Concept
Consequentia mirabilis
Overview
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 related to reductio ad absurdum, 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:
(
¬
A
→
A
)
→
A
{\displaystyle (\neg A\to A)\to A}
.
Weaker variants of the principle are provable in minimal logic, but the full principle itself is not provable even in intuitionistic logic.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
Ut enim Epicurus veretur ne si hoc concesserit, concedendum sit, fato fieri quaecunque fiant; si enim alterum ex aeternitate verum sit, esse id etiam certum; si certum, etiam necessarium; ita et necessitatem et fatum confirmari putat; sic Chrysippus metuit ne non, si non obtinuerit omne[230] quod enuncietur aut verum esse aut falsum, omnia fato fieri possint ex causis aeternis rerum futurarum.' M. Bayle observes…
Gottfried Wilhelm Leibniz, Theodicy Essays on the Goodness of God, the Freedom of Man and the Origin of Evil
States what Consequentia mirabilis is.
Talia autem neque esse neque videri possunt eorum ratione, qui illa visa, e quibus omnia decreta sunt nata, negant quicquam a falsis interesse. 28. Ex hoc illud est natum, quod postulabat Hortensius, ut id ipsum saltem perceptum a sapiente diceretis, nihil posse percipi. Sed Antipatro hoc idem postulanti, cum diceret ei, qui adfirmaret nihil posse percipi, consentaneum esse unum tamen illud dicere percipi posse, ut…
Marcus Tullius Cicero, Academica
Close in meaning to Consequentia mirabilis; the term itself is not named.
To certain facts, certain facts always do, and, as we believe, will continue to, succeed. The invariable antecedent is termed the cause; the invariable consequent, the effect. And the universality of the law of causation consists in this, that every consequent is connected in this manner with some particular antecedent, or set of antecedents. Let the fact be what it may, if it has begun to exist, it was preceded by…
John Stuart Mill, A System of Logic
States what Consequentia mirabilis is.
Related ideas
- Law of identity
- Intuitionistic logic
- Law of noncontradiction
- Law of excluded middle
- Latin
- Modus ponens
- Reductio ad absurdum
- Minimal logic
- Frege's theorem
- Peirce's law
- Classical logic
- Principle of explosion
- Conditional proof
- Disjunctive syllogism
- Contraposition
- Negation introduction
Place in the tradition
Part of Intermediate logic.