Abstract

As early as 1958 John McCarthy stressed the importance of formulating common sense knowledge, and common sense reasoning, in a rigourous manner. Today, this is considered to be the central problem in Artificial Intelligence (AI). A strong advocate of this view is Patrick Hayes, who in 1974 argued that fuzzy logic was not a useful mechanism for representing vague terms, and suggested a better formalism could be developed using Zeeman's Tolerance Geometry. Five years later, Hayes complained about AI's emphasis on toy world's and suggested that a suitable project would be to formalise our common sense knowledge of the (everyday) physical world. A project now known as Naïve Physics (NP). In this project, Hayes discussed his attempts at describing the intuitive notion of objects touching using topological techniques, and indicated that Tolerance Geometry would be a better framework for capturing this notion. This thesis investigates Hayes' suggestion of developing Tolerance Geometry into a formal framework in which one can capture such intuitive terms as bodies touching, and characterising such vague terms as being tall. The analysis in this thesis begins with a (formal) investigation of the Sorites paradox. This puzzle is singled out because it clearly illustrates the problems raised by any formal analysis of vagueness in any language. The analyses of vagueness indicate that vague predicates possess continuous interpretations, and thence demonstarte the need for a spatial structure to be incorporated into the formalised metalanguage. This metalanguage then provides the framework for the proof that the Sorites is insoluble in a logic with a truth-set given by {0,1}, but consistent in a logic with truth-set given by {0,u,1}. Furthermore, this investigation reveals that Zadeh has confused the notions of continuity and the continuum, and therefore his theory of fuzzy sets rest on a mistaken assumption