en.wikipedia.org/wiki/Reductio_ad_absurdum
2 corrections found
such as in Book 7, Proposition 33:
The proposition quoted here is not from Euclid's Book VII. It is Book XIII, Proposition 9; Book VII, Proposition 33 is a number-theory proposition about finding least numbers in the same ratio.
Full reasoning
This is a book/proposition mismatch. The text immediately following this phrase quotes Euclid's proposition about the side of the hexagon and the decagon inscribed in the same circle. In standard editions of Elements, that statement is Book XIII, Proposition 9, not Book VII.
Authoritative editions confirm both sides of the mismatch:
- Clay Mathematics' text of Book XIII, Proposition 9 gives exactly the quoted statement: "If the side of the hexagon and that of the decagon inscribed in the same circle be added together..."
- Clay Mathematics' text of Book VII, Proposition 33 is a completely different proposition: "Given as many numbers as we please, to find the least of those which have the same ratio with them."
So the article's reference to Book 7, Proposition 33 is incorrect for the proposition it quotes below.
2 sources
- Euclid's Elements, Book XIII, Proposition 9 (Clay Mathematics Institute)
Book XIII, Proposition 9: "If the side of the hexagon and that of the decagon inscribed in the same circle be added together, the whole straight line has been cut in extreme and mean ratio..."
- Euclid's Elements, Book VII, Proposition 33 (Clay Mathematics Institute)
Book VII, Proposition 33: "Given as many numbers as we please, to find the least of those which have the same ratio with them."
The principle may be formally expressed as the propositional formula ¬¬P ⇒ P, equivalently (¬P ⇒ ⊥) ⇒ P,
In this context, the formula is misidentified. ¬¬P ⇒ P is double-negation elimination (a classical proof-by-contradiction principle), not the principle of non-contradiction.
Full reasoning
In the preceding section, "the principle" refers to Aristotle's principle of non-contradiction. But the formula given here, ¬¬P ⇒ P, is not the law of non-contradiction.
Authoritative logic sources distinguish these principles:
- The principle of non-contradiction is standardly formalized as ¬(P ∧ ¬P) — i.e., it is impossible for a statement and its negation to both hold.
- ¬¬P ⇒ P is the classical principle of double-negation elimination, which logic texts also describe as equivalent to proof by contradiction / reductio ad absurdum in classical logic.
So this sentence conflates two different logical principles: the law of non-contradiction and double-negation elimination.
3 sources
- Aristotle on Non-contradiction (Stanford Encyclopedia of Philosophy)
The first version (hereafter, simply PNC) ... runs as follows: "It is impossible for the same thing to belong and not to belong at the same time to the same thing and in the same respect".
- Law of the Excluded Middle (UT Austin)
Classical logic, following Aristotle, assumes both LEM and the Principle of Noncontradiction: For any statement P: ¬( P ∧ ¬ P ).
- Classical Reasoning - Logic and Proof
Proof by contradiction is also equivalent to the principle ¬ ¬ A ↔ A. The implication from right to left holds intuitionistically; the other implication is classical, and is known as double-negation elimination.