Formalizing Approximate Objects and Theories: Some Initial Results

This paper introduces some preliminary formalizations of the approximate entities of [McCarthy, 2000]. Approximate objects, predicates, and theories are considered necessary for human-level AI, and we believe they enable very powerful modes of reasoning (which admittedly are not always sound). Approximation is known as vagueness in philosophical circles and is often deplored as a defective aspect of human language which infects the precision of logic. Quite to the contrary, we believe we can tame this monster by formalizing it within logic, and then can “build solid intellectual structures on such swampy conceptual foundations.” [McCarthy, 2000]. We first introduce various kinds of approximation, with motivating examples. Then we develop a simple ontology, with minimal philosophical assumptions, in which to cast our formalization. We present our formalization, and show how it captures some ideas of approximation.