Implication is true if both arguments are true, so by induction any function built from -> returns true when all arguments are true. Therefore, negation is not expressible.
For a characterization of completeness, see Post's theorem.
How do we show that the set of connectives is not complete?
I'm not sure how to prove it is or it is not complete. But I know how to test for adequacy. And I know the set of standard connectives is adequate , so a set of connectives is adequate if we can express all the standard connectives in terms of this set. There is a test for adequacy of connctives. But if prove that the set is not adequate, how do we know if it failed completeness?
Implication is true if both arguments are true, so by induction any function built from -> returns true when all arguments are true. Therefore, negation is not expressible.
For a characterization of completeness, see Post's theorem.