Our Response

Put brutally, our response is this. The reason that subjects perform poorly on problems like those seen in the quartet above is that their education is defective; and because their education is defective, they haven't reached Piaget's stage of formal operations. We realize, of course, that uses the term `naturally,' and we realize as well that this connotes that people, without special training, will reach the competence in question. But this is a bluff we're quite willing to call. What, precisely, does `naturally' mean? We all know that without special training humans aren't able to solve even simple arithmetic problems. For example, consider this problem:

• John is given of a chocolate chip cookie. Each of his ten friends will be content if they receive 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?

Even educated adults do poorly on this problem. (At a recent talk at a major university, S. Bringsjord found, upon presenting this problem, that a goodly number of professors had completely forgotten how to divide fractions.) Does poor performance of many subjects on a puzzle like this imply the falsity of some such proposition as the following one?

Humans naturally develop a context-free scheme at the level of elementary arithmetic.

If not, then why should fall? Perhaps, again, the problem pertains not to underlying cognitive development, but rather to education, pure and simple. If someone insists on a rather strict reading of `naturally,' according to which only a bare minimum of ``official" education is required to support the ascription of the adverb naturally, and therefore according to which is indeed taken to be false, then we will be quite content to settle for defending the view that

'

If educated in logic as they are in arithmetic, humans develop a context-free deductive reasoning scheme at the level of elementary first-order logic -- a scheme that will allow for the solving of problems like those seen in our quartet above, and significantly harder problems as well.

We see at least two general ways to reply to our view; in a nutshell they are:

1. ``You are stretching the concept of `naturally acquire' too far. ' is true, but this doesn't vindicate Piaget."
2. `` ' is false -- or at least you've done nothing to convince us that it's true."

The first complaint seems to be easy enough to handle. Clearly, ' is firmly in the spirit of Piaget, and we would be quite content with having defended him to this degree. Besides, the point of the reference in ' to arithmetic is to limit the training in logic to something well short of sustained and intense training of the sort an aspiring mathematician or logician would encounter. The training in question is supposed to be analogous to what people receive in arithmetic in the normal course of development in civilized society. The second objection is more formidable. In fact, some readers will be of the opinion that this objection is very formidable -- because apparently training in logic doesn't cause facilitation on problems like those seen in the above quartet (see [Cheng et al., 1986]). One of us (S. Bringsjord) confesses that he has long found the claim that logic training fails to facilitate on problems like our quartet nothing short of astonishing. After all, all four of the problems above (and, indeed, all logic problems at the heart of the psychology of reasoning), from the standpoint of the content of a first-course in mathematical logic, are painfully simple. What aspect of the training could be preventing facilitation?