Logical consequence#Semantic consequence

If a formula follows semantically from another, so another formula for all possible assignments always returns the same result, so the semantic entailment relation is ( engl.: semantic entailment relation ) met. Formulas on the one hand ( the premises ) are derived in a formal calculus using the approved final rules in the calculus by syntactic transformations and inferences from other sets, this corresponds to the derivative ( logic). Alternatively, a derivation is also possible by means of a truth table, that is, whether another formula for all possible assignments always returns the same result, then the semantic result is met. Both relations are equally powerful and can be converted into each other, but it varies from case to case, which is easier to close the two relations.

Definition

If all assignments that satisfy even meet, so the relation is satisfied. It is called the semantic entailment relation.

Here it is important how the result looks at various assignments, so the logical consequence or the semantics (meaning) of the formula. Therefore, this relation can be checked using a truth table.

Syntactic derivation and semantic consequence

The syntactic derivation is as follows:

Thus, the expression is valid.

The semantic consequence is, however, verified on all assignments:

Whenever is satisfied, then it is. Thus follows.

More examples

Tautology

If a formula met in all assignments, so always 'true', it is a tautology:

This is the semantic counterpart of the theorem. If a formula never met, it is a contradiction ( contradiction ).

See also

  • Propositional logic semantics #
  • Correctness ( logic)
  • Completeness ( logic)

Swell

  • Michael Huth, Mark Ryan: Logic in Computer Science - Modelling and Reasoning about Systems, 2nd edition, Cambridge University Press, 2005, ISBN 0 521 54310X site
  • Uwe Schöning: Logic for Computer Scientists, 5th edition, Oxford University Press, 2005, p 18
  • Logic
Tautology (logic) Soundness
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