Beyond Propositions

Mathematical Logic · Axiom Academy

Discover why propositional logic isn't enough and explore the power of predicates and quantifiers! The Limits of Propositional Logic In propositional logic, we work with complete statements that are either true or false. But look at these mathematical claims - can we express them with just proposition letters like P, Q, R? The key insight: replace specific values with variables! Compare how we express the same idea with and without variables: By using a variable x , we can express a whole family of related statements with a single predicate . A predicate like P(x) is like a template or function - it has a "blank" that we fill in with different values! Predicates: Properties and Relations Try different predicates on our number set! Each predicate tests whether a number has a certain property. Properties: "x is even", "x is prime" Complex conditions: "x is divisible by 3 AND x is odd" Quantifiers: Talking About All or Some Now we need a way to say "P(x) is true for ALL x" or "P(x) is true for SOME x". Meet the quantifiers! The upside-down A (∀) means "for all" or "for every". This claims the statement is true for every possible value! The backwards E (∃) means "there exists" or "for some". This claims at least one value makes the statement true! Which of these statements need predicate logic (not just propositional logic)? Variables: Placeholders like x, y, n that range over a domain Predicates: Properties like P(x), Q(x, y) that are true or false

This is the written version of the interactive lesson above. See the full Mathematical Logic course.