Here’s a tautology in propositional logic:
⊨(P → Q) ∨ (Q → R)
Try throwing that into English. Here’s a reading using some propositions I just came up with:
“I’ll die if I’m immortal, or I’ll live if I die.”
Obviously, neither of those are the case. But this formula, (P → Q) ∨ (Q → R), is both provable and self-implied in classical propositional logic.
Here’s a syntactic proof by means of natural deduction using some basic rules of inference: