Conditional reasoning and the competence-performance issue: A developmental analysis of a training task

Empirical investigations of conditional reasoning have generally found that both children and young adults perform poorly on tasks that require the selection or evaluation of those propositions that test the truth status of conditional statements (if p then q). Earlier work (D. O'Brien & W. F. Overton, Journal of Experimental Child Psychology 1980, 30, 44–60) suggested that poor performances with these tasks by young adults show improvement following the introduction of evidence that contradicts earlier faulty inferences, and this improvement generalizes to other conditional reasoning tasks. The effects of the contradiction training were not found with younger subjects. The present research is an extension of the contradiction training paradigm. Ten-, fourteen-, and eighteen-year-olds were tested to assess developmental differences in improvement with an evaluation and a conditional syllogism task. Significant improvement in performance was found for the twelfth grade students following the contradiction training, and this generalized across tasks. This effect was not found for the two younger groups. The usual poor performance of the oldest group is considered to be a false negative assessment of their conditional reasoning competency. Further, it is suggested that several correct performances of younger reasoners are false positive assessments of their conditional reasoning competency.     

The Causal Structure of Utility Conditionals

The psychology of reasoning is increasingly considering agents' values and preferences, achieving greater integration with judgment and decision making, social cognition, and moral reasoning. Some of this research investigates utility conditionals, ‘‘if p then q’’ statements where the realization of p or q or both is valued by some agents. Various approaches to utility conditionals share the assumption that reasoners make inferences from utility conditionals based on the comparison between the utility of p and the expected utility of q. This article introduces a new parameter in this analysis, the underlying causal structure of the conditional. Four experiments showed that causal structure moderated utility‐informed conditional reasoning. These inferences were strongly invited when the underlying structure of the conditional was causal, and significantly less so when the underlying structure of the conditional was diagnostic. This asymmetry was only observed for conditionals in which the utility of q was clear, and disappeared when the utility of q was unclear. Thus, an adequate account of utility‐informed inferences conditional reasoning requires three components: utility, probability, and causal structure.     

A successive-conditionalization approach to disjunctive and syllogistic reasoning

With p and q each standing for a familiar event, a disjunctive statement, “either p or q”, seems quite different from its material conditional, “if not p then q”. The notions of sufficiency and necessity seem specific to conditional statements. It is surprising, however, to find that perceived sufficiency and necessity affect disjunctive reasoning in the way they affect conditional reasoning. With B and C each standing for a category name, a universal statement, “all B are C”, seems stronger than its logically equivalent conditional statement, “if B then C”. However, the effects of perceived sufficiency or necessity were found to be as pronounced in conditional reasoning as in syllogistic reasoning. Furthermore, two experiments also showed that (a) MP (modus ponens)-comparable disjunctive reasoning was as difficult as MT (modus tollens)-comparable disjunctive reasoning, and that (b) MT-comparable syllogisms were easier to solve than MT problems in conditional reasoning.     

Logic programming,probability, and two-system accounts of reasoning: a rejoinder to Oaksford and Chater (2014)

This reply to Oaksford and Chater’s (O&C)’s critical discussion of our use of logic programming (LP) to model and predict patterns of conditional reasoning will frame the dispute in terms of the semantics of the conditional. We begin by outlining some common features of LP and probabilistic conditionals in knowledge-rich reasoning over long-term memory knowledge bases. For both, context determines causal strength; there are inferences from the absence of certain evidence; and both have analogues of the Ramsey test. Some current work shows how a combination of counting defeaters and statistics from network monitoring can provide the information for graded responses from LP reasoning. With this much introduction, we then respond to O&C’s specific criticisms and misunderstandings.     

Children's comprehension of complex sentences

Four tasks were given to children from 4–12 to test their comprehension of complex sentences containing main verbs taking underlying sentences as their complements (Sally knew that she was early). In an imperatives task, very young children interpreted only the complement verb and ignored the complex verb. In a short-term memory task, sentences with two negations usually lost the second not in recall. In direct questioning and anomaly-detection tasks, children tended to make pragmatic inferences and excessively depend on knowledge about the world, as opposed to linguistic information. Overall results showed that even sixth graders had not yet attained adult-level comprehension of complex sentences.     

Language and reasoning: a study of temporal factors

The paper is concerned with the testing of psycholinguistic hypotheses by the use of deductive reasoning tasks. After reviewing some of the problems of interpretation which have arisen with particular reference to conditional rules, an experiment is presented which measures comprehension and verification latencies in addition to response frequencies in a truth table evaluation task.The experiment tests a psycholinguistic hypothesis concerning the different usage of the logically equivalent forms of sentence: If p then q and p only if q with respect to the temporal order of the events p and q. It is proposed that the former sentence is more natural when the event p precedes the event q in time, and the latter more natural when the opposite temporal relation holds.Although significant support is found for the hypothesis in the analysis of the latency data, it is only distinguished from an alternative explanation by detailed analysis of response frequencies, thus indicating the general usefulness of the paradigm adopted.     

