MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

44 Robert Goldblattthat the formula[((p 3 ✷q) 3 ✷q) ∧ ((¬p 3 ✷q) 3 ✷q)] 3 ✷qis not a theorem of S4 (nor indeed of S5), and when added to S4 gives a systeminto which the intuitionistic logic IPC can be translated by the Gödel–McKinsey–Tarski procedures. The translation of a propositional formula is an S4-theorem iffit is a theorem of Grzegorczyk’s stronger logic, which is deductively equivalent toS4Grz.Segerberg initiated the use of truth-preserving maps between relational modelsand frames in [1968a]. Given models M and M ′ on frames S = (K, R) andS ′ = (K ′ , R ′ ) respectively, a function ϕ from K onto K ′ was called a pseudoepimorphismfrom M to M ′ if(i)(ii)xRy implies ϕ(x)R ′ ϕ(y),ϕ(x)R ′ ϕ(y) implies ∃z ∈ K(xRz & ϕ(z) = ϕ(y)), and(iii) M |= x p iff M ′ |= ϕ(x) p.For such a function every formula α has M |= x α iff M ′ |= ϕ(x) α, so if M is amodel of α, then M ′ will be also. From this it can be shown that if α is valid inS, then the existence of a function from K onto K ′ satisfying (i) and (ii) impliesthat α is valid in S ′ as well. 47The name “pseudo-epimorphism” was shortened to “p-morphism” by Segerbergin [1970; 1971] and this uninformative term has been very widely adopted, evenfor functions that are not surjective but, in place of (ii), satisfy(ii ′ )ϕ(x)R ′ w implies ∃z ∈ K(xRz & ϕ(z) = w).The notion was generalised by Johan van Benthem [1976a] to that of a “p-relation”between models, which is itself intimately related to the concept of a bisimulationrelation that has been fundamental to the study of computational processes (seesection 7.2).There is another explanation of why functions of this type are natural andimportant in the modal context. Any function ϕ : K → K ′ induces the functionϕ + : P(K ′ ) → P(K) in the reverse direction, taking each subset X of K ′ to itsinverse image {x ∈ K : ϕ(x) ∈ X}. This ϕ + is a Boolean algebra homomorphism.The conditions (i) and (ii’) are precisely what is required for it to preserve theoperators f R and f R ′, and hence be a homomorphism between the modal algebrasCm(K ′ , R ′ ) and Cm(K, R). If ϕ is surjective, then ϕ + is injective and so makesCmS ′ isomorphic to a subalgebra of CmS. Hence all modal-algebraic equationssatisfied by CmS will be satisfied by CmS ′ . But a propositional modal formulaα can be viewed as a term in the language of the algebra CmS, with α being47 A surjection between partial orderings that satisfies (i) and (ii) was defined to be stronglyisotone in [de Jongh and Troelstra, 1966], where the notion was used to demonstrate connectionsbetween partial orderings and certain algebraic models for intuitionistic propositional logic.