Consequentia mirabilis

Consequentia mirabilis (Latin for "admirable consequence"), also known as Clavius's Law, is used in traditional and classical logic to establish the truth of a proposition from the inconsistency of its negation. It is thus similar to reductio ad absurdum, but it can prove a proposition true using just its negation. It states that if a proposition is a consequence of its negation, then it is true, for consistency. It can thus be demonstrated without using any other principle, but that of consistency.

In formal notation: $(\neg A\rightarrow A)\rightarrow A$

For example: "There is no truth" (not-A), but this statement implies that it is a truth (A), therefore "there is some truth" (then A is true). Or even: "Nothing exists" implies that there is this statement, so "something there."

The most famous example is perhaps the Cartesian cogito ergo sum: Even if one can question the validity of the thinking, no one can deny the existence of thought.

Children have an uncanny appreciation for this law, and use it unwittingly on a regular basis: "I'm not talking to you" proves that, in saying this, the person is in fact talking to the other.

It is also known as Clavius' law, after the learned sixteenth century Jesuit Christopher Clavius, one of the designers of the Gregorian calendar, who first drew attention to the law in his commentary on Euclid.

See also

• Ex falso quodlibet
• Tertium non datur