Gender and height: Developmental patterns in knowledge and use of an accurate stereotype

The belief that men are taller than women is an accurate gender stereotype that is presumably learned through everyday encounters with men and women. It is unlike most gender beliefs that are difficult to quantify and verify, and that may be learned through means other than direct experience. This research makes use of several advantages available when studying an accurate stereotype (e.g., the ability to know the “truth” about judged targets). Subjects from five different age/grade levels (kindergarten, third grade, seventh grade, tenth grade, and college, N=491, 92.26% Caucasian, 7.54% Arabic, 0.2% African American) made judgments of “who is taller” among photographed male—female pairs that had actually been matched in height. Judgments were made of kindergarten, seventh grade, and college photo pairs. In a developmental context, height judgments are particularly interesting because the gender difference in height changes at different ages—specifically, girls are taller than boys around seventh grade. The data indicated that subjects were sensitive to the changing height stereotype—i.e., they judged adult males taller than adult females, but seventh grade females taller than seventh-grade males. However this was most strikingly true among subjects directly faced with this reality in their peer context—seventh graders—and among college students. Height judgments were also sensitive to an ingroup (gender) bias. The data are discussed in reference to the broader literature on stereotyping and base rate influences on judgment.     

Pragmatic reasoning schemas

We propose that people typically reason about realistic situations using neither content-free syntactic inference rules nor representations of specific experiences. Rather, people reason using knowledge structures that we term pragmatic reasoning schemas, which are generalized sets of rules defined in relation to classes of goals. Three experiments examined the impact of a “permission schema” on deductive reasoning. Experiment 1 demonstrated that by evoking the permission schema it is possible to facilitate performance in Wason's selection paradigm for subjects who have had no experience with the specific content of the problems. Experiment 2 showed that a selection problem worded in terms of an abstract permission elicited better performance than one worded in terms of a concrete but arbitrary situation, providing evidence for an abstract permission schema that is free of domain-specific content. Experiment 3 provided evidence that evocation of a permission schema affects not only tasks requiring procedural knowledge, but also a linguistic rephrasing task requiring declarative knowledge. In particular, statements in the form if p then q were rephrased into the form p only if q with greater frequency for permission than for arbitrary statements, and rephrasings of permission statements produced a pattern of introduction of modals (must, can) totally unlike that observed for arbitrary conditional statements. Other pragmatic schemas, such as “causal” and “evidence” schemas can account for both linguistic and reasoning phenomena that alternative hypotheses fail to explain.     

What causal conditional reasoning tells us about people's understanding of causality

Causal conditional reasoning means reasoning from a conditional statement that refers to causal content. We argue that data from causal conditional reasoning tasks tell us something not only about how people interpret conditionals, but also about how they interpret causal relations. In particular, three basic principles of people's causal understanding emerge from previous studies: the modal principle, the exhaustive principle, and the equivalence principle. Restricted to the four classic conditional inferences—Modus Ponens, Modus Tollens, Denial of the Antecedent, and Affirmation of the Consequent—causal conditional reasoning data are only partially able to support these principles. We present three experiments that use concrete and abstract causal scenarios and combine inference tasks with a new type of task in which people reformulate a given causal situation. The results provide evidence for the proposed representational principles. Implications for theories of the naïve understanding of causality are discussed.     

Rapid-response versus free-time selection tasks using different logical connectives

When reasoning with conditional statements (i.e., if [not] p then [not] q), people tend to display matching bias: Options that match the entities named in the rule are selected even when logically inappropriate. Three different Wason selection tasks were administered under free-time and rapid-response formats. For the latter, individual cards were presented for one second, and required a response within a further one second. Previous research using these formats (Roberts & Newton, 2001) has shown that this increases matching bias, in line with the action of preconscious heuristic processes which direct attention towards relevant aspects of a problem, but whose action can be overturned if there is sufficient time to apply analytic reasoning processes. The selection tasks administered included a standard abstract conditional task, a disjunctive version (i.e., either [not] p or [not] q), and a conditional task in which the cards showed explicitly negated values. Both conditional tasks demonstrated matching bias, but under rapid-response presentation, matching bias only increased for the standard conditional and disjunctive tasks. Overall, the data support Evans’ (e.g., 2006) heuristic-analytic framework albeit with some caveats, and it is suggested that the broad question, of whether individual selection task formats show or do not show matching bias, requires more detailed investigation.     

