Abstract: I intend to show that by generalizing the usual Boolean approach to semantics by means of polynomials over finite rings (in particular, Galois fields) it is possible to represent semantics and proof theory for several truth-functional and non-truth-functional propositional logics, bringing back a unity between logic and algebra which vindicates some intuitions by Boole and Leibniz. As motivating examples, I will discuss completely elementary ways of proving in several many-valued, paraconsistent and modal logics obtained by handling multivariable polynomials over Boolean rings. Departing from an almost obvious observation that finite functions, and even non-deterministic functions, can be represented as power series the method naturally applies to certain logics which are uncharacterizable by finite-valued semantics. (in particular to the LFI's, or ``logics of formal inconsistency'' and to some modal logics). The resulting method might be of interest for informatics and automatic proof procedures. Some challenging problems in connection to this approach, as extensions of this method to intuitionistic logic, to full quantified logic, to modal logics in general and to infinite-valued logics (as Lukasiewicz logic), as well as connections to abstract algebraic logic, will also be discussed.