Propositional logic treats a whole statement, "Socrates is mortal," as a single, indivisible unit that's simply true or false, with no way to look inside that statement to talk about the individual, Socrates, or the property, mortality, separately. Predicate logic extends that framework with two genuinely new tools: predicates, which express properties or relations separately from the specific things they're applied to, and quantifiers, which let a formal argument claim that a given property holds for every member of a group, or for at least one, distinctions ordinary propositional logic has no built-in way to express or evaluate at all.
Splitting a statement into a predicate and a subject makes its internal structure visible
Where propositional logic can only treat "Socrates is mortal" as an opaque whole, predicate logic represents it as a predicate, "is mortal," applied to a specific subject, "Socrates," making the statement's internal structure genuinely available to logical analysis for the first time. This separation is exactly what lets predicate logic express and formally evaluate the kind of argument propositional logic simply can't handle correctly: "all humans are mortal, Socrates is human, therefore Socrates is mortal" depends entirely on recognising that the same predicate, "is human," and the same predicate, "is mortal," are being applied consistently across different statements, something propositional logic's opaque, whole-statement treatment can't actually capture.
Quantifiers formalise exactly what "all" and "some" mean inside a rigorous system
The universal quantifier states that a given predicate holds for every member of some domain, formalising what "all" means, while the existential quantifier states that a predicate holds for at least one member, formalising what "some" means, and both come with precise, well-defined rules for how they can be combined and manipulated within a formal proof. This precision is exactly what makes predicate logic powerful enough to formally capture the vast majority of everyday mathematical and philosophical argument, arguments about "all," "some," "no" and "exactly one" that appear constantly in ordinary reasoning but that propositional logic, working only with whole opaque statements, simply has no vocabulary to represent correctly.
What we're still unsure about
That predicate logic can formally express and evaluate quantified claims propositional logic can't handle is well established, rigorously developed logical machinery that's underpinned mathematics and philosophy for well over a century. What's more genuinely a matter of ongoing technical interest is exactly how far predicate logic's own expressive power actually extends, since certain mathematical statements turn out to require even richer logical systems, quantifying over properties themselves rather than just over individual things, to state correctly, and logicians continue to study exactly where those further boundaries of expressive power sit, rather than predicate logic alone being sufficient to formally capture every kind of rigorous claim.
This sits inside Predicate Logic & Quantifiers, one of seven topics in Logic, one of five domains in Philosophy, one of seventeen subjects the app can quiz you on.