Showing papers by "Melvin Fitting published in 2009"

Reasoning with Justifications

Abstract: This is an expository paper in which the basic ideas of a family of Justification Logics are presented. Justification Logics evolved from a logic called \(\mathsf{LP}\) , introduced by Sergei Artemov (Technical Report MSI 95-29, 1995; The Bulletin for Symbolic Logic 7(1): 1–36, 2001), which formed the central part of a project to provide an arithmetic semantics for propositional intuitionistic logic. The project was successful, but there was a considerable bonus: \(\mathsf{LP}\) came to be understood as a logic of knowledge with explicit justifications and, as such, was capable of addressing in a natural way long-standing problems of logical omniscience. Since then, \(\mathsf{LP}\) has become one member of a family of related logics, all logics of knowledge with explicit knowledge terms. In this paper the original problem of intuitionistic foundations is discussed only briefly. We concentrate entirely on issues of reasoning about knowledge.

How True It Is = Who Says It’s True

Abstract: This is a largely expository paper in which the following simple idea is pursued. Take the truth value of a formula to be the set of agents that accept the formula as true. This means we work with an arbitrary (finite) Boolean algebra as the truth value space. When this is properly formalized, complete modal tableau systems exist, and there are natural versions of bisimulations that behave well from an algebraic point of view. There remain significant problems concerning the proper formalization, in this context, of natural language statements, particularly those involving negative knowledge and common knowledge. A case study is presented which brings these problems to the fore. None of the basic material presented here is new to this paper—all has appeared in several papers over many years, by the present author and by others. Much of the development in the literature is more general than here—we have confined things to the Boolean case for simplicity and clarity. Most proofs are omitted, but several of the examples are new. The main virtue of the present paper is its coherent presentation of a systematic point of view—identify the truth value of a formula with the set of those who say the formula is true.