Predicate Calculus (First-order Logic)

Propositional logic does not allow any reasoning based on general rules. Predicate calculus generalizes propositional logic with variables, quantifiers, and functions.

Formulas are constructed from:

• Predicates have arguments, which are terms: P(x, f(a)). Predicates are true or false.
• Terms refer to objects in the application domain:
  • Variables: x, y, z
  • Constants: John, Mary, 3, a, b. Note that a constant is generally capitalized in English: Austin can be a constant, but dog cannot. A constant is equivalent to a function of no arguments.
  • Functions: f(x) whose arguments are terms.
• Quantifiers: ∀ (``for all'') ( cf. every) and ∃ (``there exists'' or ``for some'') ( cf. some) quantify variables: ∀ x, ∃ y. If a variable is in the scope of a quantifier, it is bound; otherwise, it is free.