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COS4840 - Ontology Engineering

First Order LanguagesCreating an ontology

First order languages

A first-order language, also known as first-order logic or predicate logic, is a formal system used in mathematical logic and computer science for representing statements about objects and their relationships. In a first-order language, statements are expressed using variables, predicates, quantifiers, and logical connectives.

FOL is not concerned with the truth of statements. But rather on if one statement is true, how does that relate to other statements.

Syntax:

  1. Logical connectives: Logical connectives are used to combine or negate statements. ∧, ∨, ¬, →, ↔, (, )
  2. Quantifiers: ∀ and ∃, are used to express the scope of a statement .
  3. Variables: x, y, z, ... ranging over particulars in the domain being modelled.
  4. Constants: a, b, c, ... represent a specific element
  5. Functions: f, g, h, ... with arguements listed as f(x1, ..., xn)
  6. Relations: R, S ... with an associated arity (parameters)
  7. Disjoint: the null set or always false set ⊥

Definitions

  • Term - a variable or constant is a term. Functions are terms.
  • Atomic formula - term1 = term2 or R(t1, ..., tn)
  • Formula - a string of symbols of atomic formulae with these restrictions:
    • If f is a formula then so is ¬f (negation)
    • If f and g are formula then so is f ∧ g (conjunction)
    • If f is a formula then so is the ∃xf (existensial quantification)
  • Sentence - a formula with no free variables.

Example:

If P(x) means x is a person and Q(x) means x is an animal ¬P(x) is a function for not a person P(x) ∧ Q(x) is a function for a person AND an animal ∃xP(x) is a function for the existence of a person

Semantics:

  1. Vocabulary/intensional relations/predicates (ℝ): a set of functions, relations and constant symbols.
  2. V-structure/intensional relational structure: in a given system S, an intensional relation structure is a triple (D, W, ℝ) where:
  3. S is the specific system which we want to model.
  4. D is a set of distinguishable elements within S
  5. ℝ is a set of intensional relations on D
  6. V-formula: is a formula with all its vocabulary in V.
  7. Theory: A theory is a set of sentences (formulae with no free variables)
  8. Model: A theory mapped to specific instances of the concepts without contradictinos.

Reasoning

Deduction - moving from premise statements to a conclusion Induction - generalizing facts towards an assumption Abduction - arriving at a hypothesis

Reasoning is using logical languages to represent knowledge and rules that may be applied to the axioms so that one can construct a formal proof that the conclusion can be derived from the premises. - This can be done by hand. - It can also be automated.

See also