Paradoxes, logical
The ancient paradox of Epimenides the Cretan, who said that all Cretans were liars (i.e., absolutely incapable of telling the truth), was known under numerous variant forms in ancient and medievd times The medieval name for these was insolubilia.
A form of this paradox due to Jourdain (1913) supposes a card upon the front of which are written the words, “On the other side of this card is written a true statement” — and nothing else. It seems to be clear that these words constitute a significant statement, since, upon turning the card over one must either find some statements written or not, and, in the former case, either there will be one of them which is true or there will not. However, on turning the card over there appear the words. “On the other side of this card is written a false statement” — and nothing else. Suppose the statement on the front of the card is true, then the statement on the back must be true, and hence the statement on the front must be false. This is a proof by reductio ad absurdum that the statement on the front of the card is false. But if the statement on the front is false, then the statement on the back must be false, and hence the statement on the front must be true. Thus the paradox.
A related but different paradox is Grelling’s (1908). Let us distinguish adjectives — ie, words denoting properties — as autological or i according as they do or do not have the property which they denote (in particular, adjectives denoting properties which cannot belong to words at all will be heterological). Then, e.g., the words polysyllabic, common, significant, prosaic are autological, while new, alive, useless, ambiguous, long are heterological. On their face, these definitions of autological and heterological are unobjectionable (compare the definition of onomatopoetic as similar in sound to that which it denotes). But paradox arises when we ask whether the word heterological is autological or heterological.
That paradoxes of this kind could be relevant to mathematics first became clear in connection with the paradox of the greatest ordinal number, published by Burali-Forti in 1897, and the paradox of the greatest cardinal number, published by Russell in 1903. The first of these had been discovered by Cantor in 1895, and communicated to Hilbert in 1896, and both are mentioned in Cantor’s correspondence with Dedekind of 1899, but were never published by Cantor.
From the paradox of the greatest cardinal number Russell extracted the simpler paradox concerning the class t of all classes x such that ~ x?x. (Is it true or not that t?t?) At first sight this paradox may not seem to be very relevant to mathematics, but it must be remembered that it was obtained by comparing two mathematical proofs, both seemingly valid, one leading to the conclusion that there is no greatest cardinal number, the other to the conclusion that there is a greatest cardinal number. — Russell communicated this simplified form of the paradox of the greatest cardinal number to Frege in 1902 and published it in 1903. The sime paradox wis discovered independently by Zermelo before 1903 but not published.
Also to be mentioned are Knig’s paradox (1905) concerning the least undefinable ordinal number and Richard’s paradox (1905) concerning definable and undefinable real numbers.
Numerous solutions of these paradoxes have been proposed. Many, however, have the fault that, while they purport to find a flaw in the arguments leading to the paradoxes, no effective criterion is given by which to discover in the case of other (e.g., mathematical) proofs whether they have the same flaw.
Russell’s solution of the paradoxes is embodied in what is now known as the ramified theory of types, published by him in 1908, and afterwards made the basis of Principia Mathematica. Because of its complication, and because of the necessity for the much-disputed axiom of reducibility, this has now been largely abandoned in favor of other solutions.
Another solution — which has recently been widely adopted — is the simple theory of types (see Logic, formal, 6). This was proposed as a modification of the ramified theory of types by Chwistek in 1921 and Ramsey in 1926, and adopted by Carnap in 1929.
Another solution is the Zermelo set theory (see Logic, formal, 9), proposed by Zermelo in 1908, but since considerably modified and im proved.
Unlike the ramified theory of types, the simple theory of types and the Zermelo set theory both require the distinction (first made by Ramsey) between the paradoxes which involve use of the name relation (q.v.) or the semantical concept of truth (q.v.), and those which do not. The paradoxes of the first kind (Epimenides, Grelling’s, Knig’s, Richard’s) are solved by the supposition that notations for the name relation and for truth (having the requisite formal properties) do not occur in the logistic system set up — and in principle, it is held, ought not to occur. The paradoxes of the second kind (Burali-Forti’s, Russell’s) are solved in each case in another way. — Alonzo Church
G. Frege,
Grundgesetze der Anthmetik, vol 2, Jena, 1903 (see Appendix).
B. Russell,
The Principles of Mathematics, Cambridge, England, 1903; 2nd edn. London, 1937, and New York, 1938.
Grelling and Nelson,
Bemerkungen zu den Paradoxieen von Russell und Burali-Forti, Abhandlungen der Fries’schen Schule, n.s. vol 2 (1908), pp 301-334.
A. Rstow,
Der Lgner (Dissertation Erlangen 1908), Leipzig, 1910.
P. E. B. Jourdain,
Tales with philosophical morals, The Open Court, vol 27 (1913), pp. 310-315.