Satisfaction in First-Order Logic
- Definition: A formula is said to be satisfied by an interpretation if, under that interpretation, the formula evaluates to true.
- Symbolic Notation: M⊨ϕ, where M is an interpretation and ϕ is a formula.
Atomic Formulas
An atomic formula P(t₁, t₂, …, tₙ) is satisfied by an interpretation if the objects assigned to the terms make the predicate P true.
Complex Formulas
The satisfaction of complex formulas is determined recursively based on the satisfaction of their constituent parts, considering logical connectives and quantifiers. For example, a conjunction ϕ∧ψ is satisfied if both ϕ and ψ are satisfied.
Quantifiers
- A universally quantified formula ∀xϕ(x) is satisfied if ϕ(x) is satisfied for all objects in the domain.
- An existentially quantified formula ∃xϕ(x) is satisfied if ϕ(x) is satisfied for at least one object in the domain.
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