Is inferential reasoning just probabilistic reasoning in disguise?

Oaksford, Chater, and Larkin (2000) have suggested that people actually use everyday probabilistic reasoning when making deductive inferences. In two studies, we explicitly compared probabilistic and deductive reasoning with identical if-then conditional premises with concrete content. In the first, adults were given causal premises with one strongly associated antecedent and were asked to make standard deductive inferences or to judge the probabilities of conclusions. In the second, reasoners were given scenarios presenting a causal relation with zero to three potential alternative antecedents. The participants responded to each set of problems under both deductive and probabilistic instructions. The results show that deductive and probabilistic inferences are not isomorphic. Probabilistic inferences can model deductive responses only using a limited, very high threshold model, which is equivalent to a simple retrieval model. These results provide a clearer understanding of the relations between probabilistic and deductive inferences and the limitations of trying to consider these two forms of inference as having a single underlying process.     

Multiple layers of information processing in deductive reasoning: combining dual strategy and dual-source approaches to reasoning

The idea that inferential performance cannot be analyzed within a single model has been suggested within two theoretical contexts. The dual strategy model suggests that people reason using different approaches to processing statistical information. The dual-source model suggests that people reason probabilistically using both statistical information and some intuition about logical form. Each model suggests that people have different approaches to processing information while making inferences. The following studies examined whether these different forms of information processing were equally present during reasoning. Participants were given a series of problems designed to distinguish counterexample from statistical reasoners. They were then given a series of MP or AC inferences for which identical statistical information was provided. Results show that MP inferences were considered to be deductively valid more often than equivalent AC inferences. The effect of logical form was independent of reasoning strategy, and of relatively equivalent size for both counterexample and statistical reasoners. The second study examined explicitly probabilistic inferences, and showed smaller effects of logical form and of reasoning strategy, although with a complex set of interactions. These results show that understanding the way that people use information when making inferences requires a multidimensional approach.     

Interactions between inferential strategies and belief bias

The dual strategy model of reasoning proposed by Verschueren, Schaeken, and d’Ydewalle (Thinking & Reasoning, 11(3), 239–278, 2005a; Memory & Cognition, 33(1), 107–119, 2005b) suggests that people can use either a statistical or a counterexample-based strategy to make deductive inferences. Subsequent studies have supported this distinction and investigated some properties of the two strategies. In the following, we examine the further hypothesis that reasoners using statistical strategies should be more vulnerable to the effects of conclusion belief. In each of three studies, participants were given abstract problems used to determine strategy use and three different forms of syllogism with believable and unbelievable conclusions. Responses, response times, and feeling of rightness (FOR) measures were taken. The results show that participants using a statistical strategy were more prone to the effects of conclusion belief across all three forms of reasoning. In addition, statistical reasoners took less time to make inferences than did counterexample reasoners. Patterns of variation in response times and FOR ratings between believable and unbelievable conclusions were very similar for both strategies, indicating that both statistical and counterexample reasoners were aware of conflict between conclusion belief and premise-based reasoning.     

Everyday conditional reasoning: A working memory—dependent tradeoff between counterexample and likelihood use

Considerable evidence has revealed that working memory capacity is an important determinant of conditional reasoning performance. There are two accounts describing the conditional inference process, the probabilistic and the mental models accounts. According to the mental models account, reasoners retrieve and integrate counterexample information to attain a conclusion. According to the probabilistic account, reasoners base their judgments on probabilistic information. It can be assumed that reasoning according to the mental models process would require more working memory resources than would solving the inference on the basis of probabilistic information. By means of a verbal report study, we showed that participants with low working memory capacity more often use probabilistic information, whereas participants with higher working memory capacity are more likely to use counterexample information. Working memory capacity thus not only relates to reasoning performance, it also determines which process reasoners will engage in.     

Mental models and deduction

According to the mental-model theory of deductive reasoning, reasoners use the meanings of assertions together with general knowledge to construct mental models of the possibilities compatible with the premises. Each model represents what is true in a possibility. A conclusion is held to be valid if it holds in all the models of the premises. Recent evidence described here shows that the fewer models an inference calls for, the easier the inference is. Errors arise because reasoners fail to consider all possible models, and because models do not normally represent what is false, even though reasoners can construct counterexamples to refute invalid conclusions.     

