmission failure involve inferences from mundane beliefs (e.g., that the wall in front of you is red) to the denials of skeptical hypotheses relative to those beliefs (e.g., that the wall in front of you is not white and lit by red lights). According to the Bayesian explanation, transmission failure occurs when (i) the subject’s belief that P is based on E, and (ii) P(Q|E) < P(Q). But there are compelling cases of transmission failure where P(Q|E) > P(Q). No modifications of the Bayesian explanation are capable of accommodating such cases, so the explanation must be rejected as inadequate. Alternative explanations employing simple subjunctive conditionals are fully capable of capturing all of the paradigm cases, as well as those missed by the Bayesian explanation.     

A generalization of comparative probability on finite sets

Let ? be a binary relation on a finite algebra $\text{A}$ of events A, B,…, where A ? B is interpreted as “A is more probable than B.” Conventional subjective probability is concerned with the existence of a probability measure P on $\text{A}$ that agrees with ? in the sense that A ? B ? P(A) > P(B). Because evidence suggests that some people's comparative probability judgments do not admit an agreeing probability measure, this paper explores a more flexible scheme for representing ? numerically. The new representation has A ? B ? p(A, B) > 0, where p is a monotonic and normalized skew-symmetric function on $\text{A}$ × $\text{A}$ that replaces P's additivity by a conditional additivity property. Conditional additivity says that $\text{p(A ? B, C) + p(?, C) = p(A, C) + p(B, C)}$ whenever A and B are disjoint. The paper examines consequences of this representation, presents examples of ? that it accommodates but which violate the conventional representation, formulates axioms for ? on $\text{A}$ that are necessary and sufficient for the representation, and discusses specializations in which p in separable in its arguments.     

The parameters in the near-miss-to-Weber's law

Many empirical data support the hypothesis that the sensitivity function grows as a power function of the stimulus intensity. This is usually referred to as the near-miss-to-Weber's law. The aim of the paper is to examine the near-miss-to-Weber's law in the context of psychometric models of discrimination. We study two types of psychometric functions, characterized by the representations P_a(x)=F(ρ(a)x^γ(a)) (type A), and P_a(x)=F(γ(a)+ρ(a)x) (type B). A central result shows that both types of psychometric functions are compatible with the near-miss-to-Weber's law. If a representation of type B exists, then the exponent in the near-miss is necessarily a constant function, that is, does not depend on the criterion value used to define “just noticeably different”. If, on the other hand, a representation of type A exists, then the exponent in the near-miss-to-Weber's law can vary with the criterion value. In that case, the parameters in the near-miss co-vary systematically.     

Efficient Hessian computation using sparse matrix derivatives in RAM notation

This article proposes a new, more efficient method to compute the minus two log likelihood, its gradient, and the Hessian for structural equation models (SEMs) in reticular action model (RAM) notation. The method exploits the beneficial aspect of RAM notation that the matrix derivatives used in RAM are sparse. For an SEM with K variables, P parameters, and P′ entries in the symmetrical or asymmetrical matrix of the RAM notation filled with parameters, the asymptotical run time of the algorithm is O(P?′?K ²?+?P ² K ²?+?K ³). The naive implementation and numerical implementations are both O(P ² K ³), so that for typical applications of SEM, the proposed algorithm is asymptotically K times faster than the best previously known algorithm. A simulation comparison with a numerical algorithm shows that the asymptotical efficiency is transferred to an applied computational advantage that is crucial for the application of maximum likelihood estimation, even in small, but especially in moderate or large, SEMs.     

Causal inferences as perceptual judgments

We analyze how subjects make causal judgments based on contingency information in two paradigms. In the discrete paradigm, subjects are given specific information about the frequency a, with which a purported cause occurs with the effect; the frequency b, with which it occurs without the effect; the frequency c, with which the effect occurs when the cause is absent; and the frequency d, with which both cause and effect are absent. Subjects respond toP ₁ =a/(a+b) andP ₂ =c/(c+d). Some subjects’ ratings are just a function ofP ₁, while others are a function of ΔP =P ₁ -P ₂. Subjects’ postexperiment reports are accurate reflections of which model they use. Combining these two types of subjects results in data well fit by the weighted ΔP model (Allan, 1993). In the continuous paradigm, subjects control the purported causes (by clicking a mouse) and observe whether an effect occurs. Because causes and effects occur continuously in time, it is not possible to explicitly pair causes and effects. Rather, subjects report that they are responding to the rate at which the effects occur when they click versus when they do not click. Their ratings are a function of rates and not probabilities. In general, we argue that subjects’ causal ratings are judgments of the magnitude of perceptually salient variables in the experiment.     

