MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 45valid in the frame S precisely when the algebraic equation “α ≈ 1” is satisfied byCmS. This gives another perspective on why validity is preserved by surjectivep-morphisms.Of equal importance is the validity-preserving notion of subframe. This originatedin Kripke’s definition in [1963a] of a model structure (G, K, R) as beingconnected when K = {H : GR ∗ H}, where R ∗ is the reflexive-transitive closureof R. Lemmon adapted this in his [1966b] to the notion of the connected modelstructure S x generated from S by an element x, which is the substructure of Sbased on {y : xR ∗ y}. He observed that a formula falsified by CmS must be falsifiedby CmS x for some x. Segerberg showed in [1971, p. 36] that a model M onS can be restricted to a model M x on S x (the submodel of M generated by x)in such a way that in general M x |= y α iff M |= y α. From this it follows that anyformula valid in S will be valid in S x , and conversely a formula valid in S x for allx in S will be valid in S itself (as essentially observed by Lemmon). This notionof point-generated substructure turned out to be the relational analogue of thenotion of subdirectly irreducible algebra. Indeed the algebra CmS is subdirectlyirreducible iff S is equal to S x for some x, a fact that was first demonstrated byWim Blok [1978b, p. 12], [1980, Lemma 4.1].A frame S is a subframe of frame S ′ if it is a substructure of S ′ that is closedunder R ′ , i.e. if x ∈ K, then {y ∈ K ′ : xR ′ y} ⊆ K (some authors call thisa “generated” subframe even though there is no longer any generator involved).Then the inclusion function ϕ : K ↩→ K ′ is a p-morphism inducing ϕ + as asurjective homomorphism from CmS ′ to CmS. Since equations are preserved bysurjective homomorphisms, modal-validity is preserved in passing from S ′ to thesubframe S.The disjoint union ∐ J S j of a collection {S j : j ∈ J} of frames also preservesvalidity. The construction∐was first applied to modal model theory in [Goldblatt,1974] and [Fine, 1975b].J S j is simply the union of a collection of pairwisedisjoint copies of the S j ’s. Each S j is isomorphic to a subframe of ∐ J S j, andso the above properties of subframes guarantee that a formula is valid in ∐ J S jiff it is valid in every S j .These observations about morphisms, subframes and disjoint unions form thebasis of a theory of duality between frames and modal algebras that is discussedin section 6.5.6 METATHEORY OF THE SEVENTIES AND BEYONDThe semantic analysis of particular logics eventually gave way to investigations ofthe nature of the relational semantics itself: the strengths and limitations of itstechniques, and its relationship to other formalisms, particularly first-order andmonadic second-order predicate logic. Some of the questions raised have yet to beanswered.Throughout chapter 6 the term “logic” will always mean a normal logic.