Illusory inferences: a novel class of erroneous deductions.

The mental model theory postulates that reasoners build models of the situations described in premises, and that these models normally make explicit only what is true. The theory has an unexpected consequence: it predicts the occurrence of inferences that are compelling but invalid. They should arise from reasoners failing to take into account what is false. Three experiments corroborated the systematic occurrence of these illusory inferences, and eliminated a number of alternative explanations for them. Their results illuminate the controversy among various current theories of reasoning.     

Naive probability: a mental model theory of extensional reasoning

This article outlines a theory of naive probability. According to the theory, individuals who are unfamiliar with the probability calculus can infer the probabilities of events in an extensional way: They construct mental models of what is true in the various possibilities. Each model represents an equiprobable alternative unless individuals have beliefs to the contrary, in which case some models will have higher probabilities than others. The probability of an event depends on the proportion of models in which it occurs. The theory predicts several phenomena of reasoning about absolute probabilities, including typical biases. It correctly predicts certain cognitive illusions in inferences about relative probabilities. It accommodates reasoning based on numerical premises, and it explains how naive reasoners can infer posterior probabilities without relying on Bayes's theorem. Finally, it dispels some common misconceptions of probabilistic reasoning.     

Transitive and pseudo-transitive inferences

Given that A is longer than B, and that B is longer than C, even 5-year-old children can infer that A is longer than C. Theories of reasoning based on formal rules of inference invoke simple axioms ("meaning postulates") to capture such transitive inferences. An alternative theory proposes instead that reasoners construct mental models of the situation described by the premises in order to draw such inferences. An unexpected consequence of the model theory is that if adult reasoners construct simple models of typical situations, then they should infer transitive relations where, in certain cases, none exists. We report four studies corroborating the occurrence of these "pseudo-transitive" fallacies. Experiment 1 established that individuals' diagrams of certain non-transitive relations yield transitive conclusions. Experiment 2 showed that these premises also give rise to fallacious transitive inferences. Experiment 3 established that when the context suggested alternatives to the simple models, the participants made fewer errors. Experiment 4 showed that tense is an important aspect of meaning which affects whether individuals draw transitive conclusions. We discuss the implications of these results for various theories of reasoning.     

Disjunctive illusory inferences and how to eliminate them

The mental model theory of reasoning postulates that individuals construct mental models of the possibilities in which the premises of an inference hold and that these models represent what is true but not what is false. An unexpected consequence of this assumption is that certain premises should yield systematically invalid inferences. This prediction is unique among current theories of reasoning, because no alternative theory, whether based on formal rules of inference or on probabilistic considerations, predicts these illusory inferences. We report three studies of novel illusory inferences that depend on embedded disjunctions—for example, premises of this sort: A or else (B or else C). The theory distinguishes between those embedded disjunctions that should yield illusions and those that should not. In Experiment 1, we corroborated this distinction. In Experiment 2, we extended the illusory inferences to a more stringently controlled set of problems. In Experiment 3, we established a novel method for reducing illusions by calling for participants to make auxiliary inferences.     

Illusions in modal reasoning

According to the mental model theory, models represent what is true, but not what is false. One unexpected consequence is that certain inferences should have compelling, but invalid, conclusions. Three experiments corroborated the occurrence of such illusions in reasoning about possibilities. When problems had the heading "Only one of the premises is true," the participants considered the truth of each premise in turn, but neglected the fact that when one premise is true, the others are false. When two-premise problems had the heading "One of the premises is true and one is false," the participants still neglected the falsity of one of the premises. As predicted, however, the illusions were reduced when reasoners were told to check their conclusions against the constraint that only one of the premises was true. We discuss alternative explanations for illusory inferences and their implications for current theories of reasoning.     

