It is not the case that p

You can of course very often express the negation of p  by saying ‘it’s not the case that p’.

But some logicians in intro texts incautiously claim more — saying something like “This is cumbersome, but it works in every case” (that’s a quote, but no names, no pack drill!).

However it surely overshoots to claim that prefixing  ‘it’s not the case that’ by itself produces the negation of what you start with in every case. Here are some examples:

Jack loves Jill, or Jill is much mistaken about Jack’s feelings.
It’s not the case that Jack loves Jill, or Jill is much mistaken about Jack’s feelings.

Aren’t both  true if Jill is sadly mistaken?

Jack loves Jill and it’s not the case that Jill loves Jack.
It’s not the case that Jack loves Jill and it’s not the case that Jill loves Jack.

Aren’t both false if Jill loves Jack?

Jones, who is a Russian agent, loves caviar.
It’s not the case that Jones, who is a Russian agent, loves caviar.

Aren’t neither true if Jones isn’t a Russian agent?

So what are your favourite counterexamples to the claim that ‘it’s not the case that p’ always expresses the negation of p?

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