Intuitive biological thought: Developmental changes and effects of biology education in late adolescence

A large body of cognitive research has shown that people intuitively and effortlessly reason about the biological world in complex and systematic ways. We addressed two questions about the nature of intuitive biological reasoning: How does intuitive biological thinking change during adolescence and early adulthood? How does increasing biology education influence intuitive biological thinking? To do so, we developed a battery of measures to systematically test three components of intuitive biological thought: anthropocentric thinking, teleological thinking and essentialist thinking, and tested 8th graders and university students (both biology majors, and non-biology majors). Results reveal clear evidence of persistent intuitive reasoning among all populations studied, consistent but surprisingly small differences between 8th graders and college students on measures of intuitive biological thought, and consistent but again surprisingly small influence of increasing biology education on intuitive biological reasoning. Results speak to the persistence of intuitive reasoning, the importance of taking intuitive knowledge into account in science classrooms, and the necessity of interdisciplinary research to advance biology education. Further studies are necessary to investigate how cultural context and continued acquisition of expertise impact intuitive biology thinking.     

Implicatures et développement

We summarize our work on pragmatic inference-making in children, while generally focusing on scalar implicatures. Such inferences arise when a relatively weak term implies the rejection of a stronger one. For example, some is often understood to mean not all. While adults readily draw such implicatures, children tend to rely on the terms’ minimal, lexically encoded meanings (with which some is compatible with all). Given that children’s treatments coincide with logical ones, children end up appearing more logical than adults on standard reasoning tasks. We describe this effect in detail while showing that (a) even young children can be encouraged to carry out implicatures and, that; (b) evidence of non-pragmatic behavior is best explained as due to unavailable effort.     

Quantifying disablers in reasoning with universal and existential rules

People accept conclusions of valid conditional inferences (e.g., if p then q, p therefore q) less, the more disablers (circumstances that prevent q to happen although p is true) exist. We investigated whether rules that through their phrasing exclude disablers evoke higher acceptance ratings than rules that do not exclude disablers. In three experiments we re-phrased content-rich conditionals from the literature as either universal or existential rules and embedded these rules in Modus Ponens and Modus Tollens inferences. In Experiments 2 and 3, we also used abstract rules. The acceptance of conclusions increased when the rule was phrased with “all” instead of “some” and the number of disablers had a higher impact on existential rules than on universal rules. Further, the effect of quantifier was more pronounced for abstract rules and when tested within subjects. We discuss the relevance of phrasing, quantifiers and knowledge on reasoning.     

A necessity illusion for modal inferences from conditionals

Three experiments examined how people reason about what is possible or necessary when a conditional is true. Participants were asked to indicate whether it was necessary, possible or impossible for a specific instance to conform to one of the truth-table cases (pq, p¬q, ¬pq and ¬p¬q) (¬ = not), given the truth of the conditional. It was found that most participants, inconsistently, judged the pq case as necessary but the ¬pq or ¬p¬q cases as possible. Logically, these two kinds of judgments are contradictory. Moreover, a true conditional doesn’t imply that a specific instance under the conditional must be pq . Therefore, people demonstrate a necessity illusion for pq cases which contradicts their commitment to the possibility of ¬pq or ¬p¬q cases. Existing accounts of conditionals are unable to explain the contradiction and the necessity illusion. We propose an inference dissociation account and explore the theoretical implications of this necessity illusion.     

On the nature of mental models of conditional: The case of If,If then,and Only if

It has recently been reported that forward inferences from if p then q sentences (i.e., from antecedent to consequent) were faster than backward inferences from consequent to antecedent (Barrouillet, Grosset, & Lecas, 2000). The standard mental model theory assumes that this directionality effect is a figural effect due to the order the information enters working memory, whereas we claim that it results from the nature of the mental models that represent oriented relations from hypothetical values introduced by the word If. We tested these hypotheses in an experiment in which adult participants evaluated conditional syllogisms from either if p then q, p only if q, or p if q statements. Contrary to the predictions resulting from the standard theory, the three forms of the conditional provoked a reversed directionality effect and denial inferences took longer to endorse than affirmative inferences for all the forms of conditionals. We argue from these results that mental models of the conditional represent oriented relations instead of mere co-occurrences between events.     

Developmental changes in conditional reasoning: Linguistic or logical?

