Modus ponens
law of detachment, rule of detachment
A derivation rule in formal logical systems. The rule of modus ponens is written as a scheme
where A and B denote formulas in a formal logical system, and \supset is the logical connective of implication. Modus ponens allows one to deduce B from the premise A ( the minor premise) and A \supset B ( the major premise). If A and A \supset B are true in some interpretation of the formal system, then B is true. Modus ponens, together with other derivation rules and axioms of a formal system, determines the class of formulas that are derivable from a set of formulas M as the least class that contains the formulas from M and the axioms, and closed with respect to the derivation rules.
Modus ponens can be considered as an operation on the derivations of a given formal system, allowing one to form the derivation of a given formula B from the derivation \alpha of A and the derivation \beta of A \supset B .
Comments
The more precise Latin name of the law of detachment is modus ponendo ponens. In addition there is modus tollendo ponens, which is written as the scheme
\frac{\neg B \ A \lor B }{A} ,
where \neg stands for negation and \lor denotes the logical "or" .
References
[a1] | P. Suppes, "Introduction to logic" , v. Nostrand (1957) |
[a2] | A. Grzegorczyk, "An outline of mathematical logic" , Reidel (1974) |
Modus ponens. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modus_ponens&oldid=47879