How True It Is Who Says It's True

摘要

This is a largely expository paper in which the following simple idea ispursued. Take the truth value of a formula to be the set of agents that accept the formulaas true. This means we work with an arbitrary (finite) Boolean algebra as the truth valuespace. When this is properly formalized, complete modal tableau systems exist, and thereare natural versions of bisimulations that behave well from an algebraic point of view.There remain significant problems concerning the proper formalization, in this context, ofnatural language statements, particularly those involving negative knowledge and commonknowledge. A case study is presented which brings these problems to the fore. None ofthe basic material presented here is new to this paper--all has appeared in several papersover many years, by the present author and by others. Much of the development in theliterature is more general than here—we have confined things to the Boolean case forsimplicity 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 pointof view—identify the truth value of a formula with the set of those who say the formulais trite.