Nontransitive measurable utility

A new theory of preferences under risk is presented that does not use the transitivity and independence axioms of the von Neumann-Morgenstern linear utility theory. Utilities in the new theory are unique up to a similarity transformation (ratio scale measurement). They key to this generalization of the traditional linear theory lies in its representation of binary preferences by a bivariate rather than univariate real valued function. Linear theory obtains a linear function u on a set P of probability measures for which u(p) > u(q) if and only if p is preferred to q. The new theory obtains a skew-symmetric bilinear function φ on P × P for which φ(p, q) > 0 if and only if p is preferred to q. Continuity, dominance, and symmetry axioms are shown to be necessary and sufficient for the new representation.     

Subjunctive Conditional Probability

There seem to be two ways of supposing a proposition: supposing “indicatively” that Shakespeare didn’t write Hamlet, it is likely that someone else did; supposing “subjunctively” that Shakespeare hadn’t written Hamlet, it is likely that nobody would have written the play. Let P(B//A) be the probability of B on the subjunctive supposition that A. Is P(B//A) equal to the probability of the corresponding counterfactual, A □→B? I review recent triviality arguments against this hypothesis and argue that they do not succeed. On the other hand, I argue that even if we can equate P(B//A) with P(A □→B), we still need an account of how subjunctive conditional probabilities are related to unconditional probabilities. The triviality arguments reveal that the connection is not as straightforward as one might have hoped.     

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.     

Complexly fractionated syllogistic quantifiers

Consider syllogisms in which fraction (percentage) quantifiers are permitted in addition to universal and particular quantifiers, and then include further quantifiers which are modifications of such fractions (such as “almost 1/2 the S are P” and “Much more than 1/2 the S are P”). Could a syllogistic system containing such additional categorical forms be coherent? Thompson's attempt (1986) to give rules for determining validity of such syllogisms has failed; cf. Carnes &; Peterson (forthcoming) for proofs of the unsoundness and incompleteness of Thompson's rules. Building on Peterson (1985), the coherence of such a syllogistic can, however, be demonstrated with an algebra which provides its semantics; e.g., “almost 1/2 the S are P” is represented as “?(3(SP)?SP)”.     

Extraction of relief from visual motion

We quantified the ability of human subjects to discriminate the relative distance of two points from a slanted plane when viewing the projected velocities of this scene (orthographic projection). The relative distance from a plane (called relief) is a 3-D property that is invariant under linear (affine) transformations. As such, relief canin principle be extracted from the instantaneous projected velocity field; a metric representation, which requires the extraction of visual acceleration, is not required. The stimulus consisted of a slanted planeP (specified by three points) and two pointsQ ₁ andQ ₂ that are non-coplanar withP. This configuration of points oscillated rigidly around the vertical axis. We have measured thesystematic error andaccuracy with which human subjects estimate the relative distance of pointsQ ₁ andQ ₂ from planeP as a function of the slant of_P. The systematic error varies with slant: it is low for small slant values, reaches a maximum for medium slant values, and drops again for high slant values. The accuracy covaries with the systematic error and is thus high for small and large slant values and low for medium slant values. These results are successfully modeled by a simple relief-from-motion computation based on local estimates of projected velocities. The data are well predicted by assuming (1) a measurement error in velocity estimation that varies proportionally to velocity (Weber’s law) and (2) an eccentricity-dependent underestimation of velocity.     

