Justified common knowledge

Sergei Artemov
2006 Theoretical Computer Science  
In this paper we introduce the justified knowledge operator J with the intended meaning of J as 'there is a justification for . ' Though justified knowledge appears here in a case study of common knowledge systems, a similar approach is applicable in more general situations. First we consider evidence-based common knowledge systems obtained by augmenting a multi-agent logic of knowledge with a system of evidence assertions t: , reflecting the notion 't is an evidence for ,' such that evidence
more » ... respected by all agents. Justified common knowledge is obtained by collapsing all evidence terms into one modality J. We show that in standard situations, when the base epistemic systems are T, S4, and S5, the resulting justified common knowledge systems are normal modal logics, which places them within the scope of well-developed machinery applicable to modal logic: Kripke-style epistemic models, normalized proofs, automated proof search, etc. In the aforementioned situations, the intended semantics of justified knowledge is supported by a realization theorem stating that for any valid fact about justified knowledge, one could recover its constructive meaning by realizing all occurrences of justified knowledge modalities J by appropriate evidence terms t: . Definition 4. The language of justified common knowledge is a modal language with n + 1 modalities K 1 , . . . , K n , J . Systems T J n , S4 J n , and S5 J n are specified as T n , S4 n , and S5 n , with the modalities K 1 , . . . , K n augmented by S4 with the modality J, together with the principle: for all i = 1, . . . , n Apparently, the dummy (n + 1)st agent corresponding to J plays the role of a sceptical S4-agent who accepts facts only if they are supplied with checkable evidence. On the other hand, this agent is trusted by all other agents and is capable of internalizing and inspecting any fact actually proven in the system. As was noticed in [2,3], T J n and S4 J n correspond to McCarthy's systems with the 'any fool knows' modality (cf. [32]). Lemma 4. In each of T J n , S4 J n , and S5 J n , Proof. Immediate from K i -reflexivity and the following derivations.
doi:10.1016/j.tcs.2006.03.009 fatcat:4mxemi3reneclkeohqktuhmnam