Predicate Logic

Extends propositional logic by introducing quantifiers and predicates, providing a powerful framework for representing and reasoning about relationships and properties within a formal system.

First-Order Predicate Logic:

First-order predicate logic, also known as first-order logic or predicate calculus, is a formal system of logic that extends propositional logic by introducing quantifiers and predicates. It provides a more expressive language for representing and reasoning about relationships between objects, properties of objects, and relations between objects.

Syntax:

The syntax of first-order predicate logic includes:

• Variables: Represent objects or individuals within the domain of discourse (e.g., ( x, y, z )).

• Constants: Represent specific objects within the domain (e.g., ( a, b, c )).

• Predicates: Express properties or relations between objects (e.g., ( P(x), Q(x, y) )).

• Quantifiers: Specify the scope of variables in logical expressions:

  • Universal Quantifier (∀): Indicates that a statement holds for all objects in the domain.

  • Existential Quantifier (∃): Indicates that a statement holds for at least one object in the domain.

• Connectives: Logical operators for combining propositions:

  • Negation (¬), Conjunction (∧), Disjunction (∨), Implication (→), Biconditional (↔).

Semantics:

In first-order predicate logic, the semantics involve interpreting predicates and quantifiers over a domain of discourse. An interpretation assigns meanings to predicates and specifies the objects within the domain to which variables refer. The truth value of a formula in first-order logic is determined based on whether the formula holds true under all possible interpretations.

Properties of First-Order Logic:

First-order predicate logic has several key properties:

• Expressiveness: First-order logic can express complex relationships and statements involving quantification over objects and properties.

• Completeness: First-order logic is complete, meaning that every valid inference can be derived using its rules of inference.

• Soundness: The rules of inference in first-order logic guarantee that if a conclusion is derived, it is logically valid given the premises.

• Compactness: First-order logic exhibits the compactness property, meaning that if a set of formulas is satisfiable, then there exists a finite subset of those formulas that is also satisfiable.

Applications:

First-order predicate logic finds applications in various fields, including mathematics, computer science, linguistics, philosophy, and artificial intelligence. It serves as the basis for formalizing mathematical theories, specifying knowledge representation languages in AI, modeling natural language semantics, and reasoning about complex systems and relationships.