This chapter investigates an alternative approach to the algebraization of classical propositional logic, emphasizing the utility of Boolean pre-algebras and avoiding quotient constructions. Building on the work of Eytan and da Costa, it is shown that pre-structures—specifically Boolean pre-algebras—provide an effective framework for interpreting key logical notions such as Smullyan’s tableaux, Hintikka sets, saturated sets, and filters. The chapter formalizes logical deduction, tautologies, and realizability through the algebraic behavior of pre-orders and valuations, establishing a semantic foundation grounded in ultrafilters and homomorphic mappings. The equivalence between syntactic and semantic characterizations is demonstrated via representation theorems and the algebraic reinterpretation of tableau development and Hintikka’s Lemma. The study affirms the significance of pre-structures in modeling classical logic and opens avenues for extending these results to higher-order logics through generalizations of Lindenbaum’s theorem and other model-theoretic constructions.