Predicate Logic & Quantifiers

Propositional logic treats whole statements as atoms. Predicate logic looks inside them — at properties of things and the words 'for all' and 'there exists' — which is how you say precise things about whole collections at once.

1. Predicates: Statements With Variables A predicate is a statement with a blank — 'x is even' — that becomes true or false only once you fill the blank. Plus the 'domain': the set of things the blank is allowed to be.
2. Quantifiers: For All and There Exists ∀ ('for all') claims a predicate holds for every element of the domain; ∃ ('there exists') claims it holds for at least one. One counterexample kills a 'for all'; one example proves a 'there exists.'
3. Negating & Nesting Quantifiers To negate 'for all', flip to 'there exists a counterexample' (and vice versa) — the quantifier version of De Morgan. And when quantifiers nest, their order changes the meaning entirely.