In search of counter-examples: Deductive rationality in human reasoning

Dual-process theories come in many forms. They draw on the distinction between associative, heuristic, tacit, intuitive, or implicit processes (System 1) and rule-based, analytic, explicit processes (System 2). We present the results of contextual manipulations that have a bearing on the supposed primacy of System 1 (Stanovich & West, 2000). Experiment 1 showed that people who evaluated logically valid or invalid conditional inferences under a timing constraint (N=56), showed a smaller effect of logical validity than did people who were not placed under a timing constraint (N= 44). Experiment 2 similarly showed that stressing the logical constraint that only inferences that follow necessarily are to be endorsed (N = 36) increased the size of the validity effect, as compared to that of participants (N=33) given the standard instruction to make "logical" inferences. These findings concur with the thesis in dual-processing frameworks that "Rationality-2 processes" (Evans & Over, 1996), "test procedures" (Chater & Oaksford, 1999), or "conclusion validation processes" (Johnson-Laird & Byrne, 1991; Schroyens, Schaeken, & d'Ydewalle, 2001) serve to override the results of System 1 processes.     

More evidence for a dual-process model of conditional reasoning

Many studies have shown that the deductive inferences that people make have global properties that reflect the statistical information implicit in the premises. This suggests that such reasoning can be explained by a single, underlying probabilistic model. In contrast, the dual process model of conditional reasoning (Verschueren, Schaeken, & d'Ydewalle, 2005b) proposes that people can use either a logical, counterexample-based strategy or a probabilistic one. In two studies, we presented reasoners with sequences of affirmation-of-the-consequent inferences that differed with respect to the statistical properties of the premises, either explicitly or implicitly. As predicted by the dual-process model, an analysis of individual response patterns showed the presence of two distinct strategies, with use of the counterexample strategy being associated with higher levels of abstract-reasoning competence. Use of the counterexample strategy was facilitated by the explicit presentation of counterexample information. In a further study, we then examined explicitly probabilistic inferences. This study showed that although most reasoners made statistically appropriate inferences, the ability to make more-accurate inferences was associated with higher levels of abstract reasoning competence. These results show that deductive inferential reasoning cannot be explained by a single, unitary process and that any analysis of reasoning must consider individual differences in strategy use.     

Reasoning with conditionals: a test of formal models of four theories

The four dominant theories of reasoning from conditionals are translated into formal models: The theory of mental models (Johnson-Laird, P. N., & Byrne, R. M. J. (2002). Conditionals: a theory of meaning, pragmatics, and inference. Psychological Review, 109, 646-678), the suppositional theory (Evans, J. S. B. T., & Over, D. E. (2004). If. Oxford: Oxford University Press), a dual-process variant of the model theory (Verschueren, N., Schaeken, W., & d'Ydewalle, G. (2005). A dual-process specification of causal conditional reasoning. Thinking &Reasoning, 11, 278-293), and the probabilistic theory (Oaksford, M., Chater, N., & Larkin, J. (2000). Probabilities and polarity biases in conditional inference. Journal of Experimental Psychology: Learning, Memory, and Cognition, 26, 883-899). The first three theories are formalized as multinomial models. The models are applied to the frequencies of patterns of acceptance or rejection across the four basic inferences modus ponens, acceptance of the consequent, denial of the antecedent, and modus tollens. Model fits are assessed for two large data sets, one representing reasoning with abstract, basic conditionals, the other reflecting reasoning with pseudo-realistic causal and non-causal conditionals. The best account of the data was provided by a modified version of the mental-model theory, augmented by directionality, and by the dual-process model.     

