Approximate Objects and Approximate Theories

We propose to extend the ontology of logical AI to include approximate objects, approximate predicates and approximate theories. Besides the ontology we treat the relations among different approximate theories of the same phenomena. Approximate predicates can’t have complete if-and-only-if definitions and usually don’t even have definite extensions. Some approximate concepts can be refined by learning more and some by defining more and some by both, but it isn’t possible in general to make them well-defined. Approximate concepts are essential for representing common sense knowledge and doing common sense reasoning. Assertions involving approximate concepts can be represented in mathematical logic. A sentence involving an approximate concept may have a definite truth value even if the concept is ill-defined. It is definite that Mount Everest was climbed in 1953 even though exactly what rock and ice is included in that mountain is ill-defined. Likewise, it harms a mosquito to be swatted, although we haven’t a sharp notion of what it means to harm a mosquito. Whatif(x,p), which denotes what x would be like if p were true, is an important kind of approximate object. The article treats successively approximate objects, approximate theories, and formalisms for describing how one object or theory approximates another. Our discussion will be adequate if it has as much clearness as the subject matter admits of, for precision is not to be sought for alike in all discussions, any more than in all the products of the crafts. —Aristotle, Nicomachean Ethics