Logical consequence (FOL)
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- Note: semantic conclusion (FOL) redirects here.
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
- If [ilmath]\Gamma\models A[/ilmath] then the formula set [ilmath]\Gamma\cup\{\neg A\} [/ilmath] is not satisfiable
- Equivalent formulas - a case of valid formulas involving [ilmath]\leftrightarrow[/ilmath]
