The notion of truth-functionality seems to be essential for the analysis of logic systems, and the so-called non-truth-functional logics pose a challenge to the usual algebraically-based semantic tools.

I show that, by generalizing the Boolean setting towards polynomials over finite rings (in particular, Galois fields), we can in some sense bypass some difficulties of non-truth-functionality and offer a new view of abstract

truth-values, which brings together a unity between logic and algebra, online with some intuitions by Boole and Leibniz.

As motivating examples, I will show how semantics for several logics (as the ones by N. Belnap and J. M. Dunn, K. Gödel, J. Lukasiewicz and M. Sette) can be expressed by multivariable polynomials over Boolean rings, with happy consequences.

These ideas naturally apply to non-truth-functional paraconsistent logics (in particular to the LFI’s, or `logics of formal inconsistency’) which are uncharacterizable by finite-valued semantics.

Some challenging tasks, yet to be done, are to extend this method to full quantified logic, to higher-order logic and to modal logic, and to evaluate its connections to abstract algebraic logic.