[FOM] A Question

[hdeutsch@ilstu.edu]

Let L be a language capable of expressing its own syntax.  Add to L a  
two place predicate 'impl' satisfying

(1) impl ([A], [B]) < - > (A - > B),

where terms such as [A] are standard names of sentences A or of their  
Godel numbers.

Let p be an arbitrary sentence of the object language.  Then impl(v,  
[p]) is a formula in one free variable.  By Godel's diagonalization  
lemma, it follows that there is a sentence W such that

(2) W < - > impl ([W], [p]).

 From here on the argument for Curry's paradox (using (1) and (2)) can  
be used to deduce p.

Is this argument correct?  If so, is it "known" in the  sense at least  
of just "in the air," if not published?  If correct, does it not show  
that classical implication as a predicate expressed in terms of names  
of sentences (rather than a sentential operator) is not definable?   
Harry Deutsch




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