First Order Logic

FOL Symbols

Domain: Set of objects

• Constant: Map to a particular object in D (e.g. “Alice”)
• Variable: Map to an arbitrary object in D (e.g. “a person”)

Properties of objects: Modeled using predicates

• Predicate: p(object, …) → bool (e.g. pianist(Avery) = true, equal(Avery, Alice) = false)
• Arity: Number of argument in a predicate

Relationships between objects: Modeled using predicates and functions

• Function: f(object, …) → object (e.g. sibling(Avery) = Alice)

FOL Grammar

Term: Represent objects

• Constants & Variables are terms
• If $t_{1:n}$ are terms and $f(t_{1:n})$ is a function, then the result of $f$ is also a term

Formula: Represent true/false assertions

• p(terms) is a formula
• (¬ formula) is also a formula
• (formula {∧, ∨, →} formula) is also a formula
• (∀x formula) and (∃x formula) are also formulas