The example you give is an interesting one from a developmental psychology perspective, because it illustrates what Jean Piaget called "conservation of number," a cognitive skill that young children don't display but school-age children do.Regarding the formalization of the example in logical terms, this is not difficult, but it can be done in many different ways, and exploring these different ways brings up some interesting issues.Since the formatting of logical formulas in emails is difficult, I have put my reply to your email in HTML form and put it online at:
Thanks for the clear exposition. You have shown how the premises and conclusion can be represented in formal logic, but not how the reasoning itself is done. There are 2 issues concerning this:
1. Spontaneity: The reasoning should proceed spontaneously, after the thoughts of the premises somehow got into attention. Conventional formal logic does not tell us how to generate entailment automatically. It seems that we have to make a query like "Does there exist a person without an apple to eat?"
2. Neural plausibility: How does the brain actually do this reasoning? It seems that the sentence "5 > 4" (the number of one thing being greater than that of another thing) and the need for 1-1 correspondence directly leads to the notion that there are not *enough* of the second thing. It seems that the brain does not mentally do the 1-1 correspondence (which would be impractical when the number of items gets more than around 7).
The challenge is how to get the AGI to arrive at the conclusion in a step by step way that does not involve external knowledge (ie not explicitly teaching it the pigeonhole principle etc). This may be the key to AGI design.
The critical piece of knowledge may be represented as "A < B and 1-1 correspondence implies there is not enough A for B". The question is how does the brain (or AGI) acquire this knowledge from experience, which means generalizing from small cases of "2 > 1", "3 > 2", etc.
What I noticed is that the sentences considered so far can all be represented easily in formal logic. The problem seems to be in the automatic generation of "rules of thinking" (like the pigeonhole principle). The AGI should be able to do this spontaneously.
yky
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