Transposition (logic)

In propositional logic, transposition is a valid rule of replacement that permits one to switch the antecedent with the consequent of a conditional statement in a logical proof if they are also both negated. It is the inference from the truth of 'A implies B' the truth of 'Not-B implies not-A', and conversely. It is very closely related to the rule of inference modus tollens. It is the rule that: In propositional logic, transposition is a valid rule of replacement that permits one to switch the antecedent with the consequent of a conditional statement in a logical proof if they are also both negated. It is the inference from the truth of 'A implies B' the truth of 'Not-B implies not-A', and conversely. It is very closely related to the rule of inference modus tollens. It is the rule that: ( P → Q ) ⇔ ( ¬ Q → ¬ P ) {displaystyle (P o Q)Leftrightarrow ( eg Q o eg P)} Where ' ⇔ {displaystyle Leftrightarrow } ' is a metalogical symbol representing 'can be replaced in a proof with.' The transposition rule may be expressed as a sequent: where ⊢ {displaystyle vdash } is a metalogical symbol meaning that ( ¬ Q → ¬ P ) {displaystyle ( eg Q o eg P)} is a syntactic consequence of ( P → Q ) {displaystyle (P o Q)} in some logical system;