Reasoning with Conjunctive Causes

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 include a representation of conjunctive causes. We then demonstrate how independent vs. conjunctive representations imply sharply different patterns of reasoning (e.g., explaining away effects for independent causes as compared to exoneration effects for conjunctive causes). An experiment testing how people reason with independent and conjunctive causes found that their inferences generally matched the model’s prediction, albeit with some important exceptions. Rather than operating in a vacuum, causes frequently interact 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 enables fire given spark and fuel. In contrast, disablers interact with existing causes by preventing normal outcomes. Although 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 C1 and C2 are causes of variable E. CGMs are popular in part because they specify the causal Markov condition 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 functional relationship between an effect and its causes. For example, Figure 1A does not specify whether C1 and C2 are independent or interactive causes of E. Two possibilities are represented in Figures 1B and 1C. In these figures, we assume that C1, C2, and E are binary variables that are either present or absent. Diamonds represent independent generative causal mechanisms, processes that work to produce the effect when their causes are present. Figure 1B represents the fact that C1 and C2 are independent causes of E—that is, that E might be brought about by C1 or C2. Figure 1C represents that C1 and C2 are conjunctive causes of E—E is brought about only when C1 and C2 are both present. As mentioned, there are other ways that E might depend on an interaction between C1 and C2 (e.g., C2 might disable the causal link between E and C1), but here we focus on the contrast between independent and conjunctive causes. Below 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 alternative interpretations of Figure 1A. For example, reasoners 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 dependency network that is sensitive to causal direction (E depends on C1 and C2) but not their functional relationship. Accordingly, our goal is to first establish that reasoners indeed distinguish 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 Figure 1.