Beth definability and the Stone-Weierstrass Theorem

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 $\vDash_{\Delta}$.