MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 79modality. It was shown that J can be axiomatised by adding to the axioms andrules for IPC the three axioms∇(p → q) → (∇p → ∇q)∇∇p → ∇pp → ∇p.The last axiom allows derivation of the rule from α infer ∇α. There are a numberof alternative axiomatisations of J, one of which is to add to IPC the axioms(p → q) → (∇p → ∇q)∇∇p → ∇p∇⊤.As Macnab’s characterisation of local operators suggests, J can also be specifiedby the single axiom(p → ∇q) ↔ (∇p → ∇q).In the presence of classical Boolean logic, the middle axiom ∇∇p → ∇p in thefirst group is deducible from the other two, and the logic becomes the ratheruninteresting system K+(p → ∇p) whose only connected validating frames arethe two one-element frames S • and S ◦ (see section 6.1). But in the absence of thelaw of excluded middle we have a modal logic with many interesting models. Inparticular it has relational models based on structures S = (K, ≤, ≺) which refinethe Kripke semantics for IPC. Here ≤ is a partial ordering of K and ≺ is a binaryrelation interpreting ∇ as a universal quantifier in the familar way:M |= x ∇α iff M |= y α for all y such that x ≺ y.To ensure that M(∇α) is ≤-increasing it is required that x ≤ y ≺ z impliesx ≺ z. The logic J is characterised by the class of such frames in which ≺ is asubrelation of ≤ that is dense in the sense that x ≺ y implies ∃z(x ≺ z ≺ y).There is a canonical frame S J of this kind that characterises J, and the logic alsohas the finite model property with respect to such frames. In addition there is acharacterisation of J by neighbourhood frames (K, ≤, N) (see 5.3), where N x is afilter in the lattice of ≤-increasing subsets of K, and the following conditions hold:x ≤ y implies N x ⊆ N y ,{y : x ≤ y} ∈ N x ,{y : U ∈ N y } ∈ N x implies U ∈ N x .If ∇α is defined to be the formula ¬¬α, then the axioms of J become theorems ofIPC. Lawvere [1970] observed thatThere is a standard Grothendieck topology on any topos, namely doublenegation, which is more appropriately put into words as “it is cofinally thecase that”.