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In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables. What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for the study of various logics (in the form of classes of algebras that constitute the algebraic semantics for these deductive systems) and connected problems like representation and duality. Well known results like the representation theorem for Boolean algebras and Stone duality fall under the umbrella of classical algebraic logic.〔 Works in the more recent abstract algebraic logic (AAL) focus on the process of algebraization itself, like classifying various forms of algebraizability using the Leibniz operator. ==Algebras as models of logics== Algebraic logic treats algebraic structures, often bounded lattices, as models (interpretations) of certain logics, making logic a branch of the order theory. In algebraic logic: * Variables are tacitly universally quantified over some universe of discourse. There are no existentially quantified variables or open formulas; * Terms are built up from variables using primitive and defined operations. There are no connectives; * Formulas, built from terms in the usual way, can be equated if they are logically equivalent. To express a tautology, equate a formula with a truth value; * The rules of proof are the substitution of equals for equals, and uniform replacement. Modus ponens remains valid, but is seldom employed. In the table below, the left column contains one or more logical or mathematical systems, and the algebraic structure which are its models are shown on the right in the same row. Some of these structures are either Boolean algebras or proper extensions thereof. Modal and other nonclassical logics are typically modeled by what are called "Boolean algebras with operators." Algebraic formalisms going beyond first-order logic in at least some respects include: * Combinatory logic, having the expressive power of set theory; * Relation algebra, arguably the paradigmatic algebraic logic, can express Peano arithmetic and most axiomatic set theories, including the canonical ZFC. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「algebraic logic」の詳細全文を読む スポンサード リンク
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