<p>We present a hypersequent calculus that is sound and complete with respect to the truth-functionally contingent formulas of classical logic. We investigate its structural properties and provide a Gentzen-style cut-elimination procedure. The most notable feature of the calculus is that it jointly satisfies the subformula property and the property of <i>deductive purity</i>, to the effect that only contingent hypersequents occur in formal proofs. Moreover, since the negation of a contingent formula is also contingent, the calculus turns out to be <i>paraconsistent</i>, and since the conjunction of a formula with its own negation is not contingent, the paraconsistency is of the <i>non-adjunctive</i> kind.</p>