[FOM] Self-reference in natual languages (re >>this sentence, cannot be proven true<<)

[Hartley Slater]

Is this problem just a natural language one?

Consider a formal treatment, e.g. Albert Visser's 'Semantics and the 
Liar Paradox' in the Handbook of Philosophical Logic, Vol IV, eds 
D.M.Gabbay and F. Guenther (Reidel 1989).   Visser says (p621) 'Now 
define a formula standing for "heterological(x)" by: -Sat(x, <x>.)' 
Certainly there is the natural language problem here that the 
diagonal predicate 'is not satisfied by its own name' contains the 
pronoun 'its', and so does not express a fixed property.  But there 
are also a pair of strictly mathematical problems, as well.  For the 
thinking, from Frege, has been that predicates are functions, and 
there is no doubt that, if one has a mathematical function of two 
variables 's(x, y)', then one can obtain, by stipulative definition, 
a function of just one variable in the manner : h(x) := s(x, x). 
Furthermore, the production of a fixed property 'being heterological' 
may seem entirely unproblematic to anyone acquainted with the lambda 
calculus, since, surely, -Sat(x, <x>) iff lambda-y(-Sat(y, <y>))[x], 
and so 'is heterological' is 'lambda-y(-Sat(y, <y>)[ ]'.  But, of 
course, following either of these mathematical lines of argument, 
unlike the natural language one, leads to the well-known paradox.

I have a paper 'Frege's Hidden Assdumption' that goes into the 
mathematical problems in this case in some detail.  It is available 
on request.
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