First-order logic

Definition

First-order logic is a formal system used in mathematical logic that allows the expression of statements about objects and their relationships using quantifiers and predicates. It extends propositional logic by incorporating variables, making it possible to express more complex statements about existence and universality. This system provides a framework for constructing mathematical theories, enabling proofs, and formalizing reasoning in various domains.

5 Must Know Facts For Your Next Test

1. First-order logic can express statements about individual objects, such as 'All humans are mortal', using quantifiers to indicate scope.
2. It includes symbols for logical connectives like 'and', 'or', and 'not', which combine predicates and other logical statements.
3. The completeness theorem for first-order logic states that if a statement is true in every model, then there exists a proof for it within the system.
4. First-order logic is sound, meaning that if a statement can be proven, it is true in all models of the theory.
5. It serves as a foundation for various fields like mathematics, computer science, and artificial intelligence due to its expressive power.

Review Questions

• How do quantifiers enhance the expressive capabilities of first-order logic compared to propositional logic?
  • Quantifiers enhance first-order logic by allowing expressions about entire sets of objects rather than just specific instances. For instance, while propositional logic can only handle specific statements like 'A is true' or 'B is false', first-order logic can formulate statements such as 'For all x, P(x) is true' or 'There exists an x such that P(x) is true'. This enables more nuanced discussions about properties shared by groups of objects and relationships between them.
• Discuss the significance of the completeness theorem in the context of first-order logic and its implications for formal systems.
  • The completeness theorem is crucial because it establishes a fundamental connection between syntactic proofs and semantic truths in first-order logic. It asserts that if a formula is semantically valid—true in every model—then there exists a syntactic proof within the system. This means that first-order logic can fully capture truths about structures we wish to describe mathematically, ensuring that our reasoning aligns with actual mathematical relationships.
• Evaluate how first-order logic's structure contributes to its application in formalizing provability and understanding undecidable theories.
  • First-order logic's structured approach allows it to formalize provability by establishing clear rules for deriving conclusions from premises. This rigor aids in understanding which statements can be proven within certain systems. However, Gödel's incompleteness theorems highlight limitations by demonstrating that there are true statements about natural numbers that cannot be proven within first-order arithmetic. This interplay illustrates both the power of first-order logic in capturing mathematical truths and the inherent challenges posed by undecidable theories.

Related terms

Predicates: Functions that return true or false based on the input values, used to express properties of objects in first-order logic.

Quantifiers:

Symbols such as '∀' (for all) and '∃' (there exists) used in first-order logic to specify the quantity of instances a statement applies to.

Model Theory: A branch of mathematical logic that deals with the relationship between formal languages and their interpretations or models.