A note on incompleteness and heterologicality
Peter M. Sullivan · Analysis · 2003
Define 'het' as a predicate that truly applies to itself if and only if it does not truly apply to itself and which also truly applies to any predicate that does not truly apply to its own name. We know that the attempted definition of 'hes' is a failure, and so a fortiori is that of 'het'. Similarly, there is no Qussell class which contains itself as a member if and only if it does not contain itself as a member, so a fortiori there is no Russell Class which contains itself as a member if and only if it does not contain itself as a member and which also contains all and only non-self-membered classes (such as the class of dogs). The second conjunct in both the definition of 'het' and of the Russell class cannot revive a definition doomed to failure. Likewise, the 'definition' of n as 'n > 1 iff n 1 iff m < 1 and m is prime' is hopeless too; its final clause buys it no respectability. But 'het' is just an abbreviation for Grelling's predicate 'heterological' and since, as we have seen, this predicate has been given no meaning, sentences such as 'long is heterological', 'heterological is heterological' are meaningless; and so can be neither true nor false. But Grelling's Paradox works by deriving a contradiction from the assumption that 'heterological is heterological' is either true or false. Since, as we have seen, that assumption is untenable, we can bid a fond, if curt, farewell to Grelling.