Atomic Boolean Algebras and Classical Propositional Logic

Abstract

From the time of Boole and De Morgan, algebraic methods have been used to study logics. In this century, the customary vehicle for the application of algebra to logic has been the so-called natural or Lindenbaum algebra of the logical system being investigated.1 In the case of a formal axiomatization (logistic system) S of classical propositional logic (hereafter PL), the Lindenbaum algebra A happens to be a Boolean Algebra. (The elements of the Lindenbaum algebra A are the equivalence classes |A| of the wffs of S with respect to the relation A ↔ B, where A ↔ B if and only if there is an S-proof of B from A and of A from B, and the complement, meet, and join operations of A are defined thus: — |A| = |~A|, |A| ⋀ |B| = |A & B| and |A| ⋁ |B| = |A ⋁ B|, where the tilde, ampersand, and wedge are either primitive or defined connectives of S.)