Reasoning with Conjunctive Causes

Reasoning with Conjunctive Causes Bob Rehder (bob.rehder@nyu.edu) Department of Psychology, New York University 6 Washington Place, New York, NY 10003 USA Abstract Conjunctive causes are causes that all need to be present for an effect to occur. They contrast with independent causes that by themselves can each bring about an effect. We extend existing “causal power” representations of independent causes to in- clude a representation of conjunctive causes. We then demon- strate how independent vs. conjunctive representations imply sharply different patterns of reasoning (e.g., explaining away effects for independent causes as compared to exoneration ef- fects for conjunctive causes). An experiment testing how peo- ple reason with independent and conjunctive causes found that their inferences generally matched the model’s prediction, al- beit with some important exceptions. Figure 1. Rather than operating in a vacuum, causes frequently inter- act with other factors to produce their effects. For example, the conjunction of two or more variables is often necessary for an outcome to occur. A spark may only produce fire if there is fuel to ignite, a virus may only cause disease if one’s immune system is suppressed, the motive to commit murder may result in death only if the means to carry out the crime are available. Sometimes, conjunctive causes take the form of enablers. For example, the presence of oxygen en- ables fire given spark and fuel. In contrast, disablers interact with existing causes by preventing normal outcomes. Al- though the eight ball’s path to the side pocket may appear inevitable, it may be interrupted by an earthquake, a falling ceiling tile, or a spilled beer. The last 20 years has seen a growing interest in the role of causal knowledge in numerous areas of cognition. Many studies have investigated how causal relations are learned from observed correlations (Cheng, 1997; Gopnik et al., 2004; Griffiths & Tenenbaum, 2005; 2009; Lu et al., 2008; Sobel et al., 2004; Waldmann et al., 1995). Others have tested the impact of causal knowledge on various forms of reasoning, including inference (Rehder & Burnett, 2005; Kemp & Tenenbaum, 2009), interventions (Sloman & Lagnado, 2005; Waldmann & Hagmeyer, 2005), analogy (Holyoak et al., 2010; Lee & Holyoak, 2008), generalization (Rehder, 2006; 2009), and classification (Rehder & Hastie, 2001; Rehder 2003a; b; Rehder & Kim, 2006; 2009; 2010). But although some studies have investigated the learning of interactive causes (e.g., Novick & Cheng, 2004), their role in reasoning has received little attention. This article tests how people reason with one sort of interactive cause— conjunctions. How should one reason with conjunctive causes? One popular framework for modelling learning and reasoning with causal knowledge is Bayesian networks or causal graphical models (hereafter, CGMs). In CGMs, variables are represented as nodes and causal relations as directed edges. For example, Figure 1A presents a CGM in which variables C 1 and C 2 are causes of variable E. CGMs are popular in part because they specify the causal Markov con- dition that stipulates patterns of conditional independence between variables and which has important implications for how one learns and reasons with causal knowledge. By itself, however, a CGM says nothing about the func- tional relationship between an effect and its causes. For ex- ample, Figure 1A does not specify whether C 1 and C 2 are independent or interactive causes of E. Two possibilities are represented in Figures 1B and 1C. In these figures, we as- sume that C 1 , C 2 , and E are binary variables that are either present or absent. Diamonds represent independent genera- tive causal mechanisms, processes that work to produce the effect when their causes are present. Figure 1B represents the fact that C 1 and C 2 are independent causes of E—that is, that E might be brought about by C 1 or C 2 . Figure 1C repre- sents that C 1 and C 2 are conjunctive causes of E—E is brought about only when C 1 and C 2 are both present. As mentioned, there are other ways that E might depend on an interaction between C 1 and C 2 (e.g., C 2 might disable the causal link between E and C 1 ), but here we focus on the contrast between independent and conjunctive causes. Be- low we specify the (independent or conjunctive) functions that relate an effect and its causes and derive the different patterns of inferences implied by those functions. Other frameworks do not readily distinguish between al- ternative interpretations of Figure 1A. For example, reason- ers may treat it as an associative network, in which case they may infer one variable given others without regard to the direction of causality. Or, they may treat it as a depend- ency network that is sensitive to causal direction (E depends on C 1 and C 2 ) but not their functional relationship. Accord- ingly, our goal is to first establish that reasoners indeed dis- tinguish between independent and conjunctive causes and then determine whether they do so in the manner predicted by our proposed representation of conjunctions. Reasoning with Conjunctive Causes To test how people reason with conjunctive causes, subjects