  The Stone-Weierstrass Theorem for compact Hausdorff spaces is a basic result
of functional analysis with far-reaching consequences. We introduce an
equational logic $\vDash_{\Delta}$ associated with an infinitary variety
$\Delta$ and show that the Stone-Weierstrass Theorem is a consequence of the
Beth definability property of $\vDash_{\Delta}$, stating that every implicit
definition can be made explicit. Further, we define an infinitary propositional
logic $\vdash_{\Delta}$ by means of a Hilbert-style calculus and prove a strong
completeness result whereby the semantic notion of consequence associated with
$\vdash_{\Delta}$ coincides with $\vDash_{\Delta}$.
