Modus ponens

The modus ponens is a common already in the ancient logic circuit figure who in many logical systems (see logic calculus ) is used as a rule of inference.

The modus ponens allows to derive from two statements of the form If A, then B and A ( the two premises of the final figure) a statement of the form B ( the conclusion of the final figure).

The actual name for the mode ponens is - in contrast to the mode tollendo ponens - mode ponendo ponens. Synonym are used, among other things, the terms separation rule or Implikationsbeseitigung.

The final rule is often abbreviated to MP or MPP.

Etymology

The term mode ponens is derived from the Latin words mode ( here: final figure) and ponere (set, set ) and means releasing final figure, ie Closing figure in which a positive statement is derived.

The full Latin name, mode ponendo ponens, " final figure ( mode ) obtained by setting ( ponendo ) a statement another statement sets ( ponens ) " can be explained as that when given first premise, "If A, then B " through the " Set " ( Accept ) of the second premise, A, the following sentence from two B" set " ( derived ) is.

Formulation

From the premises

And

Follows the Conclusion

Example:

From the conditions " When it rains, the road is wet" and " It's Raining " logically follows: " The road is wet ."

Formally, the mode is ponens listed with the derivative operator as a rule of inference.

Logical forms of modus ponens

As a statement

Although the mode ponendo a ponens rule of inference, then, is a meta-linguistic concept, the term " Modus Ponens " occasionally used for object- language expressions with the following structure:

However, since inference rules and statements are completely different concepts, it is scientifically rather unfortunate to call them with the same name. Generally, the mixing property and the metalanguage is problematic and should normally be avoided.

As Subjunktionsbeseitigungsregel

As a rule of detachment in logical calculi (also: elimination rule of subjunction ( implication ) in the systems of natural deduction ) it reads as follows:

As cut rule

In metalogischer version it is the cut rule:

( | Here the double line | for padlocking of dialogue positions used.)

The fact that the cut rule is valid in the Gentz ​​type calculi, says the Gentzensche law.