Illusions in quantified reasoning: How to make the impossible seem possible, and vice versa

The mental model theory postulates that reasoners build models of the situations described in premises, and that these models normally represent only what is true. The theory has an unexpected consequence. It predicts the existence of illusions in inferences. Certain inferences should have compelling but erroneous conclusions. Two experiments corroborated the occurrence of such illusions in inferences about what is possible from disjunctions of quantified assertions, such as, "at least some of the plastic beads are not red." Experiment 1 showed that participants erroneously inferred that impossible situations were possible, and that possible situations were impossible, but that they performed well with control problems based on the same premises. Experiment 2 corroborated these findings in inferences from assertions based on dyadic relations, such as, "all the boys played with the girls."     

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.     

Diversity-Based Reasoning in Children

One of the hallmarks of inductive reasoning by adults is the diversity effect, namely that people draw stronger inferences from a diverse set of evidence than from a more homogenous set of evidence. However, past developmental work has not found consistent diversity effects with children age 9 and younger. We report robust sensitivity to diversity in children as young as 5, using everyday stimuli such as pictures of objects with people. Experiment 1 showed the basic diversity effect in 5- to 9-year-olds. Experiment 2 showed that, like adults, children restrict their use of diversity information when making inferences about remote categories. Experiment 3 used other stimulus sets to overcome an alternate explanation in terms of sample size rather than diversity effects. Finally, Experiment 4 showed that children more readily draw on diversity when reasoning about objects and their relations with people than when reasoning about objects' internal, hidden properties, thus partially explaining the negative findings of previous work. Relations to cross-cultural work and models of induction are discussed.     

The compatibility heuristic in non-categorical hypothetical reasoning: Inferences between conditionals and disjunctions

A new theory explains how people make hypothetical inferences from a premise consistent with several alternatives to a conclusion consistent with several alternatives. The key proposal is that people rely on a heuristic that identifies compatible possibilities. It is tested in 7 experiments that examine inferences between conditionals and disjunctions. Participants accepted inferences between conditionals and inclusive disjunctions when a compatible possibility was immediately available, in their binary judgments that a conclusion followed or not (Experiment 1a) and ternary judgments that included it was not possible to know (Experiment 1b). The compatibility effect was amplified when compatible possibilities were more readily available, e.g., for ‘A only if B’ conditionals (Experiment 2). It was eliminated when compatible possibilities were not available, e.g., for ‘if and only if A B’ bi-conditionals and exclusive disjunctions (Experiment 3). The compatibility heuristic occurs even for inferences based on implicit negation e.g., ‘A or B, therefore if C D’ (Experiment 4), and between universals ‘All A’s are B’s’ and disjunctions (Experiment 5a) and universals and conditionals (Experiment 5b). The implications of the results for alternative theories of the cognitive processes underlying hypothetical deductions are discussed.     

On the generation and evaluation of inferences from single premises

A theory of how individuals construct mental models to draw inferences from single premises was tested in three experiments. Experiment 1 confirmed a counterintuitive prediction that it is easier to generate inferences between conditionals and disjunctions than it is to evaluate them. Experiment 2 replicated this finding, but an advantage found in the first experiment for conditional-to-disjunction over disjunction-to-conditional inferences was removed with different sentence contents. Experiment 3 showed that disjunction-to-conditional inferences were facilitated when premises expressed familiar indicative relations, whereas conditional-to-disjunction inferences were facilitated when premises expressed causal relations. The results indicate that small changes in task format can have large effects on the strategies that people use to represent and reason about different sentential connectives. We discuss the potential for theories other than mental models to account for these results. We argue that, despite the important role played by single-premise inferences in paraphrasing logical forms during inference, mental logic theories cannot account for the results reported here.     

现实预期和意愿预期在文本阅读中的激活强度

文本阅读中,读者往往对事件的后续发展进行预期推理。预期推理有两种倾向,要么是倾向于根据客观现实条件进行的现实预期,要么是倾向于根据主观个人意愿进行的意愿预期。两个实验探讨了文本阅读中读者产生的现实预期和意愿预期的激活强度。结果发现,现实预期和意愿预期都可以在阅读中即时产生,意愿预期强于现实预期;把读者分为"情感导向型"和"非情感导向型",发现"情感导向型"的读者,阅读过程中产生的意愿预期强于现实预期,而"非情感导向型"的读者则不是。     

Young children infer causal strength from probabilities and interventions

We examine the interaction of two cues that children use to make judgments about cause-effect relations: probabilities and interventions. Children were shown a "detector" that lit up and played music when a block was placed on its surface. We varied the probabilistic effectiveness of the block, as well as whether the experimenter or the child was performing the interventions. In Experiment 1, we found that children can use probabilistic evidence to make inferences about causal strength. However, when the results of their own interventions are in conflict with the overall frequencies, preschoolers favor the results of their own interventions. In Experiment 2, children used probabilistic evidence to infer a hidden causal mechanism. Though they again gave preference to their own interventions, they did not do so when their interventions were explicitly confounded by an alternative cause.     

