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:
In Logic I at uOttawa last year, I was short on time on an exam and wrote “Magic!” as a justification for a step in natural deduction. I was going for part-marks. Fast-forward to Logic II the following semester, and my professor mentioned in class that you “can’t just write ‘magic’ as a rule of inference” if you’re stuck, as “one student did”. Ha! I promised him that day that I would either prove magic, which, I’m happy to say, I did not, or provide him with an explanation of the rules I used, which I did. My Logic II final exam ended with a page introducing a new operator, magically P, into logic.