What Anselm and Gaunilo said to each otherIn this paper the author proposes a new formal reconstruction of the debate of St. Anselm of Canterbury and Gaunilo of Marmutier on the Proslogion argument for the existence of God. The reconstruction is purely extensional (in second order predicate logic with identity) and proves both that the argument is conclusive, and that at the same time, precisely because it is valid, it leads to an infinite metalogical cycle.