Math 2.0

Mathematical logic and foundations Theorem

Double negation law

Builds on NegationLogical equivalence

Statement

In classical logic, for each proposition pp, the proposition ¬¬p\neg \neg p is logically equivalent to pp. ¬¬p≡p\neg \neg p \equiv p

Remarks

The implication p→¬¬pp \to \neg \neg p is also valid in intuitionistic logic. The other direction, ¬¬p→p\neg \neg p \to p, needs classical logic. This direction is the basis of proof by contradiction.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Proposition, concept, no evidence levelPropositionconceptConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionTruth table, concept, no evidence levelTruth tableconceptPropositional logic, concept, no evidence levelPropositional logicconceptTautology, definition, no evidence levelTautologydefinitionLogical equivalence, definition, no evidence levelLogical equivalencedefinitionDouble negation law, statement, no evidence levelDouble negation lawstatement
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. Conjunction definition
  3. Disjunction definition
  4. Implication definition
  5. Negation definition
  6. Biconditional definition
  7. Truth table concept
  8. Propositional logic concept
  9. Tautology definition
  10. Logical equivalence definition
  11. Double negation law statement

Where this leads

29 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Double_negation, lead section
About this page
Tag0NFAGWH84K
URIm2://0NFAGWH84K
Aliasesdouble-negation-elimination
Kindstatement, theorem
WritingW0 stub
MSC03B05
StatusThis copy of the site has no build data. Run m2 status 0NFAGWH84K.
UsesNegation, Logical equivalence
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