Predicate Logic Summary

Mathematical Logic · Axiom Academy

Let's review how predicate logic extends propositional logic to reason about objects, properties, and relationships. Limitations of Propositional Logic: Cannot express statements about groups or individuals like "all students study" or "some number is even" Need for Variables: Propositional logic treats entire statements as atomic units; we need to look inside them Expressing Patterns: Mathematical statements often have the form "for all x, property P holds" or "there exists an x such that Q is true" Reasoning About Collections: Need to make claims about entire sets of objects, not just individual propositions Predicates: Propositional Functions Definition: A predicate P(x) is a statement about a variable x that becomes a proposition when x is given a specific value Example: P(x) = "x is even" is not true or false until we specify x. P(4) is true, P(5) is false Multiple Variables: Predicates can have multiple variables: Q(x, y) = "x < y" or R(x, y, z) = "x + y = z" Domain of Discourse: The set of all possible values for variables; must be clearly specified Meaning: "For all" or "for every" - the statement must be true for every element in the domain Notation: ∀x P(x) means "P(x) is true for all x" Example: ∀x (x² ≥ 0) over real numbers means "every real number has a non-negative square" Falsification: To prove ∀x P(x) false, find just one counterexample where P(x) is false

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