Proofs by handling polynomials: a tool for teaching logic and metalogic

1 Polynomials as proof devices Algebraic proof systems based on formal polynomials over algebraically closed fields (the “polynomial ring calculus”) were introduced in [9] (see [10] and [11] for recent developments). Formal polynomials work as a powerful tool for logical derivation in classical and non-classical logics, in particular for propositional many-valued logics, paraconsistent logics and modal logics. Although the case of first-order logic (FOL) is still work in progress, polynomial ring calculus have been obtained for the monadic fragment of FOL and offer a nice view of syllogistic logic that permits to reassess ideas of G. Boole on the unity between algebra and logic. For the particular case of classical propositional calculus (PC) a direct formulation of propositional derivability can be obtained by translating the usual Boolean connectives as follows: Let At = {p1, p2, . . .} be the atomic sentences of PC, and ¬,∨,∧,→ the usual connectives. The translation is part of the logic folklore, and perhaps because it is so intuitive its generalization towards other logics has never been explored in full generality. The polynomial rules over Z2[X] for the case of PC are just x + x `≈ 0 and x · x `≈ x. Based on such rules and on the elementary algebraic and combinatorial properties of the ring Z2[X] it can be easily shown that φ is a PCtautology iff Π(φ) `≈ 1, or, in other words, φ is a PC-tautology iff such reduction rules end up at the element 1. For instance, the sentence α→ (¬α), supposing α atomic, is translated by Π above into x · (x+ 1) + x+ 1. The reduction rules ∗Supported by CNPq and Project LogCons-FAPESP (process 10/51038-0)