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 $\mathscr{L}$ be a first order language. Let $A\in\mathscr{L}_F$ be a formula of $\mathscr{L}$ and let $\Gamma\subseteq\mathscr{L}_F$ be a collection of formulas of $\mathscr{L}$. We say:

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

See next

• Logical consequence (AKA: Semantic conclusion)