The negated conditional: A litmus test for the suppositional conditional?

Under the suppositional account of conditionals, when people think about a conditional assertion, "if p then q," they engage in a mental simulation in which they imagine p holds and evaluate the probability that q holds under this supposition. One implication of this account is that belief in a conditional equates to conditional probability [P(q/p)]. In this paper, the authors examine a further implication of this analysis with respect to the wide-scope negation of conditional assertions, "it is not the case that if p then q." Under the suppositional account, nothing categorically follows from the negation of a conditional, other than a second conditional, "if p then not-q." In contrast, according to the mental model theory, a negated conditional is consistent only with the determinate state of affairs, p and not-q. In 4 experiments, the authors compare the contrasting predictions that arise from each of these accounts. The findings are consistent with the suppositional theory but are incongruent with the mental model theory of conditionals. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

"The meaning(s) of conditionals: Conditional probabilities, mental models, and personal utilities": Correction to Oberauer and Wilhelm (2003).

Reports an error in "The meaning(s) of conditionals: Conditional probabilities, mental models, and personal utilities" by Klaus Oberauer and Oliver Wilhelm (Journal of Experimental Psychology: Learning, Memory, and Cognition, 2003[Jul], Vol 29[4], 680-693). On page 684, Table 4, all correlations should have been identified as having a p2003-06626-018.) Five experiments were conducted to test the hypothesis that people understand conditional statements ("if p then q") as indicating a high conditional probability P(q|p). Participants estimated the probability that a given conditional is true (Experiments 1A, 1B, and 3) or judged whether a conditional was true or false (Experiments 2 and 4) given information about the frequencies of the relevant truth table cases. Judgments were strongly influenced by the ratio of pq to p?q cases, supporting the conditional probability account. In Experiments 1A, 1B, and 3, judgments were also affected by the frequency of pq cases, consistent with a version of mental model theory. Experiments 3 and 4 extended the results to thematic conditionals and showed that the pragmatic utility associated with believing a statement also affected the degree of belief in conditionals but not in logically equivalent quantified statements. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Conditionals and conditional probability.

The authors report 3 experiments in which participants were invited to judge the probability of statements of the form if p then q given frequency information about the cases pq, p?q, ?pq, and ?p?q (where ? = not). Three hypotheses were compared: (a) that people equate the probability with that of the material conditional, 1 - P(p?q); (b) that people assign the conditional probability, P(q/p); and (c) that people assign the conjunctive probability P(pq). The experimental evidence allowed rejection of the 1st hypothesis but provided some support for the 2nd and 3rd hypotheses. Individual difference analyses showed that half of the participants used conditional probability and that most of the remaining participants used conjunctive probability as the basis of their judgments. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Theories of Conditional Reasoning: A Developmental Examination of Competing Hypotheses.

Children and adolescents were presented with problems that contained deontic (i.e., if action p is taken, then precondition q must be met) or causal (i.e., if event p occurs, then event q will transpire) conditionals and that varied in the ease with which alternative antecedents could be activated. Results showed that inferences were linked to the availability of alternative antecedents and the generation of "disabling" conditions (claims that the conditionals were false under specific circumstances). Age-related developments were found only on problems involving indeterminate inferences. Correlations among inferences differed for children and adolescents. The findings provide stronger support for domain-general theories than for domain-specific theories of reasoning and suggest, under some conditions, age-related changes in the roles of implicit and explicit processing. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Putting Ifs to Work: Goal-Based Relevance in Conditional Directives.

Conditional directives are used by speakers to instruct hearers which actions are to be taken should certain events occur. The authors demonstrate that conditional directives are distinct from indicative conditionals in which speakers predict what is likely to be observed should certain events occur. The 1st set of experiments shows that goal structure determines what information speakers will select to test whether conditional directives have been followed but that these selections do not reflect their interpretations of the deontic necessity and sufficiency of the conditional relation. The 2nd set of experiments shows that formulations of conditional directives differ in how clearly speakers consider them to express their situation-specific intentions and that hearers accurately perceive what speakers intend them to do as a result of these formulations. The authors' findings illustrate a form of social rationality common in everyday interaction, which broadens normative conceptions of conditionals. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Counterfactual and Semifactual Conditionals Prime Alternative Possibilities.

