Infinitary logic and basically disconnected compact Hausdorff spaces

We extend Lukasiewicz logic obtaining the infinitary logic Infinitary Riesz Logic (IRL) whose models are algebras C(X, [0, 1]), where X is a basically disconnected compact Hausdorff space. Equivalently, our models are unit intervals in Dedekind sigma-complete Ries[spaces with strong unit. The Lindenbaum-Tarski algebra of IRL is, up to isomorphism, an algebra of [0, 1]-valued Borel functions. Finally, our system enjoys standard completeness with respect to the real interval [0, 1]