Namespaces
Variants
Actions

Modus ponens

From Encyclopedia of Mathematics
Jump to: navigation, search


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)
How to Cite This Entry:
Modus ponens. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modus_ponens&oldid=47879
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article