...theory.

Much of the theory in question arises from empirical investigation.

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...(R7#7).

The Wason card problem is often diagnosed by psychologists by appeal to rules of inference like modus ponens and modus tollens. Such diagnoses are hard to understand from the standpoint of logic. The foundational rule known as modus ponens says that given a conditional 9#9 and the antecedent p, one can infer to q. Another rule, also at the foundations of logic and mathematics (and formal disciplines generally), is modus tollens: 10#10, i.e., from a conditional and the falsity of its consequent one can infer to the falsity of its antecedent. It's hard to see why these rules are relevant. The bottom line is that (R7#7) is false in exactly one case (true antecedent, false consequent; this case is often specified using truth tables), and solving the problem consists in picking all cards that could figure in such a case.

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...ways.

See, e.g., the interesting ``postal cheat" version specified by Johnson-Laird, Legrenzi, and Lagrenzi [JOLL72].

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...improvement.

Manktelow and Evans [ME79], e.g., failed to obtain improvement in a version of the problem that used the rule: ``If I eat haddock, then I drink gin."

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...namely,

Three reactions to the results we have cited in connection with conditional and syllogistic reasoning dominate the literature:

• Domain Specificity. The basic idea here is that most humans can reason deductively only insofar as they can relate the problem at hand to specific knowledge they have acquired in the course of their experience. (Thinking back to (R11#11) and S18#18, many people are familiar with the shopping domain, and with simple zoological facts.)
• Pragmatic Reasoning Schemas. 19#19
• Mental Models. 19#19

There is no need, at least at this stage, to adjudicate these competing explanations.

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...development.

Actually, what the phenomena in question show is that humans fail to naturally acquire a formal deductive system of a type that entails success at solving abstract problems in conditional and syllogistic reasoning. We leave this issue aside herein.

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...syllogisms,

For example, out of 97 students passing S. Bringsjord's Intro to Logic course in Fall of 96, 94 solved the Wason card problem rapidly upon exit from the course. (They were not shown the Wason problem previously. Nor were they shown any similar problem.)

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...``mystery."

So for us, from the pedagogical point of view, it makes little sense to worry about the inability of the untrained to solve conditional and syllogistic reasoning problems. Of course, we concede that some problems may require more than what command over elementary formal logic can provide. (The problem shown in Appendix C, for example, requires mathematical maturity beyond what we will seek to impart via our proposed systems.) Notice that the following proposition is undoubtedly true, but no one despairs; the reaction is simply to teach division.

• Nearly all humans fail to naturally acquire a formal division system in the course of their development.

This is true because a lot of humans have a devil of a time solving simple problems like the following two without appropriate training.

• 293 students from Grover Midddle School are going on a field trip to New York City. Each bus carries 32 students. How many buses will be needed for the trip?
• John is given 20#20 of a chocolate chip cookie. Each of his ten friends will be content if they receive 21#21 of such a cookie. If John is willing to keep none for himself, and he can divide his cookie-part precisely, how many friends can he satisfy?

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...stories.

An insightful review of this book has been written by Tom Trabasso [Tra96].

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...chess:

See also [Hor87].

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...one.

SECRETS is derived from problem sets commonly used in textbooks for 2nd graders.

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...logic.

All logic problems currently in use in curricula in the United States from Kindergarten through Intro to Logic at the college level can be mapped into first-order logic, and can therefore be mapped into robust theorem-provers.

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...size=-1>YPERPROOF.

To teach deductive reasoning in introductory mathematical logic at Rensselaer, Bringsjord uses HYPERPROOF26#26 [BE94], an educational theorem-prover that allows for the construction of simple situations in a blocks world used for idealized robotics scenarios in AI. These situations are represented by both abstract geometric objects placed on an 8 by 8 grid and appropriate formulas in first-order logic. Some of the world's most robust problems in HYPERPROOF26#26 are available through Bringsjord's web site; the publisher of the software and manual, (CSLI and Cambridge University Press), currently links to this site in order to provide a service to teachers and students using the HYPERPROOF26#26 program.

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...anyone.

It is now undeniable that automated theorem provers can crack longstanding mathematical problems which logicians and mathematicians of the highest rank have utterly failed to solve: The Robbins Problem - Are all Robbins algebras Boolean? - has just been solved by EQP, a theorem proving program developed at Argonne National Laboratory. The problem has resisted human efforts for 60 years, and was studied in earnest by Alfred Tarski and his students.

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...iff

Traditional abbreviation in logic for `if and only if.'

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...CENTER>

Adapted slightly from Chapter I of Rayond Smullyan's (1992) Gödel's Incompleteness Proofs (Oxford, UK: Oxford University Press).

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