Antinomies

Abstract

An antinomy [a paradox] of logic takes place when two contradictory statements A and ~ A are derived, or equivalently A ≡ (~ A) is derived, without committing a simple logical error. In the area of deductive reasoning such a contradiction is a disaster, but, if genuine, it shows conflicts in our intuitions, incompatibility of areas of applicability of our notions. In the present exposition we shall restrict ourselves to some antinomies important for logic, leaving aside such paradoxes as those of Zeno of Elea concerning continuity and infinity (and the infinitely small), some of the Megarian paradoxes like the Heap or the Horned Man etc. Some of the paradoxes, the so-called logical or set-theoretic ones (see §2) as well as semantic ones (see §3) have been important for the development of logic and of the foundations of mathematics. Their common feature is the use of a kind of circularity in an inadmissible way. Extensive treatment of paradoxes can be found in Kleene (52), Beth (59), Fraenkel et al. (73).