Kripke Models and Master Justification

An epistemic reading of Kripke models relies on a hidden assumption of (com­mon) knowledge of the model. This observation was made in Artemov (2016a)

and leads to the following presentation of a Kripke model as a multiworld JAM? The Kripkean accessibility relation between worlds, uRv, can be recov­ered by the usual rule: What is believed at u, holds at v.

The informal construction is as follows. Let K be a Kripke model. We have to find a justification m:F for each knowledge/belief assertion ?F in K. We claim that the model K itself is such a justification. Indeed, let u b ?F in K. Then a complete description of K yields that, at state u, the agent knows/believes F because the agent knows the model K and knows that F holds at all possible worlds. So, the knowledge/belief-producing evidence for F is delivered by K itself, assuming the agent is aware of K.

Syntactically, we consider a very basic justification language in which the set of justification terms consists of just one term m, called master justification. Think of m as representing a complete description of model K = (W, R, b). Specifically, we extend the truth evaluation in K to justification assertions by stipulating that at each u ∈ W

This reading provides a meaningful justification semantics of epistemic asser­tions in K via the master justification m representing the whole of K. Because a Kripkean agent is logically omniscient, then along with K the agent knows all its logical consequences. Technically, we can assume that the description of K is closed under logical consequence and hence m is idempotent w.r.t. ap­plication, m ■ m = m. This condition manifests itself in a special form of the application principle

On the technical side, a switch from ?X to m:X is a mere transliteration that does not change the epistemic structure of a model. Finally, for each u ∈ W, we define a basic model—the maximal consistent set Γ_u in the propositional language with Tm = {m):

So, from a justification perspective, a Kripke model is a collection of basic models with master justification that represents (common) knowledge of the model.

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In which we suppress the knowledge-producing component KP to capture beliefs.

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