Contradiction
A proposition unconditionally false, central to logical proof.
In traditional logic, a contradiction arises when a proposition conflicts with itself or with established fact, serving as a tool to expose disingenuous beliefs and bias. This idea is grounded in the law of noncontradiction, which holds that the same thing cannot both belong and not belong to the same object in the same respect. The historical need for the concept of contradiction is demonstrated in Plato’s *Euthydemus* dialogue, where the character Dionysodorus denies the existence of contradiction even as Socrates contradicts him, leading to a paradox that highlights the necessity of the notion. In modern formal logic and type theory, the term is used primarily for a single proposition, often symbolized by the falsum ⊥; a proposition is a contradiction if false can be derived from it using the logic’s rules. This can be generalized to a collection of propositions that is said to contain a contradiction. In classical logic, a proposition φ is a contradiction if and only if it is provably false, and from a contradictory set of axioms, any proposition can be proven—a principle known as explosion, or *ex falso quodlibet*. In a complete logic, a formula is contradictory if and only if it is unsatisfiable. Proof by contradiction is a key technique: for consistent premises Σ and a proposition φ, Σ proves φ if and only if Σ together with the negation of φ leads to a contradiction. This method is especially useful when direct proof is difficult, as in the classic proof that √2 is irrational. Various logical systems handle contradiction differently. Minimal logic, which lacks explosion and proof by contradiction, can be extended with axioms like double-negation elimination to yield classical logic, or with *ex falso quodlibet* to yield intuitionistic logic. Other extensions include Peirce’s rule, the Gödel-Dummett axiom, the law of the excluded middle, and the weak law of the excluded middle, each producing distinct intermediate logics. In proofs, contradictions are symbolized by marks such as ↯, ⊥, or ※, often followed by Q.E.D. to indicate that an assumption has been refuted. A consistency proof for an axiomatic system requires demonstrating that no formula and its negation can both be derived.
- field
- Logic
- known_for
- Law of noncontradiction; principle of explosion; proof by contradiction
- symbol
- ⊥ (falsum)
- related_concepts
- Ex falso quodlibet, law of excluded middle, double-negation elimination
Lore & Background
In traditional logic, a contradiction arises when a proposition conflicts either with itself or with established fact, serving as a tool to detect disingenuous beliefs and bias. This concept is governed by the law of noncontradiction, which states that the same thing cannot both belong and not belong to the same object in the same respect. In modern formal logic and type theory, a contradiction is a single proposition, often denoted by the falsum symbol, that is unconditionally false—meaning false can be derived from it using the logic's rules. A collection of propositions is said to "contain" a contradiction if such a proposition follows. Historically, Plato's *Euthydemus* dialogue demonstrates the need for the notion of contradiction through a paradox: Dionysodorus denies the existence of contradiction even while being contradicted by Socrates, who asks how he can refute someone who claims false opinion is impossible. In classical logic, a proposition is a contradiction if and only if it is unsatisfiable, and from a contradictory set of axioms, any proposition can be proven—a principle known as *ex falso quodlibet* ("from falsity, anything follows"). This underpins proof by contradiction, a technique where assuming a proposition's negation leads to a contradiction, thereby proving the proposition true under consistent premises. Various logical systems handle contradiction differently: minimal logic lacks *ex falso quodlibet* and proof by contradiction, while adding double-negation elimination yields classical logic, and adding *ex falso quodlibet* yields intuitionistic logic. Symbols used to represent a contradiction in proofs include ↯, ⊥, and ※, often followed by Q.E.D. to indicate the original assumption was false.
Reader's Guide
In classical logic, a proposition φ is a contradiction if and only if φ ⊢ ⊥. Since for contradictory φ it is true that ⊢ φ → ψ for all ψ (because ⊥ ⊢ ψ), one may prove any proposition from a set of axioms which contains contradictions. This is called the 'principle of explosion', or 'ex falso quodlibet'. In a complete logic, a formula is contradictory if and only if it is unsatisfiable. For a set of consistent premises Σ and a proposition φ, it is true in classical logic that Σ ⊢ φ if and only if Σ ∪ {¬φ} ⊢ ⊥. This forms the basis of proof by contradiction, which mathematicians use extensively. Using minimal logic, various extensions yield intermediate logics: double-negation elimination yields classical logic; ex falso quodlibet yields intuitionistic logic; Peirce's rule captures proof by contradiction without explicitly referring to absurdity.
Did You Know?
- The law of noncontradiction states that 'It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect.'
- In Plato's Euthydemus, Dionysodorus denies the existence of contradiction while Socrates contradicts him.
- The principle of explosion, or ex falso quodlibet, means 'from falsity, anything follows'.
- Proof by contradiction is used to prove that √2 is irrational.
Frequently Asked Questions
What is Contradiction in logic?
Contradiction is a proposition that is unconditionally false — one from which you can derive the falsum using the rules of the system. It represents the logical bottom, the state where no consistent information can be recovered.
What symbol do logicians use for Contradiction?
In modern formal logic and type theory, it is written as the falsum symbol ⊥. This single glyph marks the proposition as the one that, once reached, lets you derive any other statement at all.
What is the principle of explosion and why does it matter for Contradiction?
Also called ex falso quodlibet, it holds that from a contradiction you can prove literally any proposition. This makes Contradiction the most destructive state in a logical system, because it erases every meaningful distinction between true and false.
How does Contradiction function inside a proof by contradiction?
You temporarily assume the negation of your target claim and push the derivation forward until ⊥ appears. Once that contradiction is reached, you discharge the assumption and conclude the original statement must hold.
What's the relationship between Contradiction and the Law of Noncontradiction?
The Law of Noncontradiction is the governing rule that no proposition and its negation can both be true at once. Contradiction (⊥) is the concrete object that surfaces inside a derivation when that law has been violated.
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