Math 2.0

Mathematical logic and foundations Theorem

Negation of a universal statement

Builds on Universal quantifierExistential quantifierNegation

Statement

Let XX be a set and let P(x)P(x) be a predicate on XX. In classical logic, the negation of ∀x∈X, P(x)\forall x \in X, \, P(x) is logically equivalent to ∃x∈X, ¬P(x)\exists x \in X, \, \neg P(x). ¬(∀x∈X, P(x))≡∃x∈X, ¬P(x)\neg \bigl( \forall x \in X, \, P(x) \bigr) \equiv \exists x \in X, \, \neg P(x)

Remarks

The direction from ∃x, ¬P(x)\exists x, \, \neg P(x) to ¬∀x, P(x)\neg \forall x, \, P(x) is valid in intuitionistic logic. The other direction needs classical logic. This law explains why one counterexample refutes a universal statement. For a finite XX, it is a form of De Morgan's laws.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Proposition, concept, no evidence levelPropositionconceptNegation, definition, no evidence levelNegationdefinitionPredicate, definition, no evidence levelPredicatedefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionNegation of a universal statement, statement, no evidence levelNegation of a univers…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Proposition concept
  2. Negation definition
  3. Predicate definition
  4. Existential quantifier definition
  5. Universal quantifier definition
  6. Negation of a universal statement statement

Where this leads

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Further reading

  1. Web https://en.wikipedia.org/wiki/Quantifier_(logic), lead section
About this page
Tag0P3A06QNDP
URIm2://0P3A06QNDP
Kindstatement, theorem
WritingW0 stub
MSC03B10
StatusThis copy of the site has no build data. Run m2 status 0P3A06QNDP.
UsesUniversal quantifier, Existential quantifier, Negation

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