Illusory inferences from a disjunction of conditionals: a new mental models account

(Johnson-Laird, P.N., & Savary, F. (1999, Illusory inferences: a novel class of erroneous deductions. Cognition, 71, 191-229.) have recently presented a mental models account, based on the so-called principle of truth, for the occurrence of inferences that are compelling but invalid. This article presents an alternative account of the illusory inferences resulting from a disjunction of conditionals. In accordance with our modified theory of mental models of the conditional, we show that the way individuals represent conditionals leads them to misinterpret the locus of the disjunction and prevents them from drawing conclusions from a false conditional, thus accounting for the compelling character of the illusory inference.     

Initial models in conditionals: evidence from priming

We examined the comprehension of different types of conditionals. We measured the reading time of sentences primed by different types of conditionals (Experiments 1 and 2). We found that the participants read not-p and not-q faster when it was primed by the conditional form p if q and they were slower to read p and q when it was primed by the conditional form p only if q. This effect disappeared in the second experiment, where the order of the elements was reversed (q and p and not-q and not-p). These results suggest that the conditional form p if q elicits an initial representation "from p to q" with two possibilities, while the conditional form p only if q elicits a reverse representation with only one possibility. The third experiment showed that there were effects of the order only for the conditional if p then q, which confirms the reverse representation hypothesis. We discuss the implications of these results for different theories of conditional comprehension.     

Conversation and conditionals

I outline and motivate a way of implementing a closest world theory of indicatives, appealing to Stalnaker’s framework of open conversational possibilities. Stalnakerian conversational dynamics helps us resolve two outstanding puzzles for a such a theory of indicative conditionals. The first puzzle—concerning so-called ‘reverse Sobel sequences’—can be resolved by conversation dynamics in a theory-neutral way: the explanation works as much for Lewisian counterfactuals as for the account of indicatives developed here. Resolving the second puzzle, by contrast, relies on the interplay between the particular theory of indicative conditionals developed here and Stalnakerian dynamics. The upshot is an attractive resolution of the so-called “Gibbard phenomenon” for indicative conditionals.     

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.     

Subjective causal networks and indeterminate suppositional credences

This paper has two main parts. In the first part, we motivate a kind of indeterminate, suppositional credences by discussing the prospect for a subjective interpretation of a causal Bayesian network (CBN), an important tool for causal reasoning in artificial intelligence. A CBN consists of a causal graph and a collection of interventional probabilities. The subjective interpretation in question would take the causal graph in a CBN to represent the causal structure that is believed by an agent, and interventional probabilities in a CBN to represent suppositional credences. We review a difficulty noted in the literature with such an interpretation, and suggest that a natural way to address the challenge is to go for a generalization of CBN that allows indeterminate credences. In the second part, we develop a decision-theoretic foundation for such indeterminate suppositional credences, by generalizing a theory of coherent choice functions to accommodate some form of act-state dependence. The upshot is a decision-theoretic framework that is not only rich enough to, so to speak, ground the probabilities in a subjectively interpreted causal network, but also interesting in its own right, in that it accommodates both act-state dependence and imprecise probabilities.

     

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.     

Mental models and the fan effect.

Explanations of data from fan effect experiments have been based on propositional network models. This article presents findings not readily predicted by such models. In particular, in three experiments we found that, during a speeded-recognition test, subjects retrieved facts about several objects associated with a single location faster than facts about several locations associated with a single object. Indeed, there was no fan effect in the former case despite the fact that there were an equivalent number of associations among concepts in both conditions. We suggest that such data are consistent with a mental model representational account.     

Bivalence and subjunctive conditionals

Writers such as Stalnaker and Dummett have argued that specific features of subjunctive conditional statements undermine the principle of bivalence. This, paper is concerned with rebutting such claims. 1. It is shown how subjective conditionals pose a prima facie threat to bivalence, and how this threat can be dissolved by a distinction between the results of negating a subjective conditional and of negating its consequent. To make this distinction is to side with Lewis against Stalnaker in a dispute about possible worlds semantics for such conditionals, and reasons are given for doing so. 2. These arguments are extended to answer Dummett's claim that behaviourist and phenomenalist analyses in terms of subjunctive conditions violate bivalence. This answer is shown to be compatible with the principle that hypothetical statements are true only in virtue of categorical facts.     

