Valid formula

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Created so I don't have to sift through notes or scour PDFs, needs more references and another book to source the info really

Note: Tautology (FOL) redirects here, as valid is another term for a tautological formula.

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]. We say:

• [ilmath]A[/ilmath] is valid if [ilmath]A[/ilmath] is satisfiable for any model of [ilmath]\mathscr{L} [/ilmath]. That is to say: [ilmath]\mathbf{M}\models_\sigma A[/ilmath] holds for any model [ilmath](\mathbf{M},\sigma)[/ilmath] of [ilmath]\mathscr{L} [/ilmath].
  • We denote this by: [ilmath]\models A[/ilmath]
  • A valid formula is also called a tautology
• Suppose that for every formula, [ilmath]B[/ilmath], in [ilmath]\Gamma[/ilmath] that [ilmath]B[/ilmath] is valid (or: [ilmath]\models B[/ilmath] holds), then we call the formula set [ilmath]\Gamma[/ilmath] valid
  • We denote this by: [ilmath]\models\Gamma[/ilmath]

See next

• Logical consequence (AKA: Semantic conclusion)