Math 2.0

Mathematical logic and foundations Definition

Logical consequence

Builds on Propositional logicWell-formed formulaStructure

Definition

Let Γ\Gamma be a set of formulas and φ\varphi a formula, all over the same signature. Then φ\varphi is a logical consequence of Γ\Gamma, written Γ⊨φ\Gamma \models \varphi, if every interpretation that makes every formula of Γ\Gamma true also makes φ\varphi true. In propositional logic an interpretation is a truth assignment to the variables. In first-order logic it is a structure together with an assignment of its elements to the variables.

Remarks

{p,  p→q}⊨q\set{p, \; p \to q} \models q: in every assignment where pp and p→qp \to q are true, qq is true by the truth table of →\to. {p→q,  q}⊭p\set{p \to q, \; q} \not\models p: the assignment p=𝖥p = \mathsf{F}, q=𝖳q = \mathsf{T} makes both premises true and pp false. ∅⊨φ\emptyset \models \varphi means that φ\varphi is valid, written ⊨φ\models \varphi. Γ⊨φ\Gamma \models \varphi holds exactly when Γ∪{¬φ}\Gamma \cup \set{\neg \varphi} is not satisfiable.

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 levelPropositionconceptStructure, concept, no evidence levelStructureconceptWell-formed formula, concept, no evidence levelWell-formed formulaconceptConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionPropositional logic, concept, no evidence levelPropositional logicconceptLogical consequence, definition, no evidence levelLogical consequencedefinition
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. Structure concept
  3. Well-formed formula concept
  4. Conjunction definition
  5. Disjunction definition
  6. Implication definition
  7. Negation definition
  8. Biconditional definition
  9. Propositional logic concept
  10. Logical consequence definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Logical_consequence
About this page
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Kinddefinition
WritingW0 stub
MSC03B05, 03B10
DefinesLogical consequence
UsesPropositional logic, Well-formed formula, Structure
Changed2 November 2026

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