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Predicates extend propositional logic by introducing variables and quantifiers. Universal quantification (∀) asserts a property holds for all values; existential quantification (∃) asserts it holds for at least one. Together they enable precise mathematical statements about sets and domains.
Translate English statements to logical notation and back. Practice negating quantified statements (∼∀x P(x) ≡ ∃x ∼P(x)). Work with nested quantifiers, paying careful attention to order.
Swapping the order of quantifiers changes meaning completely: ∀x ∃y is different from ∃y ∀x. Negation rules often trip people up.
From propositional logic you know how to reason about fixed statements like "P is true" or "P and Q are both true." But mathematical claims rarely work this way. Instead, they speak about *families* of objects: "every even number is divisible by 2," "there exists a prime larger than 1000," "for any triangle, the angles sum to 180°." These claims can't be expressed with propositional variables alone — they need predicates and quantifiers.
A predicate is a statement with a variable: P(x) = "x is even" is not itself true or false until you substitute a value for x. P(4) is true; P(7) is false. Once you have predicates, quantifiers let you make sweeping claims over a whole domain (some specified set of objects). The universal quantifier ∀x P(x) asserts that P holds for *every* object in the domain — every number, every triangle, every student in the class. The existential quantifier ∃x P(x) asserts that P holds for *at least one* object in the domain. Together they give formal logic the reach to express essentially every mathematical claim.
The most important skill with quantifiers is negation. The negation of "all swans are white" is not "no swans are white" — it's "at least one swan is not white." Formally: ¬∀x P(x) ≡ ∃x ¬P(x). Similarly, ¬∃x P(x) ≡ ∀x ¬P(x). These rules follow a simple pattern: push the negation inside the quantifier and flip ∀ to ∃ (or ∃ to ∀). This is the quantifier analogue of De Morgan's laws from propositional logic, which you already know.
Nested quantifiers are where the real subtlety lives. ∀x ∃y (x + y = 0) says "for every x, there exists a y such that x + y = 0" — true in the integers (y = −x). But ∃y ∀x (x + y = 0) says "there exists a single y that simultaneously satisfies x + y = 0 for every x" — false. The *same* predicate, the *same* two quantifiers, but reversed order produces a completely different claim. When translating mathematical proofs into logic, always read left to right: the outer quantifier's variable is chosen first, and the inner quantifier's variable may depend on that choice. Getting quantifier order right is the foundation for understanding limits, continuity, and virtually every formal definition you'll meet in advanced mathematics.