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.[1] 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. (Barnes[2] claims in passing that the term 'consequentia mirabilis' refers only to the inference of the proposition from the inconsistency of its negation, and that the term 'Lex Clavia' (or Clavius' Law) refers to the inference of the proposition's negation from the inconsistency of the proposition.)

In formal notation: ${\displaystyle (\neg A\rightarrow A)\rightarrow A}$ which is equivalent to ${\displaystyle (\neg \neg A\lor A)\rightarrow A}$.

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 (i.e. in order to justify this view); in any case, therefore, we ought to philosophise."[3]

See also

• Ex falso quodlibet
• Tertium non datur
• Peirce's law

References

1. Sainsbury, Richard. Paradoxes. Cambridge University Press, 2009, p. 128.
2. Barnes, Jonathan. The Pre-Socratic Philosophers: The Arguments of the Philosophers. Routledge, 1982, p. 217 (p 277 in 1979 edition).
3. Kneale, William (1957). "Aristotle and the Consequentia Mirabilis". The Journal of Hellenic Studies. 77 (1): 62–66. doi:10.2307/628635. JSTOR 628635.