Abstract

This essay is an informal examination of the efforts made by mathematical logicians from Russell on, to evade the paradoxes revealed by uncritical methods of definition and predication. Where systems are described it is with a minimum of formal apparatus, so that the climate is one of rather vague statement and imprecise question. Russell’s drastic banning of type–offending formulas gives the starting–point for queries about the connection of sense and formalism, and the conclusion may be summed up as the detection of a contrast between mechanisation and sensefulness which is seen as constituting a line of tension, as it were a geological weakness in formal structures, which permits the insertion of a philosophic wedge. Interesting as such an idea is, it is not clear between what regions this line has been drawn. On the one hand the undecidability results of Gödel show that formal methods can be extended indefinitely beyond the scope of what is mechanisable; on the other the author methodically refrains from defining what he means by ‘sense’. In so far as he explicitly leaves his meaning on this point to be gathered form his treatment of non–sense, and the paradoxes are here taken as representative of the latter, we might suppose that he is equating sense with consistency. For expressions leading to paradox were only ruled senseless by Russell because they lead to inconsistency. But it is fairly clear that something more epistemologically interesting is intended, since the author dismisses from consideration ‘la déductivité structurale, sens immédiat d’une vaste architecture symbolique d’abord vide, et intégralement traduisible en syntaxe’. Such translatability has, incidentally, its limits, as Gödel’s results again show, and it is our belief that the discussion might have been put in a more exact and workmanlike frame if the list of works consulted had contained more than one, and that a relatively minor, publication of Gödel.