Visual stimuli in distraction strategies for increasing pain tolerance

Recent experimental studies in pain control have questioned the value of pleasant affect in strategies employing distraction. It appears that pleasant affect may have been systematically confounded with task complexity or novelty in past research that found pleasant imagery or slides effective in increasing pain tolerance with the cold pressor test. The present study was a follow-up to a study conducted by this author (Greenstein, 1984) in which unpleasant slides had significantly increased pain tolerance above pleasant slide level. In the present study, 69 college students (35 females, 34 males) rated either the pleasant or unpleasant slides used in the original study on their perceived pleasantness, complexity, and uniqueness (novelty). Results indicated that the unpleasant slides were rated significantly more complex (P<0.001) and unique (P<0.001) than the pleasant slides. Additionally, as in the earlier study, ratings of the unpleasant slides on pleasantness deviated significantly farther from neutrality than did ratings of the pleasant slides (t=5.04,P<0.001). Thus the unpleasant slides were also perceived as being more significant (i.e., pertinent) than were the pleasant slides. The results indicate that affect was confounded with other stimulus characteristics in the Greenstein (1984) pain control study and probably in a significant number of other studies as well. Researchers are cautioned to control for the stimulus characteristics of visual distraction strategies used in pain control studies. The assumption that pleasantness,per se, contributes to strategy effectiveness is no longer tenable; future research must demonstrate an independent effect. Further, the implication that stimulus significance may increase the effectiveness of distraction strategies opens up a potentially fruitful avenue of research in pain control.     

Reasoning defeasibly about probabilities

In concrete applications of probability, statistical investigation gives us knowledge of some probabilities, but we generally want to know many others that are not directly revealed by our data. For instance, we may know prob(P/Q) (the probability of P given Q) and prob(P/R), but what we really want is prob(P/Q&;R), and we may not have the data required to assess that directly. The probability calculus is of no help here. Given prob(P/Q) and prob(P/R), it is consistent with the probability calculus for prob(P/Q&;R) to have any value between 0 and 1. Is there any way to make a reasonable estimate of the value of prob(P/Q&;R)? A related problem occurs when probability practitioners adopt undefended assumptions of statistical independence simply on the basis of not seeing any connection between two propositions. This is common practice, but its justification has eluded probability theorists, and researchers are typically apologetic about making such assumptions. Is there any way to defend the practice? This paper shows that on a certain conception of probability—nomic probability—there are principles of “probable probabilities” that license inferences of the above sort. These are principles telling us that although certain inferences from probabilities to probabilities are not deductively valid, nevertheless the second-order probability of their yielding correct results is 1. This makes it defeasibly reasonable to make the inferences. Thus I argue that it is defeasibly reasonable to assume statistical independence when we have no information to the contrary. And I show that there is a function Y(r, s, a) such that if prob(P/Q) = r, prob(P/R) = s, and prob(P/U) = a (where U is our background knowledge) then it is defeasibly reasonable to expect that prob(P/Q&;R) = Y(r, s, a). Numerous other defeasible inferences are licensed by similar principles of probable probabilities. This has the potential to greatly enhance the usefulness of probabilities in practical application.     

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.     

Demands of Justice, Feasible Alternatives, and the Need for Causal Analysis

Many political philosophers hold the Feasible Alternatives Principle (FAP): justice demands that we implement some reform of international institutions P only if P is feasible and P improves upon the status quo from the standpoint of justice. The FAP implies that any argument for a moral requirement to implement P must incorporate claims whose content pertains to the causal processes that explain the current state of affairs. Yet, philosophers routinely neglect the need to attend to actual causal processes. This undermines their arguments concerning moral requirements to reform international institutions. The upshot is that philosophers’ arguments must engage in causal analysis to a greater extent than is typical.     

Comparing the meanings of “if” and “all”

In this study, we compared the everyday meanings of conditionals (“if p then q”) and universally quantified statements (“all p are q”) when applied to sets of elements. The interpretation of conditionals was predicted to be directly related to the conditional probability, such that P(“if p then q”) = P(q|p). Quantified statements were assumed to have two interpretations. According to an instance-focused interpretation, quantified statements are equivalent to conditionals, such that P(“all p are q”) = P(q|p). According to a set-focused interpretation, “all p are q” is true if and only if every instance in set p is an instance of q, so that the statement would be accepted when P(q|p) = 1 and rejected when this probability was below 1. We predicted an instance-focused interpretation of “all” when the relation between p and q expressed a general law for an infinite set of elements. A set-focused interpretation of “all” was predicted when the relation between p and q expressed a coincidence among the elements of a finite set. Participants were given short context stories providing information about the frequency of co-occurrence of cases of p, q, not-p, and not-q in a population. They were then asked to estimate the probability that a statement (conditional or quantified) would be true for a random sample taken from that population. The probability estimates for conditionals were in accordance with an instance-focused interpretation, whereas the estimates for quantified statements showed features of a set-focused interpretation. The type of the relation between p and q had no effect on this outcome.     

