Capstone: Using Set Theory in Logic and Argument

The categorical propositions you studied earlier — 'All S are P,' 'No S are P,' 'Some S are P,' 'Some S are not P' — look like claims about English classes, but they have precise set-theoretic translations. Let S denote the set of things that are S and P denote the set of things that are P. Then 'All S are P' means S ⊆ P, 'No S are P' means S ∩ P = ∅, 'Some S are P' means S ∩ P ≠ ∅, and 'Some S are not P' means S − P ≠ ∅. Once you see those four translations, every inference in categorical logic becomes a statement about subsets and intersections.

This reframing is more than decoration. A syllogism like 'All humans are mortal; all Athenians are humans; therefore all Athenians are mortal' becomes the chain 'H ⊆ M and A ⊆ H imply A ⊆ M,' which is simply transitivity of the subset relation. Venn diagrams that you used in categorical logic are literally pictures of intersections and set differences. If you can think set-theoretically, you already know why these classical inferences are valid.

Predicate logic has the same set-theoretic backbone. The domain of discourse in a predicate-logic interpretation is just a set, and every one-place predicate P corresponds to a subset of that domain — namely the set of elements that satisfy the predicate. The universal claim ∀x (Sx → Px) says that the set of S-things is a subset of the set of P-things. The existential claim ∃x (Sx ∧ Px) says that the intersection of those two sets is non-empty. These translations are exactly the categorical ones, now lifted to the predicate-logic framework.

Relational predicates work the same way. A two-place predicate R(x, y) corresponds to a relation R ⊆ D × D, where D is the domain. A universal relational claim ∀x ∀y R(x, y) says that every pair in D × D belongs to R. An existential relational claim ∃x ∃y R(x, y) says that R is non-empty. Functions correspond to relations satisfying the well-definedness condition from the previous lesson. Every formula of predicate logic ultimately translates into a statement about sets, subsets, and relations on a domain, which is why set theory is the semantic home of predicate logic.

Once you have translated an argument into set-theoretic form, you can verify validity by applying set-theoretic reasoning. The inference 'H ⊆ M, A ⊆ H, therefore A ⊆ M' is valid because the subset relation is transitive. The inference 'S ∩ P ≠ ∅ therefore S ⊆ P' is invalid because non-empty intersection does not force inclusion — a counterexample can be built by taking S and P to overlap without one containing the other. This approach gives you a second way to check your work, complementary to formal proof in predicate logic.

The method has one more payoff. When an argument turns out to be invalid, the set-theoretic translation often makes the counterexample easier to construct. To show that a proposed inference fails, you exhibit sets with the assumed properties that nevertheless violate the conclusion, and the set-theoretic language makes those constructions transparent. 'I claim S ⊆ P follows from S ∩ P ≠ ∅? Let S = {1, 2} and P = {2, 3}. Then S ∩ P = {2} is non-empty, but S is not a subset of P because 1 ∈ S and 1 ∉ P.' A single line of concrete sets is often enough to sink a bad argument.

The reason set theory matters for logic is that it gives formal systems their meaning. Predicate logic by itself is a symbol game: strings of quantifiers, variables, and predicate letters. Set theory turns those strings into claims about collections of real objects, and the rules of inference become structural facts about subsets, intersections, and functions. A proof in predicate logic is then sound precisely because the rules respect the set-theoretic meaning; this is the deep reason soundness theorems even exist.

The mathematical content of this unit will also reappear in probability, computation, and the theory of proof itself. Cardinality and the diagonal argument are the starting points for the theory of computability and for Gödel's incompleteness theorems. Equivalence relations and partitions are the machinery of quotient constructions throughout mathematics. Order relations become the basis for lattices, Boolean algebras, and modal logic. You are building vocabulary that will pay dividends in nearly every formal field you pursue from here on.