Syntax of First-Order Logic

The syntax of first-order logic consists of symbols and rules for constructing well-formed formulas (WFFs), which are statements or formulas in the language of FOL. The syntax encompasses the language constructs used to express knowledge and relationships within a domain.

Terms in First-Order Logic

Terms represent objects or entities within the domain of discourse. In AI, terms can correspond to real-world entities, such as objects, individuals, or abstract concepts. They include:

• Constants: Specific entities, e.g., “John”, “Apple”.
• Variables: Placeholders for entities, e.g., “x”, “y”.
• Functions: Expressions applied to terms, e.g., “Age(John)”, “Parent(x)”.

Predicates in First-Order Logic

Predicates express properties, relations, or conditions that hold between objects. They describe the state of the world or assert facts about entities within the domain. Examples include:

• “IsHuman(x)”
• “IsParent(x, y)”

Quantifiers in First-Order Logic

Quantifiers in first-order logic allow for the specification of statements about the entirety or existence of objects within the domain.

• Universal quantifiers (∀): Statements that hold for all objects.
• Existential quantifiers (∃): Statements that hold for at least one object.

Connectives in First-Order Logic

Logical connectives such as conjunction (∧), disjunction (∨), implication (→), and negation (¬) enable the composition of complex statements from simpler ones. They facilitate the expression of logical relationships and constraints in AI knowledge representations.

Connectives in First-Order Logic

• Conjunction (∧):
  • Meaning: Represents logical “and” between two propositions. The conjunction of two propositions is true only if both propositions are true.
  • Example: If P(x) represents “x is red” and Q(x) represents “x is round”, then P(x)∧Q(x) represents “x is red and round”.
• Disjunction (∨):
  • Meaning: Represents logical “or” between two propositions. The disjunction of two propositions is true if at least one of the propositions is true.
  • Example: If P(x) represents “x is a cat” and Q(x) represents “x is a dog”, then P(x)∨Q(x) represents “x is either a cat or a dog”.
• Implication (→):
  • Meaning: Represents logical “if-then” relationship between two propositions. The implication P→Q is true if either Q is true or if P is false.
  • Example: If P(x) represents “x is a mammal” and Q(x) represents “x produces milk”, then P(x)→Q(x) represents “if x is a mammal, then it produces milk”.
• Negation (¬):
  • Meaning: Represents logical “not” or negation of a proposition. It reverses the truth value of the proposition.
  • Example: If P(x) represents “x is intelligent”, then ¬P(x) represents “x is not intelligent”.

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.