First-Order Logic: Assertions, Queries, and Models

Posted on Feb 15, 2025 in Social sciences

Elaborate Assertions and Queries in First-Order Logic

Sentences are added to a knowledge base using TELL, exactly as in propositional logic. Such sentences are called assertions. For example, we can assert that John is a king, Richard is a person, and all kings are persons:

TELL(KB, King(John)).

TELL(KB, Person(Richard)).

TELL(KB, ∀x King(x) ⇒ Person(x)).

We can ask questions of the knowledge base using ASK. For example:

ASK(KB, King(John))

The query returns true. Questions asked with ASK are called queries or goals. Generally speaking, any goal query that is logically entailed by the knowledge base should be answered affirmatively. For example, given the two preceding assertions, the query:

ASK(KB, Person(John))

should also return true.

We can ask quantified queries, such as:

ASK(KB, ∃x Person(x))

The answer is true, but this is perhaps not as helpful as we would like. It is rather like answering “Can you tell me the time?” with “Yes.”

If we want to know what value of x makes the sentence true, we will need a different function, ASKVARS, which we call with:

ASKVARS(KB, Person(x))

and which yields a stream of answers. In this case, there will be two answers:

{x/John} {x/Richard}

Such an answer is called a substitution or binding list. ASKVARS is usually reserved for knowledge bases consisting solely of Horn clauses, because in such knowledge bases, every way of making the query true will bind the variables to specific values. That is not the case with first-order logic; if KB has been told King(John) ∨ King(Richard), then there is no binding to x for the query ∃x King(x), even though the query is true.

A first-order logic model includes a domain of objects, relations among these objects, and functions mapping objects. Below is an example model using historical figures Richard the Lionheart and King John.

Model Domain

The domain of the model includes the following five objects:

  1. Richard the Lionheart (R)
  2. King John (J)
  3. Richard’s left leg (RL)
  4. John’s left leg (JL)
  5. A crown (C)

Relations

The model defines both binary and unary relations:

Binary Relations:

Brother: Indicates brotherhood.

{⟨R, J⟩, ⟨J, R⟩}

On Head: Indicates what object is on whose head.

{⟨C, J⟩}

Unary Relations:

Person: Indicates objects that are persons.

{R, J}

King: Indicates objects that are kings.

{J} (Assuming Richard is deceased in this context)

Crown: Indicates objects that are crowns.

{C}

Functions

Functions map objects to other objects:

  1. Left Leg:

    • LeftLeg(R) = RL
    • LeftLeg(J) = JL

Example Assertions

Using the relations and functions defined above, we can express assertions in first-order logic:

  1. Richard’s left leg is Richard’s:

    ∀x (LeftLeg(R) = x ⇔ x = RL)

  2. John is the king:

    King(J)

  3. Richard and John are brothers:

    Brother(R, J) ∧ Brother(J, R)

  4. The crown is on John’s head:

    OnHead(C, J)