Reasoning with conditionals: Does every counterexample count? It’s frequency that counts

A series of experiments investigated what determines people's degree of belief in conditionals and their readiness to draw inferences from them. Information on the frequency of exceptions to conditional rules was contrasted with information about the number of different disabling conditions causing these exceptions. Experiments 1 and 2, using conditionals with arbitrary contents, revealed a strong effect of frequency information and no effect of disabling information. Experiment 3 established that, in the absence of frequency information, the disabling condition information used in Experiments 1 and 2 affected belief in the conditionals and inference acceptance, as has been found in many previous studies (Byrne, 1989; DeNeys, Schaeken, & d'Ydewalle, 2003b). Experiment 4 extended the results of Experiments 1 and 2 to everyday conditionals. The results show that belief in a conditional, and the confidence in inferences subsequently drawn from it, both depend on the subjective conditional probability of the consequent given the antecedent. This probability is estimated from the relative frequency of exceptions regardless of what causes them.     

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.     

Testing the Mental Model Theory of Propositional Reasoning

Johnson-Laird, Byrne, and Schaeken (1992) present a theory of propositional reasoning by mental models. The present study provides a comprehensive test of the predictions of that theory for the difficulty of simple disjunctive and conditional inferences. The results are largely consistent with the complex pattern of predictions. They are discussed in the light of recently proposed modifications of the original theory.     

Individual differences in conditional reasoning: A dual-process account

Dual-process theories of conditional reasoning predict that relationships among four basic logical forms, and to intellectual ability and thinking predictions, are most evident when conflict arises between experiential and analytic processing (e.g., Stanovich & West, ). To test these predictions, 210 undergraduates were presented with conditionals for which the consequents were either weakly or strongly associated with alternative antecedents (i.e., WA and SA problems, respectively). Consistent with predictions, modus ponens inferences were not related to inferences on the uncertain forms (affirmation of the consequent, denial of the antecedent). On WA problems, modus tollens, affirmation of the consequent, and denial of the antecedent were related to each other and to verbal ability. Modus ponens was linked to verbal ability only when disabling conditions were activated. In accord with the predictions of Stanovich and West (), on most problems, thinking dispositions predicted variance in inferences independently from verbal ability. We argue that a largely automatic experiential processing system governs performance on modus ponens, unless disablers are activated. Consciously controlled analytic processing predominates on the uncertain forms and, under some conditions, on modus tollens.     

Transfer, control, and automatic processing in a complex motor task: an examination of bounce juggling

The authors evaluated the hypothesis that controlled and automatic processes are opposite ends of a continuum of learning (e.g., R. M. Shiffrin & W. Schneider, 1977) vs. an alternative, concurrent emergence hypothesis (e.g., J. M. Bebko et al., 2003; G. Logan, 1989). The authors also measured potential positive transfer effects of learning from one motor task to another. Four experienced cascade jugglers and 5 novices learned to bounce juggle, practicing regularly for 5 weeks. The experienced jugglers showed positive transfer of learning, maintaining a lead of approximately 6-10 days over the novices, even as both groups automatized the new skill. Measures of automatic and controlled processing were positively correlated, indicating that those processes emerge concurrently. The authors present a model in which controlled and automatic processes emerge orthogonally.     

Causal and conditional inferences: A comment on Cummins (1995)

Cummins (1995) offers an analysis of causal and truthfunctional sufficiency and necessity to predict and explain the effects on conditional inferences of two pragmatic factors: alternative causes and disabling conditions. However, the justification of these predictions is inconsistent. This note offers a modified analysis which puts her predictions on a sounder base: it is proposed that alternative causes and disabling conditions affect judgments of argument validity under three different models for the causal conditional.     

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.     

Probabilities and polarity biases in conditional inference

A probabilistic computational level model of conditional inference is proposed that can explain polarity biases in conditional inference (e.g., J. St. B. T. Evans, 1993). These biases are observed when J. St. B. T. Evans's (1972) negations paradigm is used in the conditional inference task. The model assumes that negations define higher probability categories than their affirmative counterparts (M. Oaksford & K. Stenning, 1992); for example, P(not-dog) > P(dog). This identification suggests that polarity biases are really a rational effect of high-probability categories. Three experiments revealed that, consistent with this probabilistic account, when high-probability categories are used instead of negations, a high-probability conclusion effect is observed. The relationships between the probabilistic model and other phenomena and other theories in conditional reasoning are discussed.     

