Modus ponens

The modus ponens is a final figure already familiar in ancient logic, which is used as a final rule in many logical systems (see logic , calculus ) . It allows a statement of form B (the conclusion of the final figure) to be derived from two statements of the form (If A, then B) and ( A) (the two premises of the final figure).

The technically correct name for the modus ponens is - in contrast to the modus tollendo ponens - modus ponendo ponens . The expressions cut-off rule or elimination of implications are used synonymously . In semi-formal calculi, the final rule is often abbreviated as MP .

etymology

The term modus ponens is derived from the Latin words modus (here: final figure) and ponere (place, set) and means final figure , i.e. H. Final figure from which a positive statement is derived.

The full Latin name, modus ponendo ponens , "final figure (modus) that sets another statement (ponens) by setting (ponendo) a statement" can be explained as follows: given the first premise, "If A, then B ", by" setting "(assuming) the second premise, A, which is" set "(derived) from the two following sentence B.

It corresponds to one of the five types of hypothetical syllogism according Chrysippus : 'When the first, the second; but the first; so the second '.

Forms and example

As a final form

Scheme

example

	${\ displaystyle A \ to B}$
	${\ displaystyle A}$
modus ponendo ponens	${\ displaystyle B}$

	When it rains, the road gets wet.
	It's raining.
modus ponendo ponens	The road is getting wet.

From the premises of the form and the conclusion is drawn . ${\ displaystyle A \ to B}$ ${\ displaystyle A}$ ${\ displaystyle B}$

Formally, the modus ponens is noted with the derivative operator as the final rule . ${\ displaystyle \ vdash}$ ${\ displaystyle A, A \ to B \ vdash B}$

As a statement

Although the modus ponendo ponens is a final rule, i.e. a metalinguistic concept, the term "modus ponens" is occasionally used for object language expressions with the following form:

(A ∧ (A → B)) → B

But since inference rules and statements are very different concepts, it is scientifically rather unfortunate to name them with the same name. In general, mixing object and metalanguage is problematic and should normally be avoided.

As a rule of subjunction elimination

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

→ Separation rule: (A → B), A ⇒ B

As a rule of thumb

In the metalogical version it is the cutting rule:

{\ displaystyle \ qquad {\ frac {\ Gamma \ | A \ qquad A, \ Delta \ | B} {\ Gamma, \ Delta \ | B}}}

(Here the double line || is used for the lockability of dialog positions .)

Gentzen's law states that the rule of intersection is permissible in the Gentzen type calculus .

Treatment in strict logic

The following paragraph describes the treatment of modus poning in Strict Logic (by Walther Brüning ):

The first premise is a condition of dyadic (ie two facts are connected) level (therefore capital letter). The second premise is a dyadic elongated henadic (i.e., relating to a matter) formula (hence lowercase letters and equations); likewise the conclusion. (The bold a is included in the conclusion because the other a of the equation is blocked by an n .). The superscript c means condicio :

	1	2	3	4th
	F G	~ F G	F ~ G	~ F ~ G
${\ displaystyle F \ supset ^ {c} G}$	u	u	N	u
${\ displaystyle F}$	a	u	a	u

${\ displaystyle G}$	a	a	u	u

The columns are separated according to the various possible combinations. The extension of the validity formulas of the premise or the conclusion results from the introduction of the second term or . Their value therefore remains the same with enlargement (equals are formed). " a " - short for affirmation (corresponds to true ); " N " - short for negation (corresponds to false ); " u " - short for indefinite ; " ~ X " - short for complement of X . ${\ displaystyle F}$ ${\ displaystyle G}$ ${\ displaystyle G}$ ${\ displaystyle F}$

Web links

Metamath link

Individual evidence

↑ See Peter Thom : Syllogismus; Syllogistics. in: Historical Dictionary of Philosophy Vol. 10, p. 695
↑ Basics of strict logic. Walther Brüning. Wuerzburg 1996.

[1] See Peter Thom : Syllogismus; Syllogistics. in: Historical Dictionary of Philosophy Vol. 10, p. 695

[2] Basics of strict logic. Walther Brüning. Wuerzburg 1996.