Parametric Constructive Kripke-Semantics for Standard Multi-Agent Belief and Knowledge (Knowledge As Unbiased Belief)

We propose parametric constructive Kripke-semantics for multi-agent KD45-belief and S5-knowledge in terms of elementary set-theoretic constructions of two basic functional building blocks, namely bias (or viewpoint) and visibility, functioning also as the parameters of the doxastic and epistemic accessibility relation. The doxastic accessibility relates two possible worlds whenever the application of the composition of bias with visibility to the first world is equal to the application of visibility to the second world. The epistemic accessibility is the transitive closure of the union of our doxastic accessibility and its converse. Therefrom, accessibility relations for common and distributed belief and knowledge can be constructed in a standard way. As a result, we obtain a general definition of knowledge in terms of belief that enables us to view S5-knowledge as accurate (unbiased and thus true) KD45-belief, negation-complete belief and knowledge as exact KD45-belief and S5-knowledge, respectively, and perfect S5-knowledge as precise (exact and accurate) KD45-belief, and all this generically for arbitrary functions of bias and visibility. Our results can be seen as a semantic complement to previous foundational results by Halpern et al. about the (un)definability and (non-)reducibility of knowledge in terms of and to belief, respectively.