MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 21in a connected model in which R is an equivalence relation, any two worldsare related. This accounts for the adequacy, for S5, of the model theory of[1959a].An illustration of the tractability of the new model theory is given by a new proofof the deduction rule in S4 that if ✷α ∨ ✷β is deducible then so is one of α andβ. If neither α nor β is derivable then each has a falsifying S4-model. Take thedisjoint union of these two models and add a new “real” world that is R-related toeverything. The result is an S4-model falsifying ✷α ∨ ✷β. This argument is mucheasier to follow than the McKinsey–Tarski construction involving well-connectedalgebras described in section 3.2., and it adapts readily to other systems.Other topics discussed include the presentation of models in “tree-like” form,and the association with each model structure of a matrix, essentially the modalalgebra of all functions ρ : K → {⊤, ⊥}, which are called propositions, with theones having ρ(G) = ⊤ being designated. A model can then be viewed as a devicefor associating a proposition H ↦→ Φ(p, H) to each propositional variable p. Thefinal section of the paper raises the possibility of defining new systems by imposingvarious requirements on R, and concludes that[i]f we were to drop the condition that R be reflexive, this would be equivalentto abandoning the modal axiom ✷A → A. In this way we could obtainsystems of the type required for deontic logic.Non-normal logics are the subject of [Kripke, 1965b], which focuses mainly onLewis’s S2 and S3 and the corresponding systems E2 and E3 of [Lemmon, 1957].The E-systems have no theorems of the form ✷α, and this suggests to Kripke theidea of allowing worlds in which any formula beginning with ✷ is false, and henceany beginning with ✸, even ✸(p ∧ ¬p), is true. A model structure now becomes aquadruple (G, K, R, N) with N a subset of K, to be thought of as a set of normalworlds, and R a binary relation on K as before, but now required to be reflexiveon N only. The semantic clause for ✷ in a model on such a structure is modifiedby stipulating thatand henceΦ(✷β, H) = ⊤ iff H is normal, i.e. H ∈ N, and Φ(β, H ′ ) = ⊤ for allH ′ ∈ K such that HRH ′ ;Φ(✸β, H) = ⊤ iff H is non-normal or else Φ(β, H ′ ) = ⊤ for someH ′ ∈ K such that HRH ′ .This has the desired effect of ensuring Φ(✷β, H) = ⊥ and Φ(✸β, H) = ⊤ wheneverH is non-normal. Thus in a non-normal world, even a contradiction is possible.These models characterise E2, and the ones in which R is transitive characteriseE3. Requiring that the “real” world G belongs to N gives models that characterise