The paper highlights the necessity to question the exclusivity of classical logic, or of approaches that are reducible to it, in the analysis of classroom proof and argumentation processes, as well as the role of the set-theoretic language as intrinsically linked to classical logic. Two examples drawn from mathematics classroom are analysed, recurring to the Ancient Indian empiricist Nyaya logic and to Peirce's non-standard quantification, associating the last to a "free logic", not axiomatizable within an axiomatic system where the specification axiom applies.