Contradiction
| Type | Logical concept |
| Symbol | ⊥ (falsum) |
| Origin | Ancient Greece |
| Chinese | 矛盾 (máodùn) |
| Japanese | 矛盾 (mujun) |
| Related | Paradox, Antinomy |
Etymology
The word contradiction derives from the Latin contradictionem (nominative contradictio), meaning "a speaking against." In Chinese and Japanese, the concept is expressed as 矛盾 (máodùn / mujun), literally "spear-shield," from an ancient story about a merchant who claimed to sell a spear that could pierce any shield and a shield that could block any spear.[1]
When asked what would happen if the spear struck the shield, the merchant had no answer. This story, recorded in Han Feizi (3rd century BCE), became the standard term for logical contradiction in East Asian languages.[2]
The Liar's Paradox
The liar's paradox is the statement "This sentence is false." Attributed to Eubulides of Miletus (4th century BCE), it is one of the oldest known logical paradoxes.[3]
If the sentence is true, then what it says must hold -- but it says it is false, creating a contradiction. If the sentence is false, then the opposite of what it says must hold -- but the opposite of "this sentence is false" is "this sentence is true," again creating a contradiction.
"A man says that he is lying. Is what he says true or false?"
-- Eubulides of MiletusModern approaches include Tarski's hierarchy of metalanguages and paraconsistent logic, which permits true contradictions (see dialetheism).
Russell's Paradox
Russell's paradox, discovered by Bertrand Russell in 1901, concerns the set of all sets that do not contain themselves.[4] Formally:
Does R contain itself? If R ∈ R, then by definition R ∉ R. If R ∉ R, then by definition R ∈ R. The paradox demonstrated that naive set theory was inconsistent, leading to the development of axiomatic set theory.
Zeno's Paradoxes
Zeno's paradoxes are a set of arguments by Zeno of Elea (c. 490-430 BCE) that appear to show that motion is impossible.[5]
The Dichotomy argument: to traverse any distance, one must first traverse half of it; to traverse that half, one must first traverse half of that; and so on infinitely. Therefore, motion requires completing infinitely many tasks in finite time.
The resolution via convergent series (developed two millennia later) shows that the infinite sum 1/2 + 1/4 + 1/8 + ... = 1, but whether this mathematical fact truly explains motion remains debated.[6]
Ship of Theseus
The Ship of Theseus is a thought experiment about identity: if all the planks of a ship are gradually replaced, is the resulting ship the same ship?[7]
An additional complication (attributed to Thomas Hobbes): if the removed planks are used to build a second ship, which ship is the "real" Ship of Theseus?
See also
References
- Han Feizi, The Collected Works, Chapter 36, "Contradictions" (3rd century BCE).
- Graham, A.C. Disputers of the Tao. Open Court, 1989.
- Mates, Benson. The Skeptic Way: Sextus Empiricus. Oxford, 1996.
- Russell, Bertrand. "Letter to Frege," 16 June 1902.
- Aristotle. Physics, Book VI.
- Salmon, Wesley. Zeno's Paradoxes. Hackett, 2001.
- Plutarch. Theseus, 75 CE.