Implicational

Implicational is an adjective relating to implication, the logical connective indicating that one statement follows from another. In academic usage it is commonly applied to the study of conditional reasoning and to formal systems in which the implication operator (often written as A → B) is central. The term is also used to describe the purely implicational fragment of propositional logic, a minimal formal language that contains only variables and the implication connective.

Purely implicational logic has a long history in proof theory and logic. It is typically treated with standard deduction rules and Hilbert-style axioms, such as the introduction rule that from A and A → B one may infer B, and axioms expressing the transitivity and currying properties of implication. In classical logic, A → B is equivalent to ¬A ∨ B; in intuitionistic logic, implication is interpreted constructively, and the law of excluded middle does not generally hold.

In algebra and semantics, implication is modeled in structures such as Boolean algebras for classical logic and Heyting algebras for intuitionistic logic, with the implicational fragment correspondingly well studied in algebraic logic and residuated lattices.

Etymology: from Latin implicatus, past participle of implicare, to fold in or entangle. The term "implicational" should not be confused with implicature, a term in linguistics describing suggested meanings beyond explicit content.