''Consequentia mirabilis'' (
Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
for "admirable consequence"), also known as
Clavius
Christopher Clavius, SJ (25 March 1538 – 6 February 1612) was a Jesuit German mathematician, head of mathematicians at the Collegio Romano, and astronomer who was a member of the Vatican commission that accepted the proposed calendar inve ...
's Law, is used in
traditional
A tradition is a belief or behavior (folk custom) passed down within a group or society with symbolic meaning or special significance with origins in the past. A component of cultural expressions and folklore, common examples include holidays or ...
and
classical logic
Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy.
Characteristics
Each logical system in this class ...
to establish the truth of a proposition from the
inconsistency of its negation. It is thus related to ''
reductio ad absurdum
In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical arguments'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absu ...
'', but it can prove a proposition using just its own negation and the concept of consistency.
For a more concrete formulation, it states that if a proposition is a consequence of its negation, then it is true, for consistency. In formal notation:
:
.
Equivalent forms
Given
being equivalent to
, the principle is equivalent to
:
.
History
''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."
Barnes 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.
[Barnes, Jonathan. ''The Pre-Socratic Philosophers: The Arguments of the Philosophers''. Routledge, 1982, p. 217 (p 277 in 1979 edition).]
See also
*''
Ex falso quodlibet''
*''
Tertium non datur''
*''
Peirce's law
In logic, Peirce's law is named after the philosopher and logician Charles Sanders Peirce. It was taken as an axiom in his first axiomatisation of propositional logic. It can be thought of as the law of excluded middle written in a form that inv ...
''
References
{{DEFAULTSORT:Consequentia Mirabilis
Theorems in propositional logic
Latin logical phrases