The Role of Spiritual Experiences and Activities in the Relationship Between Chronic Illness and Psychological Well-Being

Our research explores the correlates of spiritual experiences over a 2-year period in a sample of older adults (N = 164; mean age 81.9 years) living in a continuing care retirement community. Utilizing responses to the Daily Spiritual Experiences Scale, scores were analyzed for changes over time and for their hypothesized moderating effect in the relationship between chronic illness impact and markers of psychological well-being (as measured by the Geriatric Depression and Life Satisfaction scales). Repeated measures ANOVA indicated a significant decline (P < .01) in the reported spiritual experiences over a 2-year period of time, and t tests showed a significant difference by gender (P < .01) in years 1 and 2, with women reporting higher levels of spiritual experiences than men. Analyses found low spirituality scores associated with low life satisfaction in all years (baseline: r = ?.288, P < .01; year 1: r = ?.209, P < .05; year 2: r = ?.330, P < .001). Only weak associations were detected between low spirituality and the presence of depressive symptoms at baseline (r = .186, P < .05) and year 2 (r = .254, P < .01). Moderation effects of spirituality on the relationship between chronic illness impact and markers of psychological well-being were explored in all years, with a statistically significant effect found only for the presence of depressive symptoms in year 2. Higher impact of chronic illnesses is associated with more depressive symptoms under conditions of low spirituality. Future research may center upon longer-duration evaluation of reliance upon spiritual practices and their impact in care management models.     

Affine representations in psychophysics and the near-miss to Weber’s law

A theoretical account for the near-miss to Weber’s law in the form of a power function, with a special emphasis on the interpretation of the exponent, was proposed by Falmagne [Falmagne, J.-C. (1985). Elements of psychophysical theory. New York: Oxford University Press] within the framework of a subtractive representation, P(x,y)=F(u(x)−g(y)). In this paper, we examine a more general affine representation, P(x,y)=F(u(x)h(y)+g(y)). We first obtain a uniqueness theorem for the affine representation. We then study the conditions that force an affine representation to degenerate to a subtractive one. Part of that study involves the case for which two different affine representations co-exist for the same data. We also show that the balance condition P(x,y)+P(y,x)=1 constrains an affine representation to be a special kind of subtractive representation, a Fechnerian one. We further show that Falmagne’s power law takes on a special form for a so-called weakly balanced system of probabilities, in which case the affine representation is Fechnerian. Finally, following Iverson [Iverson, G.J. (2006a). Analytical methods in the theory of psychophysical discrimination I: Inequalities, convexity and integration of just noticeable differences. Journal of Mathematical Psychology, 50, 271-282], we generalize the Fechner method to construct the sensory scales in a weakly balanced affine representation by integrating (derivatives of) just noticeable differences.     

The effects of pain and a secondary task on postural sway during standing

BackgroundPain impairs available cognitive resources and somatosensory information, but its effects on postural control during standing are inconclusive. The aim of this study was to investigate whether postural sway is affected by the presence of pain and a secondary task during standing.MethodsSixteen healthy subjects stood as quiet as possible at a tandem stance for 30s on a force platform at different conditions regarding the presence of pain and a secondary task. Subjects received painful stimulations on the right upper arm or lower leg according to a relative pain threshold [pain 7 out 10 on a Visual Analog Scale (VAS) - 0 representing “no pain” and 10 “worst pain imaginable”] using a computer pressurized cuff. The secondary task consisted of pointing to a target using a head-mounted laser-pointer as visual feedback. Center of Pressure (COP) sway area, velocity, mean frequency and sample entropy were calculated from force platform measures.FindingsCompared to no painful condition, pain intensity (leg: VAS = 7; arm VAS = 7.4) increased following cuff pressure conditions (P < .01). Pain at the leg decreased COP area (P < .05), increased COP velocity (P < .05), mean frequency (P < .05) and sample entropy (P < .05) compared with baseline condition regardless the completion of the secondary task. During condition with pain at the leg, completion of the secondary task reduced COP velocity (P < .001) compared with condition without secondary task.InterpretationPain in the arm did not affect postural sway. Rather, postural adaptations seem dependent on the location of pain as pain in the lower leg affected postural sway. The completion of a secondary task affected postural sway measurements and reduced the effect of leg pain on postural sway. Future treatment interventions could benefit from dual-task paradigm during balance training aiming to improve postural control in patients suffering from chronic pain.