Math 2.0

Mathematical logic and foundations Definition

Logical equivalence

Builds on Truth tableBiconditionalTautology

Definition

Let φ\varphi and ψ\psi be formulas of propositional logic. The formulas φ\varphi and ψ\psi are logically equivalent when they have the same truth value for each assignment of truth values to their variables. This is written φ≡ψ\varphi \equiv \psi.

Remarks

The formulas φ\varphi and ψ\psi are logically equivalent exactly when φ↔ψ\varphi \leftrightarrow \psi is a tautology. For example, p→q≡¬p∨qp \to q \equiv \neg p \lor q. De Morgan's laws give two more examples.

Reading path

To understand this page from the ground up, read these 10 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 equivalencedefinition
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

Where this leads

39 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Logical_equivalence, lead section
About this page
Tag06GYGS41JK
URIm2://06GYGS41JK
Kinddefinition
WritingW1 stated
MSC03B05
DefinesLogical equivalence
UsesTruth table, Biconditional, Tautology

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