Advice conditionals about tips and warnings: interpretations and inferences

Two experiments examine how people interpret and reason about advice conditionals, such as tips, for example, “if you study more your grades will improve”, and warnings, for example, “if you stop exercising you will gain weight”. Experiment 1 showed that when participants reason about whether a tip or warning could be true in different situations, their judgments correspond to a biconditional or conditional interpretation on about half of all trials, but to an enabling or tautology interpretation on many others. Experiment 2 showed that participants make few modus ponens and tollens inferences from tips and warnings, and more modus ponens inferences from tips than warnings. The implications for alternative theories are discussed.     

Deductive reasoning with factual, possible, and counterfactual conditionals

We compared reasoners' inferences from conditionals based on possibilities in the present or the past (e.g., "If Linda had been in Dublin then Cathy would have been in Galway") with their inferences based on facts in the present or the past (e.g., "If Linda was in Dublin then Cathy was in Galway"). We propose that people construct a richer representation of conditionals that deal with possibilities rather than facts: Their models make explicit not only the suppositional case, in which Linda is in Dublin and Cathy is in Galway, but also the presupposed case, in which Linda is not in Dublin and Cathy is not in Galway. We report the results of four experiments that corroborate this model theory. The experiments show that reasoners make more inferences from conditionals based on possibilities rather than on facts when the inferences depend on the presupposed case. The results also show that reasoners generate different situations to verify and falsify conditionals based on possibilities and facts.     

Infants' intermodal perception of two levels of temporal structure in natural events

Infants' intermodal perception of two levels of temporal structure uniting the visual and acoustic stimulation from natural, complex events was investigated in four experiments. Films depicting a single object (single, large marble) and a compound object (group of smaller marbles) colliding against a surface in an erratic pattern were presented to infants between 3 and months of age using an intermodal preference and search method. These stimulus events portrayed two levels of invariant temporal structure: (a) temporal synchrony united the sights and sounds of object impact, and (b) temporal microstructure, the internal temporal structure of each impact sound and motion, specified the composition of the object (single vs. compound). Experiment 1 demonstrated that by 6 months infants detected a relation between the audible and visible stimulation from these events when both levels of invariant temporal structure guided their intermodal exploration. Experiment 2 revealed that by 6 months infants detected the bimodal temporal microstructure specifying object composition. They looked predominantly to the film whose natural soundtrack was played even though the motions of objects in both films were synchronized with the soundtrack. Experiment 3 assessed infants' sensitivity to temporal synchrony relations. Two films depicting objects of the same composition were presented while the motions of only one of them was synchronized with the appropriate soundtrack. Both 6-month-olds showed evidence of detecting temporal synchrony relations under some conditions. Experiment 4 examined how temporal synchrony and temporal microstructure interact in directing intermodal exploration. The natural soundtrack to one of the objects was played out-of-synchrony with the motions of both. In contrast with the results of Experiment 2, infants at 6 months showed no evidence of detecting a relationship between the film and its appropriate soundtrack. This suggests that the temporal asynchrony disrupted their detection of the temporal microstructure specifying object composition. Results of these studies support on invariant-detection view of the development of intermodal perception.     

Probabilistically Valid Inference of Covariation From a Single x,y Observation When Univariate Characteristics Are Known

Participants were asked to draw inferences about correlation from single x,y observations. In Experiment 1 statistically sophisticated participants were given the univariate characteristics of distributions of x and y and asked to infer whether a single x, y observation came from a correlated or an uncorrelated population. In Experiment 2, students with a variety of statistical backgrounds assigned posterior probabilities to five possible populations based on single x, y observations, again given knowledge of the univariate statistics. In Experiment 3, statistically naïve participants were given a problem analogous to that given in Experiment 1, framed verbally. Experiment 4 replicated Experiment 3 but added an "impossible to determine" response option. Models that rely on computing sample correlations make no predictions about these investigations. From a Bayesian perspective, participants' inferences in all four experiments tended to make probabilistically valid inferences as long as the single datum was directional. The results are discussed in light of the Brunswikian notion of vicarious functioning.     

Adjusting Social Inferences in Familiar and Unfamiliar Domains: The Generality of Response to Situational Pragmatics

Two experiments investigated the influence of situational pragmatics on the selective use of specific instances and generalized knowledge structures to make social inferences. In Experiment 1, social inferences were made in an unfamiliar domain similar in structure to a typical situation of social greetings and address, but devoid of useful cues to social schemas. Participants were told that either one or another of the features of the situation was more pragmatically important for deriving inferences about appropriate social behaviour; consistent with predictions from a computational model of analogical mapping (ACME), they made reliable inferences based on analogies to specific instances, with the situational importance of relations guiding the selection of the optimal analogue. In Experiment 2, social inferences were examined in the more familiar domain of predicting social behavior between low and high status persons and between members of an ingroup and an outgroup in Japan. The availability of specific examples was varied, as was the perceived importance of status and group membership. The situation was isomorphic to that in the first experiment, except for the availability of generalized knowledge structures to guide inferences. Participants made relatively veridical inferences that were sensitive to variations in the pragmatic importance of dimensions. Provision of specific analogues had little impact on inferences, suggesting that participants were relying instead on more general and cross-culturally applicable knowledge about adjusting social relations according to situational pragmatics.     

