MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

38 Robert Goldblattnormal (“queer”) worlds. 36 Notably absent is Kripke’s real world G ∈ K. Insteada formula α is said to be valid in S if in all models on S, α is true (i.e. assignedthe value ⊤) at all points of K.Associated with S is the modal algebra S + comprising the powerset Booleanalgebra P(K) with the additive operatorf(X) = {x ∈ K : x ∈ Q or ∃y ∈ X(xRy)}to interpret ✸. Note that f(∅) = Q, so f is a normal operator iff K has onlynormal members. Lemmon proved the result that a formula is valid in S iff itis satisfied in the algebra S + with just the element 1 (= K) designated. Thisfollows from the natural correspondence between models Φ on S and assignmentsto propositional variables in S + , under which a variable p is assigned the set{x : Φ(p, x) = ⊤} ∈ S + . The result itself is an elaboration of the construction in[Kripke, 1963a] of the matrix of propositions associated with any model structure.It remains true for S2-like systems if validity in S is confined to truth at normalworlds, and also all elements of S + that include K − Q are designated.Any finite modal algebra A = (B, f) is readily shown to be isomorphic to oneof the form S + , with S based on the set of atoms of B. Combining that observationwith McKinsey’s finite algebra constructions enabled Lemmon to deduce thecompleteness of a number of modal logics with respect to validity in their (finite)model structures. For an arbitrary A he gave a representation theorem, “due inessentials to Dana Scott”, that embeds A as a subalgebra of some S + . This wasdone by an extension of Stone’s representation of Boolean algebras, basing S onthe set K of all ultrafilters of B, with uRt iff {fx : x ∈ t} ⊆ u for all ultrafiltersu, t, while Q = {x ∈ K : f0 ∈ x}. Each x ∈ A is represented in S + by the set{u ∈ K : x ∈ u} of ultrafilters containing x, as in Stone’s theory.In the Lemmon Notes there is a model-theoretic analogue of this representationof modal algebras that has played a pivotal role ever since. Out of any normallogic Λ is constructed a modelM Λ = (K Λ , R Λ , Φ Λ )in which K Λ is the set of all maximally Λ-consistent sets of formulas, withuR Λ t iff {✸α : α ∈ t} ⊆ u iff {α : ✷α ∈ u} ⊆ t,and Φ Λ (p, u) = ⊤ iff p ∈ u. The key property of this construction is that anarbitrary formula α is true in M Λ at u iff α ∈ u. This implies that M Λ isa model of α, i.e. α is true at all points of M Λ , iff α is an Λ-theorem. ThusM Λ is a single characteristic model for Λ, now commonly called the canonicalΛ-model. Moreover, the properties of this model are intimately connected withthe proof-theory of Λ. For example, if (✷α → α) is an Λ-theorem for all α, then it36 At the time this work was done [Kripke, 1965b] had not appeared, but Lemmon had learnedabout non-normal worlds in conversation with Kripke.