This study investigated reasoning with abstract conditional sentences as a function of age. Subjects from third-grade to eleventh-grade were required to evaluate the conclusion of several conditional arguments. The results confirmed the previously established finding that performance improves with age, particularly between 11 and 15 years. This finding could be interpreted to indicate that individuals become more logical as they get older. However, another possible interpretation is that the meaning of a conditional sentence like pthenq for naive Ss may not always be given by the truth function pq true, $\text{p}\text{q}$ false, $\text{p}\text{q}$ true, and $\text{p}\text{q}$ true. Further analysis suggested that at 9 years individuals treat the connective in the sentence, ifpthenq, as if it were either a conjunctive or a biconditional, that the conjunctive meaning disappears with increasing age and after 13 years is gradually superceded by the conditional.     

Is conditional reasoning related to mathematical problem solving?

The current study aimed to investigate the relation between conditional reasoning, which is a common type of logical reasoning, and children's mathematical problem solving. A sample of 124 fourth graders was tested for their conditional reasoning skills and their mathematical problem solving skills, as well as a list of control variables (e.g., IQ, working memory, reading) and potential mediators (number sentence construction and computation). The children's ability to make modus ponens (MP) inferences significantly predicted their mathematical problem solving skills, even after controlling for the potential confounding variables. The relation was mediated by the number sentence construction skills. The findings, in addition to supporting the link between conditional reasoning and mathematics, further indicate that the ability to process relations may be the mechanism underlying the relation. Theoretical and practical implications are discussed.     

Longitudinal Analysis of Associations between 3-D Mental Rotation and Mathematics Reasoning Skills during Middle School: Across and within Genders

ABSTRACT

The development of math reasoning and 3-d mental rotation skills are intertwined. However, it is currently not understood how these cognitive processes develop and interact longitudinally at the within-person level – either within or across genders. In this study, 553 students (52% girls) were assessed from fifth to seventh grades on 3-d mental rotation spatial skills (assessed each fall) and numerical and algebraic math reasoning skills (assessed each spring). Boys outperformed girls on mental rotation tests across all three grades, and on fifth and seventh grade math reasoning tests. Consistent with the literature on between-person comparisons, there was a positive correlation between mental rotation and math reasoning skills in the full sample and for both genders. A random intercept cross-lagged panel model was used to control for these confounding group-level differences in order to isolate within-person associations between earlier and later performance. Initially in fifth grade, math reasoning predicted subsequent sixth grade mental rotation skills. By seventh grade, more advanced mental rotation skills were associated with subsequent math reasoning skills while math reasoning skills were no longer predictive of mental rotation skills. An examination of gender differences revealed that this pattern was driven by boys while girls experienced less within-person change. These findings suggest that boys may initially rely in part on their math reasoning skills to solve 3-d mental rotation tasks. However, as their 3-d mental rotation skills mature, they begin to primarily depend upon these developing spatial skills to solve math reasoning problems rather than the reverse.     

Book Review: Ecological rationality and its heuristics

In this study both adolescents with autism spectrum disorder (ASD) and typically developing controls were presented with conditional reasoning problems using familiar content. In this task both valid and fallacious conditional inferences that would otherwise be drawn can be suppressed if counterexample cases are brought to mind. Such suppression occurs when additional premises are presented, whose effect is to suggest such counterexample cases. In this study we predicted and observed that this suppression effect was substantially and significantly weaker for autistic participants. We take this as evidence that autistics are less contextualised in their reasoning, a finding that can be linked to research on autism on a variety of other cognitive tasks.     

Working memory and conditional reasoning

Little is known about the role of working memory in conditional reasoning. This paper reports three experiments that examine the contributions of the visuo-spatial scratch pad (VSSP), the articulatory loop, and the central executive components of Baddeley and Hitch's (1974) model of working memory to conditional reasoning. The first experiment employs a spatial memory task that is presented concurrently with two putative spatial interference tasks (tapping and tracking), articulatory suppression, and a verbal memory load. Only the tracking and memory load impaired performance, suggesting that these tap the VSSP and central executive, respectively. Having established the potency of these interference tasks two further experiments examined the effects of tapping and tracking (Experiment 2) and articulation and memory load (Experiment 3) on a conditional reasoning task. Neither tracking nor tapping affected the number of inferences accepted or response latency. Articulation also failed to affect conditional reasoning but memory load selectively reduced acceptance of modus tollens inferences. These results are discussed in terms of both rule-based and mental models theories of reasoning. While these data cannot discriminate between the two perspectives they provide support for one of the central assumptions in each: that some errors in reasoning are attributable directly to working memory demands. Taken together these experiments suggest that conditional reasoning requires an abstract working memory medium for representation; it does not require either the VSSP or the articulatory loop. It is concluded that the central executive provides the necessary substrate.