Definition:Satisfiable/Set of Formulas
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Definition
Let $\LL$ be a logical language.
Let $\mathscr M$ be a formal semantics for $\LL$.
A collection $\FF$ of logical formulas of $\LL$ is satisfiable for $\mathscr M$ if and only if:
- There is some $\mathscr M$-model $\MM$ of $\FF$
That is, there exists some structure $\MM$ of $\mathscr M$ such that:
- $\MM \models_{\mathscr M} \FF$
Also known as
Some sources refer to satisfiable as semantically consistent.
It is sometimes convenient to refer to satisfiability for $\mathscr M$ in a single adjective.
In such cases, $\mathscr M$-satisfiable is often seen.
Also see
Sources
- 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\text{II}.7$ First-Order Logic Semantics: Definition $\text{II}.7.13$
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.5.2$: Definition $2.42$