Arguments that propositional logic cannot see
Consider this argument: 'All humans are mortal. Socrates is a human. Therefore, Socrates is mortal.' It is obviously valid, but propositional logic cannot show that. In propositional logic, each premise is a distinct sentence letter. Premise 1 might be H, premise 2 might be S, and the conclusion might be M. The form H, S ⊢ M is not valid by any propositional rule, because propositional logic cannot see inside the sentences to notice the repeated references to humans and mortality.
Predicate logic solves this by looking inside sentences. 'All humans are mortal' becomes ∀x (Hx → Mx): for every x, if x is a human then x is mortal. 'Socrates is a human' becomes Hs, where s is a constant naming Socrates. The conclusion 'Socrates is mortal' becomes Ms. Now the inference is formally visible: instantiate the universal premise to s, apply modus ponens, and you have Ms. The sentence-internal structure is what predicate logic lets us track.
This limitation is not a defect of propositional logic; it is a consequence of the level at which propositional logic operates. Propositional logic was designed to capture relationships between whole statements — conjunction, disjunction, conditional, negation — and it does that well. But the moment an argument's validity depends on what a statement says about its subject, propositional logic has no tools to see the connection. Predicate logic was invented precisely to fill that gap.