Problem Reasoning

[Mary Zimmermann]

If people typically solve problems by recalling previous solutions, how do we ever solve a truley new problem? Why is it possible, at all, to learn to grapple with abstract problems, as we do when we attempt mathematics and logic?
In fact, we are not particularly good at solving abstract problems. To illustrate this, consider the following problem.
Suppose that I have four cards. Each card has a number on one side and a letter on the other. I place the cards on the table, so that one side of each card is exposed. The sides that you can see show the following symbols:

A 4 K 7

I assert that the numbers and letters hae been assigned according to the following rule:

"If there is a vowel on one side then there is an odd number
on the other side."

Which are the fewest cards that you must turn over to determine whether or not this rule has been followed?

[Mary Zimmermann]

Let's assign different cover stories to this problem to make it quite easy, here is an example:

Inspector Hound has been assigned to enforce the state liquor laws. The Inspector knows that you can drink alcoholic beverages only if you are over twenty-one years of age. Upon entering a restaurant Inspector Hound sees the following people:

A young man of undeterminate age with a beer.
A high school student sipping an unknown drink.
A young woman, in the 18 to 25 year olds range, with a coke.
An elderly man drinking something out of a flask.

What should the Inspector do to see if the liquor laws are being followed ?

[Mary Zimmermann]

The Inspector should find out how old the man with the beer is, and find out what the high school student is drinking.
To a logician this is something of a paradox. In terms of formal deduction, the logical problem and the drinking problem are identical:

If you are drinking liquor then you must be over 21.
If there is a vowel on one side then there must be an odd
number on the other side.

Why do the two problems differ in difficulty?

The drinking problem invites an interpretation as a "permission schema, " that can be worded roughly as:

If activity X is permitted only to people with property Y, then anyone doing X must have property Y.

The permission schema is widespread in many societies, so virtually all of us are expert in determining whether or not activities are permitted. Therefore, if a problem in logical implication can be reinterpreted as an investigation into permission, we know what to do, because we can apply schematic problem solving.

The permission schema is not the only interpretation one can give to logical implication. An alternative schema is the "causation schema":

If casual factor A is present, then result B must occur.

If DDT is released into a river, the fish will die downstream.

Here, the relevant pieces of information are clearly "Do fish die when DDT is released?" (which is equivalent to asking "If there is an A on one side of the card is there an odd number on the other?") "Are there cases where fish do not die even though DDT is released?" (equivalent to "Is there a vowel on the unexposed side of the card with the 4 exposed?"). The causation problem is easier to solve than a logically equivalent abstract problem.

Instead of storing a single set of rules for manipulating logical implication problems, the human problem solver seems to memorize many localized, logically redundant rules for dealing with different interpretations of the logical relation.

But is this really redundant? What the human information processor has done is to track off long-term memory for storing redundant rules in return for a reduction in the burden on immediate memory. Given the relative costs of long-term and immediate memory, this is a good bargain.