Mental models and probabilistic thinking

This paper outlines the theory of reasoning based on mental models, and then shows how this theory might be extended to deal with probabilistic thinking. The same explanatory framework accommodates deduction and induction: there are both deductive and inductive inferences that yield probabilistic conclusions. The framework yields a theoretical conception of strength of inference, that is, a theory of what the strength of an inference is objectively: it equals the proportion of possible states of affairs consistent with the premises in which the conclusion is true, that is, the probability that the conclusion is true given that the premises are true. Since there are infinitely many possible states of affairs consistent with any set of premises, the paper then characterizes how individuals estimate the strength of an argument. They construct mental models, which each correspond to an infinite set of possibilities (or, in some cases, a finite set of infinite sets of possibilities). The construction of models is guided by knowledge and beliefs, including lay conceptions of such matters as the “law of large numbers”. The paper illustrates how this theory can account for phenomena of probabilistic reasoning.     

Reasoning and dyslexia: A spatial strategy may impede reasoning with visually rich information

Bacon, Handley and Newstead (2003, 2004) , have presented evidence for individual differences in reasoning strategies, with most people seeming to represent and manipulate problem information using either a verbal or a spatial strategy. There is also evidence that individuals with dyslexia are inclined to conceptualise information in a visuo‐spatial, rather than a verbal, way (e.g. von Károlyi et al., 2003 ). If so, we might expect a higher proportion of individuals with dyslexia to be spatial reasoners, compared with individuals who do not have dyslexia. The study reported here directly compared strategies reported by these two groups of participants on a syllogistic reasoning task. Moreover, problem content was manipulated so that reasoning across concrete and abstract materials could be compared. The findings suggest that whilst most individuals without dyslexia use a verbal strategy, reasoners with dyslexia do tend to adopt a spatial approach, though their performance is impaired with visually concrete materials. However, when reasoning with more abstract content, they perform comparably with non‐dyslexic controls. The paper discusses these results in the light of recent research which has suggested that visual images may impede reasoning, and considers how individuals with dyslexia may differ from other reasoners.     

Different developmental patterns of simple deductive and probabilistic inferential reasoning

In three studies, we examined simple counterexample-based and probabilistic reasoning in children 6, 7, and 9 years of age. In the first study, participants were asked to make conditional (if-then) inferences under both categorical (certain or uncertain) and probabilistic instructions. Results showed that 6-year-olds respond to both forms of inference in similar ways, but whereas probabilistic conditional inferences showed little development over this period, categorical inferences clearly improved between 6 and 7 years of age. An analysis of the children's justifications indicated that performance under categorical instructions was strongly related to counterexample generation at all ages, whereas this was true only for the younger children for inferences under probabilistic instructions. These findings were replicated in a second study, using problems that referred to concrete stimuli with varying probabilities of inference. A third study tested the hypothesis that children confused probability judgments with judgments of confidence and demonstrated a clear dissociation between these two constructs. Overall, these results show that children are capable of accurate conditional inferences under probabilistic instructions at a very early age and that the differentiation between categorical and probabilistic conditional reasoning is clear by at least 9 years ofage. These results are globally consistent with dual-process theories but suggest some difficulties for the way that the analytic-heuristic distinction underlying these theories has been conceptualized.     

Suppression of valid inferences and knowledge structures: the curious effect of producing alternative antecedents on reasoning with causal conditionals

These studies looked at the difficulty that reasoners have in accepting conditional ("If P then Q") major premises that are not necessarily true empirically, as a basis for deductive reasoning. Preliminary results have shown that when reasoners are asked to produce possible alternate antecedents to the major premise ("If A then Q"), they paradoxically tend to deny the modus ponens (MP) inference ("If P is true, then Q is true"). Three studies further explored these results. The first study gave university students paper-and-pencil tests in which instructions to "suppose that the major premise is true" was followed by a request to determine the next number in a sequence, to retrieve information unrelated to the premises, or to retrieve a possible case of "If A then Q." Relative to a control group, reasoners asked to produce an alternative antecedent showed a significant tendency to deny the MP inference, whereas no such tendency was observed for the two other tasks used. A second study compared performance on a condition in which reasoners were asked to produce an alternative antecedent with that when they were given an explicit alternative. Premises used in this study were such that the latter alternative antecedent was also spontaneously produced by over 70% of reasoners. Results showed that the tendency to refuse the MP premise could not be accounted for by the specific nature of the alternative produced. A third study found that the tendency to refuse the MP inference after producing an alternative antecedent was affected by the number of "disabling conditions" (i.e., conditions that allow "P to be true" and "Q to be false") available for the major premise. These results are interpreted as being consistent with a model that supposes that logical reasoning requires selective inhibition of real-world knowledge.     

