Conditional Reasoning

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CORRECTED MARCH 18, 2010; SEE LAST PAGEJournal of Experimental Psychology: Learning, Memory, and Cognition 2010, Vol. 36, No. 2, 298 323 2010 American Psychological Association 0278-7393/10/$12.00 DOI: 10.1037/a0018705

Conditional Reasoning in Context: A Dual-Source Model of Probabilistic InferenceKarl Christoph Klauer, Sieghard Beller, and Mandy Hu tterAlbertLudwigsUniversita t FreiburgA dual-source model of probabilistic conditional inference is proposed. According to the model, inferences are based on 2 sources of evidence: logical form and prior knowledge. Logical form is a decontextualized source of evidence, whereas prior knowledge is activated by the contents of the conditional rule. In Experiments 1 to 3, manipulations of perceived sufficiency and necessity mapped on the parameters quantifying prior knowledge. Emphasizing rule validity increased the weight given to form-based evidence relative to knowledge-based evidence (Experiment 1). Manipulating rule form (onlyif vs. ifthen) had a focused effect on the parameters quantifying form-based evidence (Experiment 3). The model also provides a parsimonious description of data from the so-called negations paradigm and adequately accounts for polarity bias in that paradigm (Experiment 4). Relationships to alternative conceptualizations of conditional inference are discussed. Keywords: conditional reasoning, probabilistic reasoning, dual-process model

Recent years have seen an increased interest in the role of prior knowledge in conditional reasoning. Such work is characterized by a couple of features: Prior knowledge is activated either through the use of contents with which participants have prior experience (e.g., Beller, 2008; Beller & Spada, 2003; Cummins, Lubart, Alksnis, & Rist, 1991; Thompson, 1994) or by providing explicit prior information about the relationships between the antecedent and consequent of a conditional rule, often in the form of bivariate frequency information (e.g., Evans, Handley, & Over, 2003; Oaksford, Chater, & Larkin, 2000; Oberauer & Wilhelm, 2003). In contrast, earlier work frequently relied on so-called abstract materials, for which little prior information is presumably available. Simultaneously, instructions in earlier work tended to stress the notion of logical necessity, according to which a conclusion can be drawn only if it logically follows from the given premises, whereas the recent work more frequently relies on a graded response format in which the acceptability, plausibility, or probability of the conclusion given the premises has to be assessed (e.g., Liu, Lo, & Wu, 1996; Oaksford et al., 2000). We refer to the recent line of research as research on probabilistic conditional inference. In this article, we propose a dual-source model of probabilistic conditional inference. According to the model, inferences are based on two sources of evidence: logical form and prior knowledge. Logical form is a decontextualized source of evidence, whereas prior knowledge is activated by the contents of

the conditional rule. The dual-source hypothesis is contrasted with the view that probabilistic conditional reasoning draws primarily on prior knowledge. In this view, exemplified by Oaksford et al.s (2000) probabilistic model of conditional inference, the role of the conditional rule is to alter the knowledge base from which inferences are derived. In the introduction, we review existing evidence for two qualitatively distinct modes of reasoning, one based on logical form, the other based on prior knowledge. This is followed by a review of research on the role of logical form in probabilistic conditional inference. The research suggests that both prior knowledge and logical form play a role in such inferences. This hypothesis is then formally specified by means of the dual-source model. In four experiments, the dual-source model is evaluated in terms of its ability to fit the data, in terms of whether the effects of experimental manipulations targeted at specific model parameters indeed affect these parameters as expected, and in terms of whether the model adequately reproduces critical data patterns observed in probabilistic conditional inference. In each experiment, the performance of the dual-source model is compared with that of Oaksford et al.s one-source model. Four conditional inferences are typically studied for a conditional rule of the form if p then q: Modus ponens (MP): Given the rule and p, it follows that q. Modus tollens (MT): Given the rule and not-q, it follows that not-p. Affirmation of the consequent (AC): Given the rule and q, it follows that p. Denial of the antecedent (DA): Given the rule and not-p, it follows that not-q.298

