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AI: first-order logic

While propositional logic assumes the world contains facts, first-order logic assumes the world contains objects, relations, functions, like natural language.

Syntax

  • Constants: Poly, 2, refer to objects
  • Predicates: brother, larger than, refer to relations
  • functions: sqrt, refer to functional relations
  • variables: x, y
  • connectives: ^ ⇒\Rightarrow⇒
  • equality: =
  • quantifiers: ∃\exists∃, ∀\forall∀

Sentences

  • atomic sentence = predicate(term1, …, termN) or term1 = term2. where term = function(term1, …, termN) or constant or variable
  • complex sentences are made from atomic sentences using connectives

Universal quantification

∀\forall ∀ Typically, ⇒\Rightarrow⇒ is the main connective with ∀\forall∀, avoid ∧\wedge∧.

Existential quantification

∃\exists ∃ Typically, ∧\wedge∧ is the main connective with ∀\forall∀, avoid ⇒\Rightarrow⇒.

Properties of quantifiers

  • ∀x∀y\forall x \forall y∀x∀y is same as ∀y∀x\forall y \forall x∀y∀x
  • ∃x∃y\exists x \exists y∃x∃y is same as ∃y∃x\exists y \exists x∃y∃x
  • duality:
  • ∀xRelation(object)\forall x Relation(object)∀xRelation(object) is same as ¬∀x¬Relation(object)\neg\forall x \neg Relation(object)¬∀x¬Relation(object)

Universal instantiation

Summary

  • objects and relations are semantic primitives
  • increase expressive power

Contents

  1. 1. Syntax
  2. 2. Sentences
  3. 3. Universal quantification
  4. 4. Existential quantification
  5. 5. Properties of quantifiers
  6. 6. Universal instantiation
  7. 7. Summary

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