Predicate Logic & Quantifiers
Propositional Logic gave you AND, OR, and NOT - but it treats every
statement as a single sealed atom. "All users have a password" is only P to it; it can't see the
all, can't see the users, can't reason about them one by one. That's a ceiling you hit fast, because
almost everything worth saying is a statement about a collection: every request, some account, no
file.
Predicate logic raises that ceiling. It cracks statements open to talk about properties of things
(predicates) and adds two small words that do enormous work: for all (∀) and there exists (∃).
With them you can say exactly what's true of an entire set, spot when someone overclaims ("all?
really?"), and negate a sweeping statement correctly. If you've ever written .all(), .any(),
.every(), or .some(), you've already used this - here's the logic underneath.
How to read this
- Want the two power words? Phase 2 is ∀ and ∃ - the heart of the guide.
- Want it solid? Read in order - Phase 1 explains predicates, which the quantifiers act on.
The phases
- Predicates: Statements With Variables - a statement with a blank in it, and the "domain" it ranges over.
- Quantifiers: For All and There Exists - ∀ and ∃, what makes each true, and the counterexample that kills a "for all."
- Negating & Nesting Quantifiers - how to negate "for all" / "there exists" correctly, and why the order of nested quantifiers changes the meaning.
This builds on Propositional Logic and uses the idea of a set from Sets, Relations & Functions. The Logic track continues into proof and spotting fallacies.