Karl Christoph Klauer, Sieghard Beller, and Mandy Hu tter, Institut fu r Psychologie, AlbertLudwigsUniversita t Freiburg, Freiburg, Germany. The research reported in this article was supported by Grant Kl 614/31-1 from the Deutsche Forschungsgemeinschaft to Karl Christoph Klauer. Correspondence concerning this article should be addressed to Karl Christoph Klauer, Institut fu r Psychologie, Sozialpsychologie und Methodenlehre, AlbertLudwigsUniversita t Freiburg, D-79085 Freiburg, Germany. E-mail: klauer@psychologie.uni-freiburg.de

A DUAL-SOURCE MODEL

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Each of these consists of the major premise (i.e., the rule), the minor premise (e.g., p for MP), and a conclusion (e.g., q for MP). Under a traditional interpretation of the conditional rule ( p is sufficient, but not necessary for q; Evans & Over, 2004), MP and MT are logically valid inferences, whereas AC and DA are not logically valid. Let us briefly review results obtained with abstract or arbitrary rule contents and instructions stressing logical necessity. Endorsement rates are typically close to 100% for MP, whereas the AC, DA, and MT inference rates vary from 23%, 17%, and 39% to 89%, 82%, and 91%, respectively (Schroyens, Schaeken, & dYdewalle, 2001). Across studies, MP is accepted significantly more frequently than MT and than AC; MT and AC are accepted significantly more frequently than DA; the difference between MP and MT is significantly and substantially larger than that between AC and DA (Schroyens et al., 2001); and the difference between AC and DA is often not significant in individual studies (Evans, 1993; OBrien, Dias, & Roazzi, 1998). Taken together, acceptance rates tend to be ordered as MP MT AC DA. Some of the variability in the inference rates reflects the fact that participants sometimes adopt a biconditional rather than conditional interpretation of if p then q. The biconditional interpretation ( p is sufficient and necessary for q) justifies acceptance of all four inferences. But many procedural variations seem to play a role in shaping the profile of acceptance rates (Evans & Over, 2004, Chapter 3; Schroyens et al., 2001). As pointed out by Schroyens et al. (2001), these results are consistent with a (revised version of the) mental model theory (Johnson-Laird, Byrne, & Schaeken, 1992) and the theory of mental rules (e.g., Rips, 1994). Studies using contents for which prior knowledge is available also address these inferences and typically require ratings of plausibility, probability, or confidence in the truth of the conclusion. In such studies, there is even more variability in the profiles of ratings over the four inferences, but two major variables account for much of it: perceived sufficiency of p for q and perceived necessity of p for q. Perceived sufficiency and necessity have been assessed in different ways; one possibility is to have participants rate sentences such as It is necessary for p to happen in order for q to happen (perceived necessity) and p happening is enough to ensure that q will happen (perceived sufficiency; Thompson, 1994, p. 745). Thompson (1994) systematically chose contents differing in perceived sufficiency and necessity from different domains dealing with causal relationships, permissions, obligations, and definitions. Across these domains, she consistently found strong effects of perceived sufficiency on the acceptance of MP and MT and strong effects of perceived necessity on DA and AC: Perceived sufficiency and MP and MT acceptability are monotonically related as are perceived necessity and DA and AC acceptability. Figure 1 (middle panel) illustrates the typical results with data compiled from experiments by Liu (2003; Experiments 1a and 2), which we use as an illustrative example throughout the introduction. Perceived sufficiency and necessity were varied in three steps: high (H), medium (M), and low (L), creating six rules HL, ML, LL, LH, LM, and LL, with the first letter referring to degree of sufficiency and the second letter to degree of necessity. Contents with high sufficiency of p for q (HL) received the highest MP and MT ratings, followed by contents with medium sufficiency (ML), whereas contents with high necessity of p for q (LH)

received the highest AC and DA ratings, followed by contents with medium necessity (LM). A distinction related to perceived necessity and sufficiency is that between alternative antecedents and disabling conditions. An alternative antecedent is an event distinct from p which is sufficient for q; for example, an alternative antecedent for the rule If a stone is thrown at a window, it will break is to fire a gun at a window. A disabling condition is a condition that prevents q from happening in the presence of p (e.g., the window is made of Plexiglas). Alternative antecedents thus undermine perceived necessity, and disabling conditions undermine perceived sufficiency. Perceived necessity and availability of disablers as well as perceived sufficiency and availability of alternatives are relatively highly correlated (Verschueren, Schaeken, & dYdewalle, 2005a), and they engender similar effects on the profiles of ratings over