Hypothetical and certain inferences from conditional arguments read in texts

In 3 experiments, affirmative and hypothetical probes were presented after narrative texts containing conditional arguments. According to the data, readers represented modus ponens deductions as certain, except when it was only a weakly necessary cause of a given effect. They represented any logically invalid inferences resulting from affirming the conditional consequent as hypothetical, except when it was the effect of strongly sufficient cause. Accordingly, readers must be processing conditional syntax as an asymmetric constraint. However, the underlying causal knowledge can be sufficient either to discredit or warrant the inferences. Thus according to the theory of natural logic, readers can draw formal deductions and be convinced of their necessity. This provides further evidence that readers can represent their inferences as hypothetical (N. Campion, 2004).     

The mental representation of causal conditional reasoning: mental models or causal models

In this paper, two experiments are reported investigating the nature of the cognitive representations underlying causal conditional reasoning performance. The predictions of causal and logical interpretations of the conditional diverge sharply when inferences involving pairs of conditionals—such as if P₁then Q and if P₂then Q—are considered. From a causal perspective, the causal direction of these conditionals is critical: are the P_icauses of Q; or symptoms caused byQ. The rich variety of inference patterns can naturally be modelled by Bayesian networks. A pair of causal conditionals where Q is an effect corresponds to a “collider” structure where the two causes (P_i) converge on a common effect. In contrast, a pair of causal conditionals where Q is a cause corresponds to a network where two effects (P_i) diverge from a common cause. Very different predictions are made by fully explicit or initial mental models interpretations. These predictions were tested in two experiments, each of which yielded data most consistent with causal model theory, rather than with mental models.     

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.     

The specificity of terms affects conditional reasoning

Conditional inferences can be phrased with unspecific terms (“If a person is on a diet, then the person loses weight. A person is on a diet. The person loses weight”) or specific terms (“If Anna is on a diet, then Anna loses weight. Anna is on a diet. Anna loses weight”). We investigate whether the specificity of terms affects people's acceptance of inferences. In Experiment 1, inferences with specific terms received higher acceptance ratings than inferences with unspecific terms. In Experiments 2 and 3, we used the same problems as in Experiment 1 but also problems with unspecific terms in the conditional and specific terms in the categorical and vice versa. When the conditional and the categorical had the same specificity, results were as in Experiment 1. When the specificity of the conditional and the categorical mismatched, acceptance ratings were lower. Our results illustrate the importance of phrasing on reasoning.     

Updating meta-analytic research findings: Bayesian approaches versus the medical model

The authors examine 3 methods of combining new studies into existing meta-analyses: (a) adding the new study or studies to the database and recalculating the meta-analysis (the medical model); (b) using the Bayesian procedure advocated by F. L. Schmidt and J. E. Hunter (1977) and F. L. Schmidt, J. E. Hunter, K. Pearlman, and G. S. Shane (1979) to update the meta-analysis; and (c) using the Bayesian methods advocated by these authors and M. T. Brannick (2001) and M. T. Brannick, S. M. Hall, and Y. Liu (2002) to estimate study-specific parameters. Method b was found to severely overweight new studies relative to the previous studies contained in the meta-analysis, and Method c was found to do the same while also requiring an assumption with a low prior probability of being correct, causing the method to violate Bayesian principles. The authors present an alternative Bayesian procedure that does not suffer from these drawbacks and yields meta-analytic results very similar to those obtained with the medical model. They recommend use of the medical model or this alternative Bayesian procedure.     

