In this chapter, we will learn to further decompose statements in terms of their verbs (called predicates) and their nouns (called individuals). This leads to predicate logic (also called first-order logic).
As a motivation of why we want more expressive power, suppose we wanted to translate the following statements to propositional logic:
All humans are mortal.
Socrates is a human.
Socrates is mortal.
Unfortunately, each statement would be a propositional atom:
p: All humans are mortal.
q: Socrates is a human.
r: Socrates is mortal.
But what if we wanted to prove that given the premises: “All humans are mortal” and “Socrates is a human”, that the conclusion “Socrates is mortal” naturally followed? This logical argument makes sense – Socrates is a human, and all such individuals are supposed to be mortal, so it should follow that Socrates is mortal. If we tried to write such a proof in propositional logic, though, we would have the sequent:
p, q ⊢ r
…and we clearly don’t have enough information to complete this proof.
We need a richer language, which we will get with predicate logic.