MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 11method due to Lindenbaum that was explained to him by Tarski and which appliesto any propositional calculus that has the rule of uniform substitution for variables.Taking (K, − , ∗ , ·) as the algebra of formulas, with −α = ¬α, ∗ a = ✸αand α · β = α ∧ β, and with D as the set of S2-theorems, gives a characteristicS2-matrix which satisfies all but the last normality condition on D. Since thatcondition is needed to make the matrix into a Boolean algebra, it is imposed byidentifying formulas α, β whenever (α ⇔ β) ∈ D. The resulting quotient matrixis the one desired, and is what is now widely known as the Lindenbaum algebra ofthe logic. Its designated elements are the equivalence classes of the theorems.Now if α is a formula that not an S2-theorem, then there is some evaluationin this Lindenbaum algebra that fails to satisfy α. Let x 1 , . . . , x n be the valuesof all the subformulas of α in this evaluation, and let K 1 be the Boolean subalgebragenerated by the n + 1 elements x 1 , . . . , x n , ∗ 0. Then K 1 has at most 2 2n+1members. Define an element of K 1 to be designated iff it was designated in theambient Lindenbaum algebra. McKinsey showed how to define an operation ∗ 1 onK 1 such that ∗ 1 x = ∗ x whenever x and ∗ x are both in K 1 :∗1 x = ∏ { ∗ y ∈ K 1 : x ≤ y ∈ K 1 }.The upshot was to turn K 1 into a finite S2-matrix in which the original falsifyingevaluation of α can be reproduced.This same construction shows that S4 has the finite model property, with theminor simplification that the element ∗ 0 does not have to be worried about, since∗ 0 = 0 in any normal S4-matrix (so the computable upper bound becomes 2 2n ).The Lindenbaum algebra for S4 has only its greatest element designated, i.e.D={1}, because (α 3 β)∧(β 3 α) is an S4-theorem whenever α and β are, puttingall theorems into the same equivalence class. This is a fact that applies to anylogic that has the rule of Necessitation, and it allows algebraic models for normallogics to be confined to those that just designate 1.3.2 Topology for S4Topological interpretations of modalities were given in a paper of Tang Tsao-Chen [1938], which proposed that “the algebraic postulates for the Lewis calculusof strict implication” be the axioms for a Boolean algebra with an additionaloperation x ∞ having x ∞ · x = x ∞ and (x · y) ∞ = x ∞ · y ∞ . The symbol ✸ wasused for the dual operation ✸x = −(−x) ∞ . The notation ⊢ x was defined to meanthat 1 ∞ ≤ x, and it was shown that ⊢ x holds whenever x is any evaluation ofa theorem of S2. In effect this says that putting D = {x : 1 ∞ ≤ x} turns one ofthese algebras into an S2-matrix. In fact if 1 ∞ = 1, or equivalently ✸0 = 0, it alsosatisfies S4. But S4 was not mentioned in this paper.A “geometric” meaning was proposed for the new operations by taking x ∞ to bethe interior of a subset x of the Euclidean plane, in which case ✸x is the topologicalclosure of x, i.e. the smallest closed superset of x. If the greatest element 1 of the