The Review of Symbolic Logic

Research Article

DEFINING KNOWLEDGE IN TERMS OF BELIEF: THE MODAL LOGIC PERSPECTIVE

JOSEPH Y. HALPERNa1 c1, DOV SAMETa2 c2 and ELLA SEGEVa3 c3

a1 Computer Science Department, Cornell University

a2 The Faculty of Management, Tel Aviv University

a3 Faculty of Industrial Engineering and Management, Technion - Israel Institute of Technology

Abstract

The question of whether knowledge is definable in terms of belief, which has played an important role in epistemology for the last 50 years, is studied here in the framework of epistemic and doxastic logics. Three notions of definability are considered: explicit definability, implicit definability, and reducibility, where explicit definability is equivalent to the combination of implicit definability and reducibility. It is shown that if knowledge satisfies any set of axioms contained in S5, then it cannot be explicitly defined in terms of belief. S5 knowledge can be implicitly defined by belief, but not reduced to it. On the other hand, S4.4 knowledge and weaker notions of knowledge cannot be implicitly defined by belief, but can be reduced to it by defining knowledge as true belief. It is also shown that S5 knowledge cannot be reduced to belief and justification, provided that there are no axioms that involve both belief and justification.

(Received January 06 2008)

Correspondence:

c1 COMPUTER SCIENCE DEPARTMENT, CORNELL UNIVERSITY, ITHACA, NY 14853 E-mail: halpern@cs.cornell.edu

c2 THE FACULTY OF MANAGEMENT, TEL AVIV UNIVERSITY, TEL AVIV 69978, ISRAEL E-mail: samet@post.tau.ac.il

c3 FACULTY OF INDUSTRIAL ENGINEERING AND MANAGEMENT, TECHNION - ISRAEL INSTITUTE OF TECHNOLOGY, TECHNION CITY, HAIFA 32000, ISRAEL, E-mail: esegev@ie.technion.ac.il