Strategies during complex conditional inferences

In certain contexts reasoners reject instances of the valid Modus Ponens and Modus Tollens inference form in conditional arguments. Byrne (1989) observed this suppression effect when a conditional premise is accompanied by a conditional containing an additional requirement. In an earlier study, Rumain, Connell, and Braine (1983) observed suppression of the invalid inferences “the denial of the antecedent” and “the affirmation of the consequent” when a conditional premise is accompanied by a conditional containing an alternative requirement. Here we present three experiments showing that the results of Byrne (1989) and Rumain et al. (1983) are influenced by the answer procedure. When reasoners have to evaluate answer alternatives that only deal with the inferences that can be made with respect to the first conditional, then suppression is observed (Experiment 1). However, when reasoners are also given answer alternatives about the second conditional (Experiment 2) no suppression is observed. Moreover, contrary to the hypothesis of Byrne (1989), at least some of the reasoners do not combine the information of the two conditionals and do not give a conclusion based on the combined premise. Instead, we hypothesise that some of the reasoners have reasoned in two stages. In the first stage, they form a putative conclusion on the basis of the first conditional and the categorical premise, and in the second stage, they amend the putative conclusion in the light of the information in the second premise. This hypothesis was confirmed in Experiment 3. Finally, the results are discussed with respect to the mental model theory and reasoning research in general.     

career beliefs and job satisfaction in adults with symptoms of attention‐deficit/hyperactivity disorder

The authors evaluated dysfunctional career beliefs and subsequent low job satisfaction in adults reporting significant symptoms of attention‐deficit/hyperactivity disorder (ADHD). Participants (N= 81) completed the Adult Attention Deficit Disorders Evaluation Scale (S. B. McCarney & P. D. Anderson, 1996), the Career Thoughts Inventory (J. P. Sampson, G.W. Peterson, J. G. Lenz, R. C. Reardon, & D. E. Saunders, 1996), and the Minnesota Satisfaction Questionnaire (D. J. Weiss, R. V. Dawis, G. W. England, & L. H. Lofquist, 1967). Regression analyses confirmed that ADHD symptoms were significantly predictive of dysfunctional career beliefs, decision‐making confusion, commitment anxiety, and external conflict.     

Conditional reasoning by mental models: chronometric and developmental evidence

The aim of this article is to verify two predictions resulting from the mental models theory of conditional reasoning. First, the denial of antecedent (DA) and modus tollens (MT) inferences should take longer to verify than modus ponens (MP) and affirmation of consequent (AC) because the former require subjects to flesh out the initial model whereas the latter do not. This prediction was confirmed in two reaction time experiments in adults. In line with Evans' proposal (Evans, J. St. B. T. (1993). The mental model theory of conditional reasoning: critical appraisal and revision. Cognition, 48, 1-20), there was a strong directionality effect: inferences from antecedent to consequent (MP and DA) took less time to verify than the inferences in the opposite direction (AC and MT). Second, the development of conditional reasoning should result from the increasing capacity to construct and coordinate more and more models. As a consequence, the pattern of conditional inference production should evolve with age from a one-model conjunctive pattern (production of MP and AC more frequent than DA and MT) to a three-model conditional production pattern (higher production rate for MP and MT than for DA and AC). This prediction was confirmed using an inference production task in children, adolescents, and adults.     

Processing of logically valid and logically invalid conditional inferences in discourse comprehension

Two competing theories of processing of conditionals (if-then) were tested. Syntactic theories posit that people only draw inferences conforming to the logically valid modus ponens (MP) schema. Mental models theories predict that people draw MP and invalid affirming-the-consequent (AC) inferences. Three experiments tested these predictions. Participants read short stories that conformed to either the MP or AC form but without conclusions, and they completed either priming or recognition tasks. Results indicate that both MP and AC inferences occur during discourse processing: MP and AC premise forms prime their respective conclusions, participants erroneously judged that they had read the conclusions to MP and AC arguments, and AC inferences did not stem from a biconditional interpretation of conditionals. Findings support mental models theories.