The authors examined in 3 experiments the comprehension of counterfactuals, such as "If it had rained, the plants would have bloomed," and semifactuals, such as "Even if it had rained, the plants would have bloomed," compared with indicative conditionals, "If it rained, the plants bloomed." The first experiment showed that people read the negative conjunction, "not p and not q" faster when it was primed by a counterfactual than when it was primed by an indicative conditional. They read the affirmative conjunction, "p and q" equally quickly when it was primed by either conditional. The 2nd experiment showed that people read the negated-antecedent conjunction, "not p and q" faster when it was primed by a semifactual conditional. The 3rd experiment corroborated these results in a direct comparison of counterfactuals and semifactuals. The authors discuss the implications of the results for the mental representations of different conditionals. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The meaning(s) of conditionals: Conditional probabilities, mental models, and personal utilities.

[Correction Notice: An erratum for this article was reported in Vol 29(6) of Journal of Experimental Psychology: Learning, Memory, and Cognition (see record 2007-16865-001). On page 684, Table 4, all correlations should have been identified as having a pp then q") as indicating a high conditional probability P(q|p). Participants estimated the probability that a given conditional is true (Experiments 1A, 1B, and 3) or judged whether a conditional was true or false (Experiments 2 and 4) given information about the frequencies of the relevant truth table cases. Judgments were strongly influenced by the ratio of pq to p?q cases, supporting the conditional probability account. In Experiments 1A, 1B, and 3, judgments were also affected by the frequency of pq cases, consistent with a version of mental model theory. Experiments 3 and 4 extended the results to thematic conditionals and showed that the pragmatic utility associated with believing a statement also affected the degree of belief in conditionals but not in logically equivalent quantified statements. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Reasoning counterfactually: Making inferences about things that didn't happen.

The authors investigated the relationship between reasoners' understanding of subjunctive conditionals (e.g., if p had happened, then q would have happened) and the inferences they were prepared to endorse. Reasoners who made a counterfactual interpretation of subjunctive statements (i.e., they judged the statement to imply that p and q did not happen) endorsed different inferences than those who did not. Those who made a counterfactual interpretation were more likely to (a)judge the situation in which p and q occurred to be inconsistent with the conditional statement and (b) make negative inferences such as modus tollens (i.e., – q therefore – p ). These findings occurred with familiar and unfamiliar content, affirmative and negative conditionals, and conditional and biconditional relations. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Perceived necessity explains the dissociation between logic and meaning: The case of "Only If."

The purpose of this article was to investigate why sentences of the form "p only if q" often appear to have a different meaning than sentences of the form "if p, then q," despite their logical equivalence. The results of 3 experiments indicate that when "if…then" statements were equated with respect to necessity and temporal relations, different pragmatic relations (such as permission, causation, etc.) elicited similar "only if" judgments. However, different necessity relations elicited different "only if" judgments, regardless of the type of pragmatic relation expressed in the "if then" statement. These data suggest that "only if" judgements are primarily mediated by necessity and temporal relations and that pragmatic contexts may play a more indirect role, such as in the interpretation of necessity and temporal relations. Suggestions for how these findings might be incorporated into pragmatic scheme theory (P. W. Cheng & K. J. Holyoak; see record 73:29075; P. W. Cheng, K. J. Holyoak, R. E. Nisbett, & L. M. Oliver; see record 73:8193) and mental models theory (P. N. Johnson-Laird & R. M. J. Byrne, 1991) are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Conditional reasoning in children.

Conditional reasoning ability is analyzed in the context of Piaget's logical operational system. It is argued that facility in concrete operations is necessary and sufficient for competence in the simple syllogistic forms, though dealing with conditional statements in certain more complex contexts requires formal operations. Data from 3 studies are presented in support of these assertions. Using a total of 120 Ss in Grades 1–5, results of Exps I and II show that presentation of statements in a concrete conversational context elicited substantially more correct conditional reasoning than has been found in earlier studies. Results of Exp III, which used a total of 154 Ss in Grades 2, 4, 6, and 8, indicate that a distinction is drawn between comprehension that p (not p) entails the possibility of q (given If p, then q) and comprehension that q entails the possibility of p. The former is less difficult than the latter, and only the latter requires concrete operations. A methodological implication of these studies is that in the assessment of logical competence, results depend heavily on the exact manner of presentation, as well as on the specific logical operation being assessed. Implications for the theory of formal operations and for the study of traditional syllogistic reasoning are also discussed. (19 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Conversational comprehension processes are responsible for reasoning fallacies in children as well as adults: If is not the biconditional.

