Predicates and Variables

Mathematical Logic · Axiom Academy

LESSON Predicates and Variables Moving beyond propositional logic to express statements about objects and their properties through predicates, variables, and domains of discourse. Unlike propositions (which are always true or false), predicates have an indeterminate truth value until we specify what the variables represent. P(x) has no truth value until we specify x Variables in predicates are placeholders that range over a specific set of values called the domain of discourse (or simply the domain ). 3. Predicates with Multiple Arguments Predicates can express relationships between multiple objects by taking multiple arguments. A predicate with n arguments is called an n-ary predicate. P(x) - unary (1 argument): expresses a property of x P(x, y) - binary (2 arguments): expresses a relationship between x and y P(x, y, z) - ternary (3 arguments): relates three objects P(x₁, x₂, ..., xₙ) - n-ary: relates n objects Q(x, y): "x < y" (binary relation on numbers) L(x, y): "x loves y" (binary relation on people) B(x, y, z): "z is between x and y" (ternary relation) S(a, b, c, d): "a + b = c + d" (4-ary relation) To evaluate a predicate, we substitute specific values from the domain for the variables. This process is called instantiation and transforms the predicate into a proposition. The distinction between formulas with free variables and those without is fundamental in predicate logic.

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