TR-2006004: Justified Knowledge Is Sufficient

Three formal approaches to public knowledge are “any fool” knowledge by McCarthy (1970), Common Knowledge by Halpern and Moses (1990), and Justified Knowledge by Artemov (2004). We compare them to mathematically address the observation that the light-weight systems of Justified Knowledge and ‘any fool knows’ suffice to solve standard epistemic puzzles for which heavier solutions based on Common Knowledge are offered by standard textbooks. Specifically we show that epistemic systems with Common Knowledge modality C are conservative with respect to Justified Knowledge systems on formulas χ∧Cφ → ψ, where χ, φ, and ψ are C-free. We then notice that formalization of standard epistemic puzzles can be made in the aforementioned form, hence each time there is a solution within a Common Knowledge system, there is a solution in the corresponding Justified Knowledge system. 1 Multi-agent Logics The logics Tn, S4n, and S5n are logics in which each of the finitely many (n) agents has a knowledge operator Ki which is T, or S4, or S5 respectively. We only consider cases where all agents’ modalities are of the same logical strength. Definition 1. The formal systems for Tn, S4n, and S5n are as follows: propositional logic plus for Ki, i = 1, 2, . . . , n we have