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could be derived in two steps, first conjoining A and B, and then using modus ponens to derive the conclusion. However, the inference is so easy that PSYCOP has a single formal rule for drawing the inference (a conjunctive form of modus ponens).

Formal rule theorists try to postulate psychologically plausible rules of inference and a mechanism for using them to construct mental proofs. One problem is that unless certain rules, such as the rule for introducing "and" (see Table I), are constrained, they can lead to futile derivations:

The brakes are on. The switches are on.

.'. The brakes are on and the switches are on. .'. The brakes are on and the brakes are on and the switches are on.

.'. The brakes are on and the brakes are on and the brakes are on and the switches are on.

and so on ad infinitum. One solution is to incorporate the effects of such rules within other rules. In computer programs, however, a rule of inference can be used in two ways: either to derive a step in a chain of inference leading forward from the premises to the conclusion or to derive a step in a backward chain leading from the conclusion to a subgoal of proving its required premises. PSYCOP allows the dangerous rules to be used only in such backward chains, and thereby prevents them from yielding futile steps. PSYCOP therefore has three sorts of rules: those that it uses only forwards, such as the conjunctive rule for modus ponens; those that it uses only backwards, such as the rule for conditional proof; and those that it uses in either direction, such as the rule for modus ponens. A corollary is that reasoners should make modus tollens inferences only when they are given the putative conclusion, or when they can guess the conclusion and then try to prove it.

Given an inference to evaluate, PSYCOP always halts after a finite number of steps either with a proof of the conclusion or else in a state in which it has unsuccessfully tried all its possible derivations. Hence, the theory implies that people infer that a conclusion is invalid only if they fail to prove it. They carry out an exhaustive search of all possible derivations, and only then do they judge that the conclusion does not follow from the premises. However, valid inferences exist that PSYCOP cannot prove. If its exhaustive search has failed to find a proof, then there are two possibilities. Either the inference is invalid, or it is valid but beyond the competence of PSYCOP to prove. A psychological corollary is that people should never know for certain that an inference is invalid.

Formal rule theories postulate that the difficulty of a deduction depends on the number of steps in its derivation and the availability and ease of use of the required rules of inference. Modus ponens is easy because it depends on a single rule; modus tollens is more difficult because it depends on a chain of inferences. Formal rule theorists have corroborated their theories in experiments using large batteries of deductions. They estimate post hoc the probability of the correct use of each rule of inference. When these empirical estimates are combined appropriately for each inference, they yield a satisfactory fit with the difficulty of the inferences in the battery.

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