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A refutation of a collection of logical assumptions is a derivation of a contradiction from them. To prove a logical formula $\phi$ by refutation in classical propositional logic, add its negation $\neg \phi$ to the assumptions and derive falsity. The correctness of this inference follows from the rules of the chosen proof system.

For example, the clauses $p\vee q$, $\neg p$ and $\neg q$ have a resolution refutation: the first two resolve to $q$, and $q$ with $\neg q$ yields the empty clause. No truth assignment satisfies the three original clauses. This distinguishes the semantic claim of unsatisfiability from a syntactic certificate establishing it. In automated proving, producing a refutation is useful because another program can check the individual inference steps without repeating the search.

A logical formula or a collection of formulas is unsatisfiable when no permitted interpretation makes all of it true. The permitted interpretations come from the chosen semantics: Boolean valuations for propositional logic, or structures and variable assignments for first-order logic. Unsatisfiability is therefore a semantic property; a refutation is a syntactic proof of that property.

For example, $p\wedge \neg p$ is unsatisfiable because neither truth value of $p$ makes both parts true. The logical formula $p\vee q$ is satisfiable, although it is false in the valuation assigning false to both letters. Thus being false in one interpretation does not establish unsatisfiability. In classical propositional logic, $\phi$ is valid exactly when $\neg \phi$ is unsatisfiable. Likewise, $\Gamma \models \phi$ exactly when $\Gamma \cup \{\neg \phi \}$ is unsatisfiable. This turns entailment questions into searches for countermodels or contradiction certificates.