Syllogistic reasoning time: Disconfirmation disconfirmed

University of Saskatchewan, Saskatoon, Saskatchewan, Canada Models of deductive reasoning typically assume that reasoners dedicate more logical analysis to unbelievable conclusions than to believable ones (e.g., Evans, Newstead, Allen, & Pollard, 1994; Newstead, Pollard, Evans, & Allen, 1992). When the conclusion is believable, reasoners are assumed to accept it without much further thought, but when it is unbelievable, they are assumed to analyze the conclusion, presumably in an attempt to disconfirm it. This disconfirmation hypothesis leads to two predictions, which were tested in the present experiment: Reasoners should take longer to reason about problems leading to unbelievable conclusions, and reasoners should consider more models or representations of premise information for unbelievable conclusions than for believable ones. Neither prediction was supported by our data. Indeed, we observed that reasoners took significantly longer to reason about believable conclusions than about unbelievable ones and generated the same number of representations regardless of the believability of the premises. We propose a model, based on a modified version of verbal reasoning theory (Polk & Newell, 1995), that does not depend on the disconfirmation assumption.     

Integrating physical constraints in statistical inference by 11-month-old infants

Much research on cognitive development focuses either on early-emerging domain-specific knowledge or domain-general learning mechanisms. However, little research examines how these sources of knowledge interact. Previous research suggests that young infants can make inferences from samples to populations (Xu & Garcia, 2008) and 11- to 12.5-month-old infants can integrate psychological and physical knowledge in probabilistic reasoning (Teglas, Girotto, Gonzalez, & Bonatti, 2007; Xu & Denison, 2009). Here, we ask whether infants can integrate a physical constraint of immobility into a statistical inference mechanism. Results from three experiments suggest that, first, infants were able to use domain-specific knowledge to override statistical information, reasoning that sometimes a physical constraint is more informative than probabilistic information. Second, we provide the first evidence that infants are capable of applying domain-specific knowledge in probabilistic reasoning by using a physical constraint to exclude one set of objects while computing probabilities over the remaining sets.     

A Process Model of Causal Reasoning

How do we make causal judgments? Many studies have demonstrated that people are capable causal reasoners, achieving success on tasks from reasoning to categorization to interventions. However, less is known about the mental processes used to achieve such sophisticated judgments. We propose a new process model—the mutation sampler—that models causal judgments as based on a sample of possible states of the causal system generated using the Metropolis–Hastings sampling algorithm. Across a diverse array of tasks and conditions encompassing over 1,700 participants, we found that our model provided a consistently closer fit to participant judgments than standard causal graphical models. In particular, we found that the biases introduced by mutation sampling accounted for people's consistent, predictable errors that the normative model by definition could not. Moreover, using a novel experimental methodology, we found that those biases appeared in the samples that participants explicitly judged to be representative of a causal system. We conclude by advocating sampling methods as plausible process-level accounts of the computations specified by the causal graphical model framework and highlight opportunities for future research to identify not just what reasoners compute when drawing causal inferences, but also how they compute it.     

Strategies in sentential reasoning

Four experiments examined the strategies that individuals develop in sentential reasoning. They led to the discovery of five different strategies. According to the theory proposed in the paper, each of the strategies depends on component tactics, which all normal adults possess, and which are based on mental models. Reasoners vary their use of tactics in ways that have no deterministic account. This variation leads different individuals to assemble different strategies, which include the construction of incremental diagrams corresponding to mental models, and the pursuit of the consequences of a single model step by step. Moreover, the difficulty of a problem (i.e., the number of mental models required by the premises) predisposes reasoners towards certain strategies. Likewise, the sentential connectives in the premises also bias reasoners towards certain strategies, e.g., conditional premises tend to elicit reasoning step by step whereas disjunctive premises tend to elicit incremental diagrams.