Proposes that children do poorly on reasoning from premises of the form if p then q not because they construe if as a biconditional but rather because they use discourse comprehension processes that lead them to accept the invited inferences if not p then not q and if q then p. This hypothesis predicts that children should respond appropriately to premises in which the invited inferences are countermanded. In Exp I, 24 undergraduates and 44 10-yr-olds were given conditional reasoning problems. Some of these had a major premise consisting of a single if–then sentence, while others had a more elaborate major premise in which the invited inferences were explicitly countermanded. In Exp II, Ss were 24 undergraduates, 20 10-yr-olds, and 34 7-yr-olds. In some problems the major premise consisted of a single if–then statement; in others, the major premise consisted of 3 such statements, 2 of which shared the same consequent, thus implicitly countermanding the invited inferences. In both experiments, all age groups committed the fallacies in the simple condition but not in the more complex condition. It is concluded that children's representation of if distinguishes necessary from merely invited inferences. Data suggest a collection of countermandable context-dependent inferences of varying degrees of invitingness associated with if. (19 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

On the probability of making Type I errors.

A statistical test leads to a Type I error whenever it leads to the rejection of a null hypothesis that is in fact true. The probability of making a Type I error can be characterized in the following 3 ways: the conditional prior probability, the overall prior probability, and the conditional posterior probability. In this article, we show (a) that the alpha level can be equated with the 1st of these and (b) that it provides an upper bound for the second but (c) that it does not provide an estimate of the third, although it is commonly assumed to do so. We trace the source of this erroneous assumption first to statistical texts used by psychologists, which are generally ambiguous about which of the 3 interpretations is intended at any point in their discussions of Type I errors and which typically confound the conditional prior and posterior probabilities. Underlying this, however, is a more general fallacy in reasoning about probabilities, and we suggest that this may be the result of erroneous inferences about probabilistic conditional statements. Finally, we consider the possibility of estimating the (posterior) probability of a Type I error in situations in which the null hypothesis is rejected and, hence, the proportion of statistically significant results that may be Type I errors. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The role of classification on testing conditional sentences in the context of an empirical task.

An analysis of a disjunctive concept based on 2 binary dimensions reveals that the relationships between the individual stimuli and classification are implicatory; these relationships may be expressed in conditional sentences of the form "if p then q." A 4-phase experiment with 40 students (average age 17 yrs 6 mo) was conducted to determine the effect on testing the truth value of conditional sentences of (a) prior experience in classifying the instances of the conditional, and (b) knowledge that such classifications would be of empirical value in a subsequent concept learning task. Phase I involved classifying individual stimuli for a subsequent concept learning problem; Phase II was the problem itself; Phase III was testing the truth value of 4 conditional sentences which, depending on whether they were true or false, described respectively the implicatory or tautologous relationships inherent in a concept learning task that followed (Phase IV). It is concluded that both prior experience with classifying the components of an implicatory rule and knowledge of the empirical relevance of such classifications to a subsequent task were necessary to the emergence of logical reasoning in the Phase III selection task. (14 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

What we think we do (to each other): How personality can bias behavior schemas through the projection of if–then profiles.

People's knowledge about others includes not only person schemas about the typical traits of others but also behavior schemas about the likely interpersonal consequences of different behaviors. In this article, it is argued that perceiver effects can be interactive at the level of behavior schemas. A person's own personality configuration of if–then responses in social interactions (Mischel & Shoda, 1995) may contribute to that person's beliefs about the meaning and impact of relational behaviors more generally. In consequence, people who experience strong (or weak) responses to behaviors that vary along a particular trait dimension, such as warmth–coldness, may expect others to experience similarly strong (or weak) responses to those same kinds of behaviors. In 3 studies, people who were high in trait communion expected others to respond more strongly to behaviors that varied in warmth–coldness than did people who were low in trait communion, and people who were low in trait agency expected others to respond more strongly to behaviors that varied in assertiveness–unassertiveness than did people who were high in trait agency. Studies 2 and 3 provided evidence that participants' behavior schemas were based on assumptions derived from their own if–then personality profiles. (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

Conditional reasoning: The necessary and sufficient conditions.

