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In mathematical logic, the implicational propositional calculus is a version of classical propositional calculus which uses only one connective, called implication or conditional. In formulas, this binary operation is indicated by "implies", "if ..., then ...", "→", "", etc.. ==Virtual completeness as an operator== Implication alone is not functionally complete as a logical operator because one cannot form all other two-valued truth functions from it. However, if one has a propositional formula which is known to be false and uses that as if it were a nullary connective for falsity, then one can define all other truth functions. So implication is virtually complete as an operator. If ''P'',''Q'', and ''F'' are propositions and ''F'' is known to be false, then: *¬''P'' is equivalent to ''P'' → ''F'' *''P'' ∧ ''Q'' is equivalent to (''P'' → (''Q'' → ''F'')) → ''F'' *''P'' ∨ ''Q'' is equivalent to (''P'' → ''Q'') → ''Q'' *''P'' ↔ ''Q'' is equivalent to ((''P'' → ''Q'') → ((''Q'' → ''P'') → ''F'')) → ''F'' More generally, since the above operators are known to be functionally complete, it follows that any truth function can be expressed in terms of "→" and "''F''", if we have a proposition ''F'' which is known to be false. It is worth noting that ''F'' is not definable from → and arbitrary sentence variables: any formula constructed from → and propositional variables must receive the value true when all of its variables are evaluated to true. It follows as a corollary that is not functionally complete. It cannot, for example, be used to define the two-place truth function that always returns ''false''. 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「implicational propositional calculus」の詳細全文を読む スポンサード リンク
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