First-Order Logic in Ontologies

Formalizing Ontology Knowledge

First-Order Logic (FOL) provides a formal system for knowledge representation, forming the foundation for many ontology languages like OWL and description logics.

Core Concepts

1. Syntax and Semantics

  • Terms: Constants, variables, and functions
  • Predicates: Represent properties and relations
  • Quantifiers: Universal (\(\forall\)) and existential (\(\exists\))
  • Connectives: \(\land\) (and), \(\lor\) (or), \(\rightarrow\) (implies), \(\neg\) (not), \(\leftrightarrow\) (iff)

2. Expressivity

FOL allows expressing complex relationships:

  • Class subsumption
  • Property characteristics
  • Complex class descriptions
  • Property restrictions

Python Implementation

Using SymPy for FOL operations:

from sympy import symbols
from sympy.logic.boolalg import Implies, And, Or, Not

# Define propositional variables for demonstration
P, Q, R = symbols('P Q R')

# Example FOL patterns using propositional logic
print("First-Order Logic Patterns with SymPy")
print("=" * 40)

# Example 1: Implication - "If P then Q"
implication = Implies(P, Q)
print(f"1. Implication: P → Q")
print(f"   SymPy representation: {implication}")

# Example 2: Universal pattern - "P and Q implies R"
universal_pattern = Implies(And(P, Q), R)
print(f"\n2. Universal Pattern: (P ∧ Q) → R")
print(f"   SymPy representation: {universal_pattern}")

# Example 3: Disjunction - "P or Q"
disjunction = Or(P, Q)
print(f"\n3. Disjunction: P ∨ Q")
print(f"   SymPy representation: {disjunction}")

# Example 4: Complex logical expression
complex_expr = And(Implies(P, Q), Implies(Q, R), P)
print(f"\n4. Complex Expression: (P→Q) ∧ (Q→R) ∧ P")
print(f"   SymPy representation: {complex_expr}")

# Check if complex expression implies R (modus ponens chain)
conclusion = Implies(complex_expr, R)
print(f"\n5. Logical Inference Check:")
print(f"   Does our complex expression imply R?")
print(f"   {conclusion}")

# Simplify the conclusion
from sympy.logic.boolalg import simplify_logic
simplified = simplify_logic(conclusion)
print(f"   Simplified form: {simplified}")

# Demonstrate ontology-style reasoning
print(f"\n6. Ontology-Style Reasoning:")
print(f"   In plant ontology terms:")
print(f"   - P could represent 'Plant has fungal disease'")
print(f"   - Q could represent 'Plant shows symptoms'") 
print(f"   - R could represent 'Plant needs fungicide treatment'")
print(f"   - Pattern (P→Q)∧(Q→R)∧P allows us to infer R")

Reasoning with FOL

Theorem Proving

from sympy.logic.inference import valid

# Define some axioms and a theorem
P, Q, R = symbols("P Q R")
axioms = [
    Implies(P, Q),  # If P then Q
    Implies(Q, R),  # If Q then R
    P,  # P is true
]

theorem = R  # We want to prove R

# Check if the theorem follows from the axioms
is_valid = valid(Implies(And(*axioms), theorem))
print(f"The theorem is {'' if is_valid else 'not '}valid")

Key Research Papers

  • “Combining Existential Rules and Description Logics”
    • arXiv:1505.00326
    • Investigates combining existential rules with description logics
  • “A Description Logic Primer”

Practical Applications

  1. Ontology Engineering

    • Define complex class expressions
    • Specify property restrictions
    • Create inference rules

  2. Knowledge Base Completion

    • Infer new facts from existing ones
    • Detect inconsistencies
    • Answer complex queries

Exercises

  1. Express the following in FOL:

    • “Every student who is enrolled in a course is taught by some professor”
    • “There is a course that is taught by all professors”

  2. Implement the above statements in Python using SymPy.

Further Reading