Classical propositional logic is truth-functional, that is the truth of propositions are determined by the assignment of truth-values taken from {false,true} to the atomic propositions. And it is this that gives the semantics for the logic.
But noting that {false, true} is actually a boolean algebra, and in fact the smallest one, can we generalise to a B-valent classical propositional logics by taking values from some boolean algebra B, which need not be finite?
It seems to me that the usual completeness and soundness theorems should hold in this context. Is that right?