Conditional and Syllogistic Reasoning

Cognitive research on deductive reasoning is almost invariably confined to conditional and syllogistic reasoning. One thrust of the proposed research is to investigate deductive reasoning that is more like the sophisticated, modern forms of thought a contemporary student of logic would be expected to master, but in this section we follow much of the literature and restrict our attention to conditional and syllogistic inference. Accordingly, let's start with the time-honored ``Wason card problem" [WJL72]. Suppose that you are presented with four cards, each of which bears a character:

Suppose, as well, that each card has a letter on one side and a number on the other. Now, your task is to pick the card or cards you would turn over to gather conclusive evidence on the following rule:

(R )

If a card has a vowel on one side, then it has an even number on the other side.

Less than 5% of the educated adult population can solve this problem. (We have replicated the result countless times over the past 15 years, with subjects ranging from 8th grade students to illustrious members of the Academy.) About 30% of subjects do turn over the E card, but that isn't enough: the 7 card must be turned over as well. The mathematical reason why is simple enough. The rule in question is a so-called conditional, that is, a proposition having an if-then form, which is sometimes symbolized as

A conditional is false if and only if its antecedent, p, is true, while its consequent, q, is false. So, if the E card has an odd number on the other side, (R ) is overthrown. However, if the 7 card has a vowel on the other side, this too would be a case sufficient to refute (R ).

Subjects do much better on more ``concrete" versions of the Wason card problem. For example, D'Andrade [D'A82] asked subjects to imagine that they managed a store operating with the following rule.

(R )

If a purchase exceeds $30, then the receipt must have the manager's signature on the back.

Many subjects realized the need to check the amounts on receipts with no signature. (This is isomorphic to checking the 7 card in the abstract version of the problem.) The same result can be obtained in other ways. However, not all concrete versions of the problem result in significant performance improvement.

When we move from conditional reasoning to syllogistic reasoning, the situation is strikingly similar.

Aristotle discovered and proved 14 valid forms of syllogistic inference (out of a space of 256 forms). While his system admits of an elegant coding (devised by Medieval logicians; cf. [Gly92]), for our purposes we need only refer to a fragment of the formalism. To start, consider the following syllogism.

Many people can see that S is a valid inference. But as is the case with the 7 card in the Wason problem, many people have trouble judging syllogisms like the following:

However, subjects can improve their performance if they look to concretize a syllogism and think ``like skeptics." For example, not many subjects fail to see that S is invalid when they consider the following instantiation of it.

On the other hand, for subjects in command of first-order logic ( ) and associated mathematical techniques, forms like S can be directly evaluated with ease. For example, in this logic S becomes

This can then be evaluated and shown to be false via proofs that are a routine part of our Introduction to Logic course at RPI.

From these phenomena regarding conditional and syllogistic reasoning we distill our first three desiderata, namely,

Desideratum 1: No Natural Scheme

An intelligent tutoring system in this domain must be sensitive to the fact that nearly all humans fail to naturally acquire a formal deductive system in the course of their development.

Desideratum 2: Concretization Can Improve Performance

An intelligent tutoring system in this domain must be sensitive to the fact that humans are at least passably good at negogiating some concretized reasoning problems that would otherwise be expressed in formal logic.

Desideratum 3: Not All Concretization Works

Some concretized problems in deductive reasoning fail to result in improved performance; an intelligent tutoring system must be responsive to this fact (e.g., it might allow for experimentation designed to test hypotheses offered to account for the phenomenon in question).

To these three we would add a fourth, which aries from our own investigations, namely, that while humans don't naturally acquire a formal deductive scheme,

Desideratum 4: A Deductive Scheme is Powerful

When students do acquire a deductive scheme, they easily solve the Wason card problem, syllogisms of any form, and problems of much greater difficulty; accordingly, a good intelligent tutoring system in this domain should strive to give students such a scheme.

Students armed with a formal deductive scheme are not only able to solve the Wason problem and all syllogisms, but they are able to solve all those deductive reasoning problems in New York State's math curriculum (see Appendix A), harder problems of the same form (i.e., those involving tricky proofs in the propositional calculus; see Appendix B), and problems calling for very sophisticated deductive reasoning, for example the following ``mystery."

Someone who lives in Dreadsbury Mansion killed Aunt Agatha. Agatha, the butler, and Charles live in Dreadsbury Mansion, and are the only people who live therein. A killer always hates his victim, and is never richer than his victim. Charles hates no one that Aunt Agatha hates. Agatha hates everyone except the butler. The butler hates everyone not richer than Aunt Agatha. The butler hates everyone Agatha hates. No one hates everyone. Agatha is not the butler.

Now, given the above clues, there is a bit of a disagreement between three Norwegian (and hence not exceedingly bright) detectives: Inspector Bjorn is sure that Charles didn't do it. Is he right? Inspector Reidar is sure that it was a suicide. Is he right? Inspector Olaf is sure that the butler, despite conventional wisdom, is innocent. Is he right?