Logical consequence (FOL)

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Note: semantic conclusion (FOL) redirects here.

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 $\mathscr{L}$. If for any model, $(\mathbf{M},\sigma)$, of $\mathscr{L}$ we have:

• if $\mathbf{M}\models_\sigma\Gamma$ holds then $\mathbf{M}\models_\sigma A$ holds (recall $\mathbf{M}\models_\sigma A$ denotes satisfiability)

then we say that $A$ is a logical consequence or semantic conclusion of $\Gamma$.

• We may denote this by: $\Gamma\models A$, also we may say that $\Gamma\models A$ is valid

See next

• If $\Gamma\models A$ then the formula set $\Gamma\cup\{\neg A\}$ is not satisfiable
• Equivalent formulas - a case of valid formulas involving $\leftrightarrow$