Causal conditionals and counterfactuals

Causal counterfactuals e.g., 'if the ignition key had been turned then the car would have started' and causal conditionals e.g., 'if the ignition key was turned then the car started' are understood by thinking about multiple possibilities of different sorts, as shown in six experiments using converging evidence from three different types of measures. Experiments 1a and 1b showed that conditionals that comprise enabling causes, e.g., 'if the ignition key was turned then the car started' primed people to read quickly conjunctions referring to the possibility of the enabler occurring without the outcome, e.g., 'the ignition key was turned and the car did not start'. Experiments 2a and 2b showed that people paraphrased causal conditionals by using causal or temporal connectives (because, when), whereas they paraphrased causal counterfactuals by using subjunctive constructions (had…would have). Experiments 3a and 3b showed that people made different inferences from counterfactuals presented with enabling conditions compared to none. The implications of the results for alternative theories of conditionals are discussed.     

Mental models and omissive causation

Some causal relations refer to causation by commission (e.g., “A gunshot causes death”), and others refer to causation by omission (e.g., “Not breathing causes death”). We describe a theory of the representation of omissive causation based on the assumption that people mentally simulate sets of possibilities—mental models—that represent causes, enabling conditions, and preventions (Goldvarg & Johnson-Laird, 2001). The theory holds that omissive causes, enabling conditions, and preventions each refer to distinct sets of possibilities. For any such causal relation, reasoners typically simulate one initial possibility, but they are able to consider alternative possibilities through deliberation. These alternative possibilities allow them to deliberate over finer-grained distinctions when reasoning about causes and effects. Hence, reasoners should be able to distinguish between omissive causes and omissive enabling conditions. Four experiments corroborated the predictions of the theory. We describe them and contrast the results with the predictions of alternative accounts of causal representation and inference.     

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.     

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.     

Conversational scorekeeping and conditionals

David Lewis has recently developed the notion of conversational score-keeping as a way of explaining the acceptability of utterances in various contexts and the manner in which this acceptability changes in a rule-governed manner. I will expand Lewis's discussion by showing how the acceptibility of conditionals is linked to conversational score. In particular, I will argue that at least one controversial issue concerning the logic of conditionals, the interpretation and use of conditionals with disjunctive antecedents, may be resolved by applying Lewis's notion of an accommodation rule for conversational scorekeeping.1|This paper has benefitted from the helpful criticisms of two anonymous referees.     

Heuristic and analytic processes in mental models for conditionals: An integrative developmental theory

This article presents a developmental dual-process theory of the understanding of conditionals that integrates Evans’ heuristic–analytic theory within the revised mental model theory of conditional proposed by Barrouillet, Gauffroy, and Lecas (2008). According to this theory, the interpretation of a conditional sentence is driven by unconscious and implicit heuristic processes that provide individuals with an initial representation that captures its meaning by representing the cases that make it true. This initial model can be enriched with additional models (a process named fleshing out within the mental model theory) through the intervention of conscious and demanding analytic processes. Being optional, these processes construct representations of cases that are only compatible with the conditional, leaving its truth-value indeterminate when they occur. Because heuristic processes are relatively immune to developmental changes, while analytic processes strongly develop with age, the initial model remains stable through development whereas the number of additional models that can be constructed increases steadily. Thus, the dual-process mental model theory predicts in which cases conditionals will be deemed true, indeterminate, or false and how these cases evolve with age. These predictions were verified in children, adolescents and adults who were asked to evaluate the truth value and the probability of several types of conditionals. The results reveal a variety of developmental trajectories in the way different conditionals are interpreted, which can all be accounted for by our revised mental model theory.