Consequentia mirabilis

The Consequentia mirabilis (Admirable Inference), also known as Clavius's Law , is used in classical logic to prove the validity of an assertion from the invalidity of its negation. The argument is related to the reductio ad absurdum ; however, in the case of the Consequentia mirabilis one only has to show that the assertion follows from its negation. This method was used by Christophorus Clavius in the context of the publication of the elements of Euclid. Giovanni Girolamo Saccheri later used this method in an original way in his studies of the syllogism .

The proof method

One can put the argument informally into words as follows:

• "If an assertion follows from its opposite, then it is correct."

Today the "Consequentia mirabilis" is mostly formulated as a formula of classical propositional logic . In this formulation it says that the formula

• ${\displaystyle (\neg p\to p)\to p}$

a tautology is what can be seen from the following truth table :

p	~ p	~ p → p	(~ p → p) → p
w	f	w	w
f	w	f	w

The accuracy of the formula can also be purely syntactically z. B. derive with the help of the Principia Mathematica calculus for propositional logic: ${\displaystyle (\neg p\to p)\to p}$

• By definition is ${\displaystyle p\to q:=\neg p\lor q}$
• So it applies ${\displaystyle \neg p\to p=p\lor p}$
• The first of these axioms is ${\displaystyle p\lor p\to p}$

The claim is thus proven.

example

• “There is no truth” ( ), but this is asserted to be true ( p ), so “there is some truth” (so p is true). ${\displaystyle \neg p}$
• Correspondingly: From the statement "I do not claim anything" it follows that I do claim something (namely the claim that I am claiming nothing).

swell

• William Kneale : Aristotle and the Consequentia Mirabilis . In: Journal of Hellenic Studies . Volume 77, 1957, pp. 62-66.

Individual evidence

1. ↑ Cyril FA Hoormann, Jr .: A Further Examination of Saccheri's use of the "consequentia mirabilis". Notre Dame Journal of Formal Logic Volume XVII, Number 2, April 1976, p. 239-274
2. ↑ s. the axiom system in propositional logic # axioms