Russell, the Liar and Contingent Kripke: Slaying the Dragons of Paradox


Pat Hayes

Institute for Human and Machine Cognition


Paradoxes are sentences that seem to be false if they are true, and true if they are false. For millennia they were regarded as amusing puzzles, but during the 20th century became centrally important to the foundations of logic, mathematics and philosophy of language: Russell's set-theoretic paradox and, later, Goedel's famous incompleteness theorem - both variations on the ancient liar paradox - are considered to mark ground-breaking fault lines in modern thought; and Tarski's theorem, a direct consequence, seems to establish absolute limitations on the expressiveness of logics by showing that they cannot fully describe their own notion of truth. More recently, a variety of new paradoxes have been identified and analyzed: there seem to be infinitely many possible ways to construct paradoxical sentences.

During the last year, with colleagues in the IKRIS project, we have developed IKL, a variation on classical first-order logic which can talk about its own propositions and has a complete, unrestricted 'truth predicate', apparently contradicting Tarski's theorem. All the classical paradoxes can be written in IKL, but they are no longer paradoxical: they appear there simply as contradictions, which seem "paradoxical" because they have the superficial form of definitions. The semantics of IKL provide a new way to resolve the paradoxes, by giving up not on the notion of truth, but on the notion that everything that looks like a definition must define something.

This talk will review and survey the paradoxes, explain how they arise, and how IKL handles them, and why this important for knowledge interchange. The talk is self-contained and does not require any background in logic, but if you aren't used to thinking logically it might make your head spin :-)

Pat Hayes received a BA in mathematics from Cambridge University and a PhD in Artificial Intelligence from Edinburgh. He has held academic positions in computer science at the University of Essex (England), in philosophy at the University of Illinois and as the Luce Professor of cognitive science at the University of Rochester. He has been a visiting scholar at Universite de Geneve and the Center for Advanced Study in the Behavioral Studies at Stanford, and has directed applied AI research at Xerox-PARC, SRI and Schlumberger, Inc. At various times, Pat has been secretary of AISB, chairman and trustee of IJCAI, associate editor of Artificial Intelligence, a governor of the Cognitive Science Society and president of AAAI.

Pat's current research interests include knowledge representation and automatic reasoning, especially the representation of space and time; the semantic web; ontology design; and the philosophical foundations of AI and computer science. He also restores antique mechanical clocks, remodels old houses, draws portraits and enjoys arguing with anyone about almost anything. Pat is a charter Fellow of AAAI and of the Cognitive Science Society, and has professional competence in domestic plumbing, carpentry and electrical work