Infants learn about objects from statistics and people

In laboratory experiments, infants are sensitive to patterns of visual features that co-occur (e.g., Fiser & Aslin, 2002). Once infants learn the statistical regularities, however, what do they do with that knowledge? Moreover, which patterns do infants learn in the cluttered world outside of the laboratory? Across 4 experiments, we show that 9-month-olds use this sensitivity to make inferences about object properties. In Experiment 1, 9-month-old infants expected co-occurring visual features to remain fused (i.e., infants looked longer when co-occurring features split apart than when they stayed together). Forming such expectations can help identify integral object parts for object individuation, recognition, and categorization. In Experiment 2, we increased the task difficulty by presenting the test stimuli simultaneously with a different spatial layout from the familiarization trials to provide a more ecologically valid condition. Infants did not make similar inferences in this more distracting test condition. However, Experiment 3 showed that a social cue did allow inferences in this more difficult test condition, and Experiment 4 showed that social cues helped infants choose patterns among distractor patterns during learning as well as during test. These findings suggest that infants can use feature co-occurrence to learn about objects and that social cues shape such foundational learning in distraction-filled environments.     

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.     

Temporal order judgments with amnesia

The hypothesis that patients with anterograde amnesia fail to recognize the temporal relations among recognizable past events was evaluated. Experiment 1 showed that order recognition for words in a list was at chance levels even though event recognition was significantly above chance. The same pattern did not obtain for intact adults. Experiment 2 examined event and order recognition for news events. Amnesics' event recognition for news events was quite accurate and comparable to controls, yet their order recognition was significantly below that of controls, again at chance levels. It is argued that the failure to encode the temporal relations among events has broad consequences for memory performance in general.     

Inferences from disclosures about the truth and falsity of expert testimony

Participants acting as mock jurors made inferences about whether a person was a suspect in a murder based on an expert's testimony about the presence of objects at the crime scene and the disclosure that the testimony was true or false. Experiment 1 showed that participants made more correct inferences, and made inferences more quickly, when the truth or falsity of the expert's testimony was disclosed immediately after the testimony rather than when the disclosure was delayed. Experiment 2 showed no advantage for prior disclosure over immediate disclosure. Experiment 3 showed that the pattern of inferences when there was no disclosure mirrored the pattern when it was disclosed that the expert's testimony was true rather than false. Participants made more correct inferences from true conjunctions than disjunctions, and from false disjunctions than conjunctions. We discuss the implications for theories of the mental representations and cognitive processes that underlie human reasoning.     

Mental models and temporal reasoning

We report five experiments investigating reasoning based on temporal relations, such as: “John takes a shower before he drinks coffee”. How individuals make temporal inferences has not been studied hitherto, but we conjectured that they construct mental models of events, and we developed a computer program that reasons in this way.As the program shows, a problem of the form:

1. a before b

2. b before c

3. d while b

4. e while c

What is the relation between d and e?

where a, b, c, etc. refer to everyday events, calls for just one model, whereas a problem in which the second premise is modified to c before b calls for multiple models because a may occur before c, after c, or at the same time as c.

Experiments 1–3 showed that problems requiring one mental model elicited more correct responses than problems requiring multiple models, which in turn elicited more correct answers than multiple model problems with no valid answers. Experiment 4 contrasted the predictions of the model theory with those based on formal rules of inference; its results corroborated the model theory. Experiment 5 confirmed that a premise leading to multiple models took longer to read than the corresponding premise in one-model problems, and that latency to respond correctly was greater for multiple-model problems than for one-model problems. We conclude that the experiments corroborate the mental model theory.     

Assessing the empirical validity of the "take-the-best" heuristic as a model of human probabilistic inference

The boundedly rational 'Take-The-Best" heuristic (TTB) was proposed by G. Gigerenzer, U. Hoffrage, and H. Kleinb?lting (1991) as a model of fast and frugal probabilistic inferences. Although the simple lexicographic rule proved to be successful in computer simulations, direct empirical demonstrations of its adequacy as a psychological model are lacking because of several methodical problems. In 4 experiments with a total of 210 participants, this question was addressed. Whereas Experiment 1 showed that TTB is not valid as a universal hypothesis about probabilistic inferences, up to 28% of participants in Experiment 2 and 53% of participants in Experiment 3 were classified as TTB users. Experiment 4 revealed that investment costs for information seem to be a relevant factor leading participants to switch to a noncompensatory TTB strategy. The observed individual differences in strategy use imply the recommendation of an idiographic approach to decision-making research.