16 - Logics and State Spaces

Summary

In Lecture 8 we saw how to construct state spaces from classifications and vice versa. In Lecture 12 we saw how to associate a canonical logic Log(S) with any state space S. In this lecture we study the relation between logics and state spaces in more detail. Our aim is to try to understand how the phenomena of incompleteness and unsoundness get reflected in the state-space framework. We will put our analysis to work by exploring the problem of nonmonotonicity in Lecture 19.

Subspaces of State Spaces

Our first goal is to show that there is a natural correspondence between the subspaces of a state space S and logics on the event classification of S. We develop this correspondence in the next few results.

Definition 16.1. Let S be a state space. An S-logic is a logic ℒ on the event classification Evt(S) such that Log(S) ⊑ ℒ.

The basic intuition here is that an S-logic should build in at least the theory implicit in the state-space structure of S. We call a state σ of S ℒ-consistent if {σ}⊬_ℒ and let Ω_ℒ the set of ℒ-inconsistent states.

Proposition 16.2.If S is a state space and ℒ is an S-logic, then ⊬_ℒΩℒ. Indeed, Ω_ℒ is the smallest set of states such that ⊢_ℒΩ_ℒ.

Proof. To prove the first claim, let (Г, Δ) be any partition of the types of Evt(S) with Ω_ℒ∈Δ. We need to see that Г⊢_ℒΔ.