How We Detect Logical Inconsistencies

How do individuals detect inconsistencies? According to the theory described in this article, they search for a possibility represented in a mental model, in which each proposition in a description is true. If they find such a possibility, the description is consistent; otherwise, it is inconsistent. Evidence corroborates the theory. The evaluation of consistency is easy when the first possibility generated from the start of a description fits later propositions in the description; it is harder when this possibility does not fit later propositions, and individuals have to look for an alternative possibility. The theory postulates that models represent what is true, not what is false. As a result, individuals succumb to systematic illusions of consistency and of inconsistency. KEYWORDS—inconsistency; mental models; reasoning Do I contradict myself? Very well then I contradict myself, (I am large, I contain multitudes.) –Walt Whitman, Leaves of Grass In Daniel Defoe’s novel Robinson Crusoe, the hero pulls off his clothes and swims out to his wrecked ship. Soon after he climbs on board, Defoe writes, he fills his pockets with ‘‘bisket.’’ But if he has no clothes, how can he have pockets? The two propositions are inconsistent. When enough text separates such details, no one notices their inconsistency. Otherwise, the detection of inconsistency seems trivial. That may be why psychologists have long neglected the topic (but cf. Black, Freeman, & Johnson-Laird, 1986). In this article, we explain why inconsistencies are not trivial, describe a theory of the mental processes underlying their detection, and outline some corroboratory evidence. THE LOGIC AND PSYCHOLOGY OF INCONSISTENCY In general, a set of propositions is consistent if there is at least one possibility in which they are all true, and it is inconsistent if they cannot all be true. Inconsistency is devastating in logic, because any proposition whatsoever follows from an inconsistency (e.g., Jeffrey, 1981). Inconsistency is also serious in life, because it is dangerous to believe what is false. It can lead to disaster. As Perrow (1984) remarked about collisions at sea, ‘‘Captains still inexplicably turn at the last minute and ram each other. We hypothesized that they built perfectly reasonable mental models of the world, which work almost all the time, but occasionally turn out to be almost an inversion of what really exists’’ (p. 230). Similarly, the Chernobyl catastrophe was exacerbated by the engineers’ failure to believe that the reactor had been destroyed, even though firemen showed them chunks of graphite that they had found. (Graphite is inside a nuclear reactor to moderate its reactions.) The ability to detect inconsistencies is, accordingly, a hallmark of rationality. Psychologists need to understand how people detect them, and what can go wrong in the process. In the Crusoe example, inconsistency occurs between one proposition and another. But, consider the following set of propositions: If the reactor is intact, then it is safe. If the reactor is safe, then no graphite is outside it. The reactor is intact, and some graphite is outside it. Together, the three propositions are inconsistent; that is, they cannot all be true. But if any one of the three propositions is dropped, the remaining pair is consistent. A large set of propositions can be inconsistent, but again, if any one of them is dropped, the remaining set is consistent. In general, the detection of inconsistency makes bigger and bigger demands on time and memory as the number of distinct atomic propositions in the set of propositions increases. (An atomic proposition is one that does not contain negation or any sentential connective such as ‘‘and’’ or ‘‘or.’’) These demands can increase so that no feasible computational system, not even a computer that is as big as the universe and runs at the speed of light, could yield a result. A set of, say, 100 atomic propositions allows for 2 possible states of affairs, because each proposition can be either true or false. This number is vast, and, in the worst case, a test of the consistency of the beliefs containing these propositions calls for checking every possibility. If one possibility can be checked in a millionth of a second, it would take more than 40 thousand million million years to examine all the possibilities. Your beliefs depend on many more than 100 propositions, so how do you maintain consistency among them? One answer is that you keep Address correspondence to P.N. Johnson-Laird, Department of Psychology, Princeton University, Princeton, NJ 08544. CURRENT DIRECTIONS IN PSYCHOLOGICAL SCIENCE Volume 13—Number 2 41 Copyright r 2004 American Psychological Society them segregated into separate sets. Your beliefs about, say, Walt Whitman have nothing to do with your beliefs about golf. In this way, you have some chance of maintaining consistency within sets, though inconsistencies may arise from one set to another. Even with separate small sets of beliefs, the psychological problem remains: How do you determine whether a set is consistent? A MODEL THEORY OF THE EVALUATION OF CONSISTENCY One way to evaluate consistency is to rely on formal rules of inference from logic. Here is a typical formal rule: