Validity in First-Order Logic
- Definition: A formula is considered valid if it is satisfied by every interpretation, meaning it holds true universally.
- Symbolic Notation: ⊨ϕ, meaning ϕ is valid.
Examples
- ∀x(P(x)→Q(x)) is valid if, under every interpretation, whenever P(x) holds true, Q(x) also holds true.
- ∃xP(x) is satisfied if there exists at least one object in the domain for which P(x) holds true.
Relationship between Satisfaction and Validity
- A formula is valid if and only if its negation is unsatisfiable. In other words, a formula is valid if there is no interpretation that makes it false.
- If a formula is valid, it is satisfied by every interpretation.
- If a formula is satisfied by a specific interpretation, it does not necessarily mean it is valid unless it holds true under all possible interpretations.
Syntax and Semantics of First-Order Logic in AI
First-order logic (FOL), also known as first-order predicate logic, is a fundamental formal system used in mathematics, philosophy, computer science, and linguistics for expressing and reasoning about relationships between objects in a domain. In artificial intelligence (AI), first-order logic (FOL) serves as a cornerstone for representing and reasoning about knowledge. Its syntax and semantics provide a robust framework for encoding information in a precise and structured manner, enabling AI systems to perform tasks such as automated reasoning, planning, and natural language understanding.
This article provides an in-depth overview of FOL’s syntax, semantics, and applications in AI.
Table of Content
- Syntax of First-Order Logic
- Quantifiers in First-Order Logic
- Well-Formed Formulas (WFFs) in First-Order Logic
- Semantics of First-Order Logic
- Satisfaction in First-Order Logic
- Validity in First-Order Logic
- Applications of First-Order Logic in AI
- Conclusion