The history of Boolean algebras goes back to George Boole (Boole [2]). Boole
stated a list of algebraic identities governing the “laws of thought”,i.e. of classical
propositional logic. Boole had in mind two interpretations for his identities. One is
from logical systems and the other from the “algebra of classes”. Boole’s observation
amounted, in algebraic language, to saying that his identities held true under
both interpretations. Some 50 years later, the completeness theorem for propositional
logic, saying that every identity valid in the two-element Boolean algebra is
derivable from Boole’s axioms, and Stone’s representation theorem, which asserts
that every Boolean algebra is isomorphic to an algebra of sets, together proved
that Boole’s identities give in fact a complete axiomatization for both of his interpretations.
As we shall see in this article, the proofs of both results are closely
connected.
The article is devided into 4 sections. Section 2 will provide the necessary definitions
and notations, and will familiarize the readers with the algorithms through
some examples. In section 3 and 4, we will prove Stone’s representation theorem
(set-theoretical version) and the completeness theorem for propositional logic in
terms of Boolean algebra respectively. We assume the readers with some backgrounds
in propositional logic.
For a more detailed history of Boole and Boolean algebras, see Machale [3].
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