Consequentia mirabilis


Consequentia mirabilis, 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:
which is equivalent to.
Consequentia mirabilis was a pattern of argument popular in 17th century Europe that first appeared in a fragment of Aristotle's Protrepticus: "If we ought to philosophise, then we ought to philosophise; and if we ought not to philosophise, then we ought to philosophise ; in any case, therefore, we ought to philosophise."
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 that they are thinking.