MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

76 Robert Goldblattmodal formulas such that α is provable in IPC iff α τ is PA-valid. In fact α τ isPA-valid iff it is ω-valid [Goldblatt, 1978, theorem 5].Research into the modal logic of provability since the 1970s has contributedmuch to our understanding of the phenomena of self-reference and diagonalisationthat underly the incompleteness of PA and other systems. An account of theorigins of the subject has been given by George Boolos and Giovanni Sambin[1991], and extensive expositions are provided in the books of Boolos [1979; 1993]and Craig Smoryński [1985]. The most recent survey is that of Giorgi Japaridzeand Dick de Jongh [1998].7.6 Grothendieck Topology as Intuitionistic ModalityBy composing his semantic analysis of S4 with the McKinsey–Tarski translation ofIPC into S4, Kripke [1965a] derived a relational model theory for intuitionistic logicbased on structures S = (K, R) in which R is a quasi-ordering, i.e. reflexive andtransitive. He interpreted the members of K informally as “evidential situations”temporally ordered by R. His paper presented a semantics for predicate logic,proving completeness by the method of tableaux 66 . It also showed that attentioncan be confined to structures that are partially ordered, i.e. antisymmetric as well.By identifying elements x, y ∈ K whenever xRy and yRx we pass to a partiallyordered quotient S ′ which validates the same intuitionistic formulas as S. Morestrongly, any model on S has an equivalent model on S ′ . This contrasts with themodal semantics on these structures: it can happen that S ′ validates the modalaxiom Grz while S does not (see section 5.3).Segerberg [1968b] studied the propositional fragment of this model theory, usingonly partially ordered frames from the outset. He constructed canonical modelsand applied the filtration method to prove the finite model property for a numberof logics, including some that are weaker than or independent of IPC. The fact thatIPC is characterised by the finite partially ordered frames, which also characteriseS4Grz under the modal semantics, provides a clear picture of why IPC translatesinto S4Grz and not just S4.Here is a brief description of the relational models for IPC. Given a partialordering S = (K, ≤), a subset X of K will be called increasing if it is closed“upwards” under the ordering, i.e. whenever x ∈ X and x ≤ y, then y ∈ X. Thedefinition of a model M = (S, Φ) requires that the set {x ∈ K : Φ(p, x) = ⊤} beincreasing for all propositional variables p. Formally this requirement is dictatedby the modal translation of p as ✷p, while informally it conveys the idea that oncep is established as true in a given evidential situation then it remains true in thefuture. The truth conditions for implication and negation areM |= x α → β iff for all y ≥ x, if M |= y α then M |= y β,M |= x ¬α iff for all y ≥ x, not M |= y α.66 An extension of intuitionistic predicate logic that is incomplete for Kripke’s semantics wasfound by Hiroakira Ono [1973], and an incomplete extension of intuitionistic propositional logicwas obtained by Valentin Šehtman [1977].