Logical consequence (FOL)

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Definition

Let [ilmath]\mathscr{L} [/ilmath] be a first order language. Let [ilmath]A\in\mathscr{L}_F[/ilmath] be a formula of [ilmath]\mathscr{L} [/ilmath] and let [ilmath]\Gamma\subseteq\mathscr{L}_F[/ilmath] be a collection of formulas of [ilmath]\mathscr{L} [/ilmath]. If for any model, [ilmath](\mathbf{M},\sigma)[/ilmath], of [ilmath]\mathscr{L} [/ilmath] we have[1]:

  • if [ilmath]\mathbf{M}\models_\sigma\Gamma[/ilmath] holds then [ilmath]\mathbf{M}\models_\sigma A[/ilmath] holds (recall [ilmath]\mathbf{M}\models_\sigma A[/ilmath] denotes satisfiability)

then we say that [ilmath]A[/ilmath] is a logical consequence[1] or semantic conclusion[1] of [ilmath]\Gamma[/ilmath].

  • We may denote this by: [ilmath]\Gamma\models A[/ilmath][1], also we may say that [ilmath]\Gamma\models A[/ilmath] is valid[1]

See next

References

  1. 1.0 1.1 1.2 1.3 1.4 Mathematical Logic - Foundations for Information Science - Wei Li

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