Kit Fine's situation semantics

Kit Fine gave an invited talk at the Third Interdisciplinary Ontology Conference on Sunday, titled "A State-Based Approach to the Frame Problem". He gives a kind of situation semantics to the classical propositional language, and then extends it to a hybrid modal language for reasoning about change.

A model for the propositional language is an ordered triple (S,≤,[ ]), where (S,≤) is a partially ordered set of states (his term for partial situations), subject to certain conditions, and [ ] is a valuation function assigning each propositional variable a pair ([p]⁺,[p]^-) of sets of verifying and falsifying states, subject to certain natural conditions. The recursive clauses for ∧ and ∨ are interesting:

s verifies B ∧ C if for some t and u, t verifies B, u verifies C, and s = t+u (the l.u.b of {t,u}).
s falsifies B ∧ C if s falsifies B or s falsifies C or for some t and u, t falsifies B, u falsifies C, and s = t+u.

s verifies B ∨ C if s verifies B or s verifies C or for some t and u, t verifies B, u verifies C, and s = t+u.
s falsifies B ∨ C if for some t and u, t falsifies B, u falsifies C, and s = t+u.

Let (S,≤) be the upper semi-lattice generated by three states s₁, s₂, s₃. Then (S,≤) is a state space according to Fine's definition. Let p₁, p₂, p₃ be propositional variables, and for each i = 1,2,3, let

[p_i]⁺ = { s_i }, [p_i]^- = ∅.

Then (S,≤,[ ]) is a model. The formula (p₁ ∧ p₂) ∨ p₃ is verified by the states s₁+s₂, s₃, and s₁+s₂+s₃, but not by any other states. This means that the recursive clauses do not preserve the following condition on the sets of verifying and falsifying states of a formula:

s verifies A, u verifies A, and s ≤ t ≤ u imply t verifies A.
s falsifies A, u falsifies A, and s ≤ t ≤ u imply t falsifies A.