Quantified Ontologies

Understanding quantifiers in ontologies

Quantifiers play a crucial role in ontological reasoning, allowing us to express general statements about classes and properties.

Universal Quantifier (\(\forall\))

The universal quantifier (\(\forall\)) states that a property holds for all members of a domain.

Example in First-Order Logic:

\(\forall x (Human(x) \rightarrow Mortal(x))\)

This reads as “For all x, if x is a Human, then x is Mortal.”

Existential Quantifier (\(\exists\))

The existential quantifier (\(\exists\)) states that there exists at least one member of the domain for which a property holds.

Example in First-Order Logic:

\(\exists x (Human(x) \land Professor(x))\)

This reads as “There exists an x such that x is a Human and x is a Professor.”

Practical Implementation

Using Python with SymPy

from sympy.logic import *
from sympy import symbols, Implies, And, Or

# Define predicates and variables
x, y = symbols("x y")
Human, Mortal, Professor = symbols("Human Mortal Professor", cls=Function)

# Universal quantification: ∀x(Human(x) → Mortal(x))
universal_statement = Implies(Human(x), Mortal(x))

# Existential quantification: ∃x(Human(x) ∧ Professor(x))
existential_statement = And(Human(x), Professor(x))

print("Universal statement:", universal_statement)
print("Existential statement:", existential_statement)

Key Research Papers

  • “Quantification and Aggregation over Concepts of the Ontology”
    • arXiv:2202.00898
    • Investigates quantification over sets of concepts in ontologies
  • “Non-contractive logics, Paradoxes, and Multiplicative Quantifiers”
    • arXiv:2209.11592
    • Explores non-contractive logics with multiplicative quantifiers

Exercises

  1. Express the following in first-order logic:

    • “Every student takes some course”
    • “There is a course that is taken by all students”

  2. Implement the above statements in Python using SymPy.

Further Reading