Resolving Infinitary Paradoxes

Graph normal form, GNF, [1], was used in [2, 3] for analysing paradoxes in propositional discourses, with the semantics – equivalent to the classical one – defined by kernels of digraphs. The paper presents infinitary, resolution-based reasoning with GNF theories, which is refutationally complete for the classical semantics. Used for direct (not refutational) deduction it is not explosive and allows to identify in an inconsistent discourse, a maximal consistent subdiscourse with its classical consequences. Semikernels, generalizing kernels, provide the semantic interpretation. §1. Motivation and overview. An informal discourse, represented by just writing its statements in some logical language, can be analyzed for consistency or validity, but hardly for paradoxicality. For paradox does not amount to the inconsistency of the discourse but of its truth-theory, which means here, roughly, the collection of T-schemata for discourse’s statements, [3]. There is nothing paradoxical about a ∧ ¬a. Its propositional T-schema, f ↔ (a ∧ ¬a), is unproblematic, classifying this statement, called now f, as false. When there are no references between statements, the truth-theory becomes such a trivially satisfiable repetition of each statment in an equivalence to its unique identifier. When statements refer to statements, identifiers become essential already for their representation. The truth-teller becomes at once t ↔ t, the liar l ↔ ¬l, and the truth-theory may become inconsistent. Classical provability of everything from such an inconsistent theory makes all its statements, so to speak, equally paradoxical. This is easily found unsatisfactory. The discourse D, to the left below, consists of Yablo’s paradox and three statements (a)-(c). Its truth-theory T is given to the right: (Y) Yablo’s paradox {yi ↔ ∧ j>i ¬yj | i ∈ N} (a) All statements in (Y) are false. a↔ ∧ i∈N ¬yi (b) All statements in (Y) and (c) are false. b↔ (¬c ∧ ∧ i∈N ¬yi) (c) Earth is round. c↔ 1 ((c) is true) One can accept that (a) is a paradox because of (Y), though even this could be disputed. It is a bit harder to accept paradoxicality of (b) which, denying a true claim (c), can be considered false, irrespectively of (Y). But even granting that (b) is a (part of the) paradox, too, there seems to be no reason whatsoever why Yablo’s paradox should affect also the indisputability of Earth’s roundness. c © 0000, Association for Symbolic Logic 0022-4812/00/0000-0000/$00.00