Five experiments, with 163 university students, investigated 2 theories of conditional reasoning (CDR). The pragmatic schema theory posits that CDR is mediated by context-sensitive inference rules. According to the contextual cuing theory, inferences are based on a mental model that represents necessity and sufficiency relations. Both schematic relations and necessity relations predicted responses on a 4-card selection task. In contrast, after the effects of perceived necessity had been partialled out, schematic relations did not predict responses to either a conditional arguments task or a task in which Ss judged the similarity of "if then" and "only if" statements. Findings question the assumption that reasoning is mediated by schematic rules, which presumably apply regardless of task. A reconceptualization of the pragmatic reasoning schema is proposed. (French abstract) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Optimal data selection in the reduced array selection task (RAST).

The predictions of M. Oaksford and N. Chater's (see record 1995-08271-001) optimal data selection (ODS) model for the reduced array selection task (RAST) were tested in 4 experiments. Participants tested a hypothesis, if p then q, by selecting cards showing q or not q instances. In Experiment 1, where selections were made from different sized stacks of q and not q cards, as P(q) increased, not q card selections rose, and q card selections fell, as predicted. Experiment 2 controlled for the possibility that stack height influenced responses; these results were also consistent with ODS. Experiment 3, which controlled further for this possibility, replicated Experiment 1. Experiment 4 addressed a final issue concerning the medium P(q) condition by concentrating on initial card selections; the results were again consistent with ODS. Although generally consistent with the ODS model, these experiments also suggest some interesting revisions. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

User settings of cue thresholds for binary categorization decisions.

The output of binary cuing systems, such as alerts or alarms, depends on the threshold setting—a parameter that is often user-adjustable. However, it is unknown if users are able to adequately adjust thresholds and what information may help them to do so. Two experiments tested threshold settings for a binary classification task based on binary cues. During the task, participants decided whether a product was intact or faulty. Experimental conditions differed in the information participants received: all participants were informed about a product’s fault probability and the payoffs associated with decision outcomes; one third also received information regarding conditional probabilities for a fault when the system indicated or did not indicate the existence of one (predictive values); and another third received information about conditional probabilities for the system indicating a fault, in the instance of the existence or lack thereof, of an actual fault (diagnostic values). Threshold settings in all experimental groups were nonoptimal, with settings closest to the optimum with predictive-values information. Results corresponded with a model describing threshold settings as a function of the conditional probabilities for the different outcomes. From a practical perspective, results indicate that predictive-values information best supports decisions about threshold settings. Consequently, for users to adjust thresholds, they should receive information about predictive-values, provided that such values can be computed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Classical statistical hypotheses testing within the context of Bayesian theory.

A portion of results that are judged significant on the basis of classical statistical tests will be due to chance. The conditional probability of error in the presence of statistical significance depends upon the significance level employed, the power of the test, and the prior probability that a valid null hypothesis was chosen for testing. Bayesian theory provides a logical model for the design of experiments in which classical hypothesis testing is to be used. In this manner, Bayesian objectives can be realized while the safeguards of classical statistical hypothesis testing are retained. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Aging and Conditional Probability Judgments: A Global Matching Approach.

Age differences in bias in conditional probability judgments were investigated based on predictions derived from the Minerva-Decision Making model (M. R. P. Dougherty, C. F. Gettys, & E. E. Ogden, 1999), a global matching model of likelihood judgment. In this study, 248 younger and older adults completed frequency judgment and conditional probability judgment tasks. Age differences in the frequency judgment task are interpreted as an age-related deficit in memory encoding. Older adults' stronger biases in the probability judgment task point to age differences in criterion setting. Age-related biases were eliminated when age groups were equated on memory encoding by means of study time manipulation. The authors conclude that older adults' stronger judgment biases are a function of memory impairment. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

What college students really think about ethnic student organizations.

In this study, the authors examined European, Hispanic, and African American college students' attitudes toward ethnic student organizations (ESOs). Based on data from Study 1 (N = 750), it was found that students across ethnic groups expressed uncertainty about whether ESOs were beneficial/necessary, fair/acceptable, and about their interest in joining an ESO. As a group, Hispanic and African American students did not believe that ESOs contributed to racial or ethnic separatism on campus, whereas European Americans expressed uncertainty about that possibility. In Study 2, a separate sample of students (N = 631) was randomly assigned to read the mission statement of a White, Hispanic, or African American ESO. Consistent with asymmetry theory, students in general judged the White American ESO as significantly less beneficial/necessary, less fair/acceptable, and as contributing more to racial/ethnic separatism than the Hispanic and African American ESOs. European, Hispanic, and African American students viewed their own ESO most favorably, although European American students were more consistent in their appraisals of ESOs irrespective of the ethnic focus of the ESO. Implications of these findings are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)