Disjunctive illusory inferences and how to eliminate them

SANGEET KHEMLANI 0 P. N. JOHNSON-LAIRD 0 0 Princeton University , Princeton, New Jersey 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 disjunctionsfor 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. - Even if you know nothing about memory or servers, you can grasp that the conclusion follows from the premises; that is, the inference is valid: If its premises are true, its conclusion must be true too (Jeffrey, 1981). The ability to make valid deductions is a cornerstone of rationality, yet no consensus exists about the logical competence of naive individuals (i.e., those with no training in logic). Some theorists have even argued that deductive ability is not part of their thinking. In the case of inferences based on quantifiers such as all and some, individuals are instead supposed to rely on the atmosphere of the premises: They select conclusions that match the quantifiers in the premises (e.g., Wetherick & Gilhooly, 1995). Another view is that deductive validity is the wrong criterion to assess rationality, because everyday reasoning is probabilistic (e.g., Oaksford & Chater, 2001, p. 349). Similarly, Hertwig, Ortmann, and Gigerenzer (1997, pp. 105106) wrote that those who study first-order logic or variants thereof, such as mental rules and mental models, ignore the ecological and social structure of environments. Still others argue that deductive reasoning depends on pragmatic schemas for specific contents (e.g., Cheng & Holyoak, 1985) or on innate modules adapted to deal with specific contents, such as checking for cheaters (e.g., Cosmides, Tooby, Fiddick, & Bryant, 2005). One counterexample to all of these views is the worldwide popularity of Sudoku puzzles. Their solutions depend solely on pure deduction, and so it is useless to rely on atmosphere, probabilities, fast and frugal heuristics, or content-specific modules. Naive individuals soon learn that Sudoku puzzles, which are quite remote from the ecological structure of daily life, call for deduction (Lee, Goodwin, & Johnson-Laird, 2008). As the popularity of these puzzles shows, individuals enjoy exercising this abstract ability, which is presumably a prerequisite for the development of logic, math, and science. Psychological theories of deduction fall into two broad categories: those based on formal rules akin to those in the proof theory of logic (e.g., Braine & OBrien, 1998; Rips, 1994) and those based on models akin to those in the semantic theory of logic (e.g., Johnson-Laird & Byrne, 1991; Polk & Newell, 1995). Formal rule theories postulate that deduction is a process of proof in which the rules are used to derive conclusions from the premises. A precursor to reasoning is, accordingly, the recovery of the logical form of premises to allow the application of rules. In logic, the logical form of a sentence is transparent, because it is defined by the syntax of the formal language; in daily life, sentences themselves are not normally true or false; rather, the propositions that they are used to express are. Hence, a major and unsolved problem is the recovery of the logical form of propositions. Ripss (1994) computer implementation of his theory finesses the problem by calling for users to input the logical forms of premises. Modern logicians are often skeptical about the role of logical form in everyday reasoning. Barwise (1989, p. 4), for example, wrote, I find the notion [of logical form] unilluminating. Within the model-theoretic tradition, valid entailments are valid not in virtue of form, but in virtue of content. The model-theoretic tradition to which he refers relies on abstract models, such as truth tables, to specify the meanings of logical terms and to determine the validity of inferences that hinge on them. The theory of mental models, which was inspired in part by this tradition in logic (Johnson-Laird, 1983), is based on the notion that individuals reason, both deductively and inductively, on the basis of possibilities. They use the meanings of sentences and their knowledge to envisage what is possible, given the propositions expressed in the premises, and they represent these possibilities in mental models. A conclusion that holds in all of these models is necessary; a conclusion that holds in at least one model is possible; and a conclusion that holds in most models is probable (Johnson-Laird & Byrne, 1991). Likewise, a conclusion is invalid if it has a counterexamplethat is, a possibility in which the premises hold but the conclusion does not. Mental models differ from other proposed sorts of mental representation, because models are as iconic as possible: Their structures correspond to the structure of what they represent. They can likewise unfold in time kinematically to simulate sequences of events (Johnson-Laird, 1983). But, they can also contain symbols that are not iconic, such as a symbol for negation (Khemlani, Orenes, & JohnsonLaird, 2009). The model theory provides an explanation of how individuals make deductions, inductions, explanatory abductions, probabilistic inferences, and inferences to default conclusions that hold in the absence of evidence to the contrary (Johnson-Laird, 2006). In the next section, we describe in more detail the operation of forming models, and show how it leads to the prediction of systematic fallacies that are compellingfor example, so-called illusory inferences. Previous studies have demonstrated their occurrence, but they have tended to rely on conditionals of the form if A then B (Johnson-Laird & Savary, 1999). The meanings of conditionals are controversial, and critics have argued that the illusory conclusions are, in fact, valid (Handley, Evans, & Thompson, 2006; Rips, 1997; Stenning & van Lambalgen, 2008). Although we do not accept this argument, in our present study, we used novel instances of illusory inferences that are simpler than those previously studied and that use connectives with uncontroversial meanings, such as disjunctions and conjunctions. In the article, we (...truncated)