Consequentia mirabilis

The consequentia mirabilis ( admirable conclusion ), also known as Clavius ​​- law, is used in classical logic to prove the validity of an assertion of the invalidity of their negation. The argument is related to the reductio ad absurdum; However, one need only show at the consequentia mirabilis that the assertion follows from its negation. This method was used by Christopher Clavius ​​in the publication of Euclid's Elements. Later, Giovanni Girolamo Saccheri made ​​in his investigations of the syllogism in an original way by this method use.

The method of proof

You can see the argument informally as follows put into words:

• "If a claim follows from its opposite, then it is right."

Today the " consequentia mirabilis " is usually formulated as a formula of classical propositional logic. In this formulation, it states that the formula

Is a tautology, which can be seen from the following truth table:

The correctness of the formula can also be derived purely syntactic eg using the Principia Mathematica calculus for propositional logic:

• Is as defined
• So true
• The first of these axioms is

The assertion is proved.

Example

• "There is no truth " ( ), but this is indeed asserted as true ( p), that " there is some truth " (ie, p is true).
• Accordingly: From the statement, " I say nothing," follows that I still say something (namely, the claim that I say nothing ).

Swell

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