In fact, as you might have suspected already, there is a bug with our scheme of inference using imagery. And this bug stems from the realization of the previous section, that we cannot substitute different representations for one another. When I started the description, ``imagine a rectangle...'', you were imagining a specific rectangle. Your image did not capture the concept of a rectangle in general. The answer to that particular puzzle happens to hold for any rectangle, so you were able to solve it. But it didn't have to be that way. The typical mistake of geometry students is to draw an equilateral triangle instead of ``any'' triangle and be misled by their own drawing into wrong conclusions. Good thing we brought this up. I was almost suggesting a constant time algorithm for the NP-hard problem of inference.
Whenever you pass information from one representational system to another, knowledge gets lost. You have to add defaults to visualize a concept, or you have to choose particular abstractions to verbalize a picture.
To conclusively prove a statement including abstract concepts like triangle using pictures, one would have to consider the picture of every possible triangle. Typically, however, a few carefully selected examples suffice to span the whole set and convince us of a proposition. How to select the right examples is a hard problem, because it partly depends on the question asked. This is the heuristic component of our ``constant time'' algorithm, and it may fail at times.
The fact that some mechanism of inference is not complete, does not render it useless. It may be better to have a system that works 99% of the time, and then try to focus on learning the exceptions.
This starts to look more like inductive inference rather than deductive inference. The difference from the standard idea of ``induction'' is that the mind has the ability to generate its own examples. Instead of going out to a field and try to find flying birds, you can just imagine them to infer they don't touch the ground.
In fact, if we were to do deduction based on explicit rules, we would have to acquire those rules using induction. We would have to see enough examples of flying birds and falling rocks etc. to compile the rules about them. If Cyc did not start with the idea of typing in the knowledge, but to discover the regularities watching the real world, it would have to come up with this mechanism in the first place.
Similarly, if the primitive symbols were not typed in initially but Cyc had to come up with its own concepts, the team would have to come up with perceptual systems. Then the implicit representations and the imagination perception loop would come more naturally.
Polya distinguishes between demonstrative reasoning and plausible reasoning, and further claims that plausible reasoning is the only means by which we can acquire new knowledge. For example, what you turn in with your mathematics problem set (hopefully) is demonstrative reasoning, but the activity you engaged in the night before, to understand new concepts from examples, trying out special cases, searching for similar solution patterns in your head is plausible reasoning. The problem of AI concerns the latter, dependence on pure symbols belongs to the former.
``We secure our knowledge by demonstrative reasoning, but we support our conjectures by plausible reasoning. A mathematical proof is demonstrative reasoning, but the inductive evidence of the physicist, the circumstantial evidence of the lawyer, the documentary evidence of the historian, and the statistical evidence of the economist belong to plausible reasoning.''``The difference between two kinds of reasoning is great and manifold. Demonstrative reasoning is safe, beyond controversy, and final. Plausible reasoning is hazardous, controversial and provisional. Demonstrative reasoning penetrates the sciences just as far as mathematics does, but it is in itself (as mathematics is in itself) incapable of yielding essentially new knowledge about the world around us. Anything new that we learn about the world involves plausible reasoning, which is the only kind of reasoning for which we care in everyday affairs. Demonstrative reasoning has rigid standards, codified and clarified by logic (formal or demonstrative logic), which is the theory of demonstrative reasoning. The standards of plausible reasoning are fluid, and there is no theory of such reasoning that could be compared to demonstrative logic in clarity or would command comparable consensus.'' [Polya, 1954]