Satisfiable formula

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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 $(\mathbf{M},\sigma)$ is a model of $\mathscr{L}$ such that:

• $A_{\mathbf{M}[\sigma]}=T$ (where $T$ represents "true" as an element of set of truth values and $A_{\mathbf{M}[\sigma]}$ represents the semantics of the formula $A$)

Then the formula, $A$, is said to be satisfiable under the model $(\mathbf{M},\sigma)$, or the model $(\mathbf{M},\sigma)$ satisfies $A$, or that $A$ is satisfiable if the model is understood; we denote this by:

• $\mathbf{M}\models_\sigma A$, if $A$ is a sentence then we may write just: $\mathbf{M}\models A$

If for every formula, $B$, in the collection $\Gamma$ of formulas of $\mathscr{L}$ under the model $(\mathbf{M},\sigma)$ we have $\mathbf{M}\models_\sigma B$ then:

• We may say the formula set $\Gamma$ is satisfiable under the model $(\mathbf{M},\sigma)$, the model $(\mathbf{M},\sigma)$ satisfies $\Gamma$ or $\Gamma$ is satisfiable if the model is understood, we denote this:
  • $\mathbf{M}\models_\sigma\Gamma$, if though every formula in $\Gamma$ is a sentence then we may write: $\mathbf{M}\models\Gamma$

See next

• Valid formula (AKA: Tautology)
  • Logical consequence (AKA: Semantic conclusion)