MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 13The proof of result (5) involved taking the direct product of two closure algebrasthat each reject one of the equations σ = 0 and τ = 0, and then embedding thisdirect product into another closure algebra that is well-connected, meaning that ifx and y are non-zero elements, then Cx · Cy ≠ 0. The result itself is equivalent tothe assertion that if the equation Iσ + Iτ = 1 is satisfied by all closure algebras,then so is one of the equations σ = 1 and τ = 1, where I = −C− is the abstractinterior operator dual to C. This is an algebraic version of one of the facts aboutS4 stated in [Gödel, 1933] (see later in this section).In a sequel article [1946], McKinsey and Tarski studied the algebra of closed (i.e.Cx = x) elements of a closure algebra. These form a sublattice with operationsx . y = C(x·−y) and ⊖x = 1 . x = C−x. An axiomatisation of these algebras wasgiven in the form of an equational definition of certain Brouwerian algebras of thetype (K, + , · , . , 1), and a proof that every Brouwerian algebra is isomorphic toa subalgebra of the Brouwerian algebra of closed sets of some topological space.Results were proven for Brouwerian algebras that are analogous to results (1)–(5)above for closure algebras, with the analogue of (5) being:1. If the equation σ · τ = 0 is satisfied by all Brouwerian algebras, then so isone of the equations σ = 0 and τ = 0.Brouwerian algebras are so named because they provide models of the intuitionisticpropositional calculus IPC. This works in a way that is dual to the method that hasbeen described for evaluating modal formulas, in that 0 is the unique designatedelement; ∧ is interpreted as the lattice sum/join operation + ; ∨ is interpreted aslattice product/meet · ; → is interpreted as the operation ÷ defined by x÷y = y . x;and ¬ is interpreted as the unary operation x ÷ 1 = ⊖x.The algebra of open (i.e. Ix = x) elements of a closure algebra also form asublattice that is a model of intuitionistic logic. It relates more naturally to theBoolean semantics in that 1 is designated and ∧ and ∨ are interpreted as · and+. Implication is interpreted by the operation x ⇒ y = I(−x + y) = −C(x · −y)and negation by −x = x ⇒ 0 = I−x. This topological interpretation had beendeveloped in the mid-1930’s by Tarski [1938] and Marshall Stone [1937–1938] whoindependently observed that the lattice O(S) of open subsets of a topological spaceS is a model of IPC under the operations just described. Tarski took this furtherto identify a large class of spaces, including all Euclidean spaces, for which O(S)exactly characterises IPC.The abstract algebras (K, + , · , ⇒ , 0) that can be isomorphically embeddedinto ones of the type O(S) form an equationally defined class. They are commonlyknown as Heyting algebras, or pseudo-Boolean algebras. The relationship betweenBrouwerian and Heyting algebras as models is further clarified by the descriptionof Kripke’s semantics for IPC given in section 7.6.McKinsey and Tarski applied their work on the algebra of topology to S4 andintuitionistic logic in their paper [1948], which uses closure algebras with just onedesignated to model S4, and Brouwerian algebras in the manner just explainedto model Heyting’s calculus. Using various of the results (1)–(4) above, it follows