...logic.

That is, at the level of the propositional calculus plus command over some set of simple operations involving the quantifiers `some' ( in FOL) and `all' ( in FOL)). The proposition , or at least a thesis very close to it (more about variants on below), is articulated and defended by Piaget and Inhelder in [Inhelder and Piaget, 1958].

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... .

For example, as Johnson-Laird (sarcastically?) says:

It seems that adult subjects in the selection task have not reached the Piagetian level of formal operations. Yet they are supposed to have attained it around the age of 12. ([Johnson-Laird, 1995], 133)

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...[Johnson-Laird and Savary, 1995].

As cognoscenti know, Johnson-Laird worked with Wason in the ``early days" to devise the experiments taken by nearly all to overthrow the likes of (e.g., see [Wason and Johnson-Laird, 1972]).

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...task.

Wason first described this problem in print in [Wason, 1966].

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...justification.

This is Wason's [Wason, 1977] THOG problem, with the shape-names changed to work alongside a representation of this problem in the HYPERPROOF system [Barwise and Etchemendy, 1994] S. Bringsjord uses to teach logic, automated theorem-proving, and AI. HYPERPROOF is discussed below.

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...answer.

Actually, in 12 years of presenting the selection task and the THOG problem, S. Bringsjord has never obtained only a 5% differential between it and P1. In his experience at Brown University and Rensselaer, students find the THOG problem much easier than the selection task. E.g, in General Psychology of Fall 1997, 10% of the students (at the start of the semester) solved the selection task, but over 30% solved the THOG. In a recent offering of (S. Bringsjord's) Introduction to Logic Programming, 80% solved THOG, 60% the selection task.

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...[Johnson-Laird and Savary, 1995]:

One of us (Noel) notes that this particular problem may be confusing to people with some background in computer programming. (Do you see why?) We are planning an experiment running in parallel to J-L's in which the phrase `or else if' is replaced with `or' and the phrase `, but not both' is added at the end.

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...section:

As we say, these are the main responses. One response we leave aside holds that there is a procedure for ``checking for cheaters" that has evolved in us as a result of natural selection (e.g. see [Cosmides, 1989]). This response appeals to much of the same experimental data appealed to by proponents of the Pragmatic Reasoning Schemas Response.

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...will

Correctly, in S. Bringsjord's opinion: he thinks that mental models is provably reducible to semantic tableaux.

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...resolution.)

What we have just said is in no way offered as a thorough description of the mental models approach.

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...consider),

Here is the tip of the iceberg of problems we see:

• The permission schema reaction fails to account (e.g.) for consistently better-than-guessing performance on syllogisms. Subjects can be given a choice between four conclusion of the A, E, I or O form, as well as ``no conclusion follows," producing a chance rate of 20% for correct solution. A correct response rate of more than 50% is common; one of us (S. Bringsjord) has replicated this rate many times in my psychology, logic, and computer science classes.
• Pragmatic reasoning schemas by definition fail to apply to (or to explain competence with) abstract reasoning problems having nothing to do with pragmatic reasoning.
• Both the mental logic and mental models approach fails to ``scale up" to logic problems more difficult than the simple ones that dominate the psychology of reasoning literature.

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...question.

We realize also that part of what we've been calling the `received view' is that some such term is part of the thesis at stake. For example, in their discussion of the psychology of reasoning, Stillings et al. [Stillings et al., 1995] opine that a proposition virtually identical to is ``obviously" overthrown by the fact that the vast majority of subjects fail to solve problems like those visited in our quartet.

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...problem.

Here's another problem with a slightly different twist:

• 293 students from Grover Middle 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?

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...society.

This is as good a place as any to say that at least one of us (S. Bringsjord) does happen to believe that careful reading of Piaget's work on the issues before us - e.g., [Inhelder and Piaget, 1958] and [Beth and Piaget, 1966] - reveals that a sucessful defense of ' constitutes his vindication in the area of logical reasoning, but exegesis of his writings will have to wait for a time when we have more space.

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...class.

The training in question, unfortunately, is probably equivalent to that required and supplied in a number of states as part of their K-12 math curricula. For example, students in New York State are taught to manipulate symbols in the propositional calculus, but they are not taught any of the things we enumerate immediately below in my hypotheses H1-H3. To make the point a bit more focussed, students in New York State are taught how to respond to questions like

Given the statements

b

which one of the following statements must also be true? (Check the correct answer.)

c

h

a

none of the above

But they are not taught how to disprove the incorrect answers here, nor how to use established diagrammatic techniques to carry out proofs and disproofs at this level, nor how to take an English word (logic) problem and transform it into a formal representation that can then be used to mechanically generate a solution in the form of a proof.

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...correct.

A fuller description of the experiments conducted, as well as those we're planning, is forthcoming.

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...Mystery:

This problem, posed in HYPERPROOF, can be obtained from S. Bringsjord's web site (under Intro to Logic). The problem formalized for and solved by OTTER can be found on the site as well (under Intro to Logic Programming).

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...arithmetic.

We have no doubt that you have by this time solved P2, the THOG problem. But perhaps you would like the answer to P3? (If not, stop reading this footnote now.) Since the disjunction in P3 is a so-called exclusive one, we can represent it by It is easy to prove that one of the conditionals must be false. (Intuitively, if both can't be true, but one of them is true, this leaves one as false.) But since (as follows from the truth-table for ) the only way a conditional can be false is if the antecedent is true while the consequent is false, it follows immediately that there cannot possibly be an ace in the hand.

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