First-Order Logic

In propositional logic we can represent the statements which are either True or False. Propositional logic is not enough to represent the complex statements or natural language statements. The Propositional logic has very limited power to express. Consider the following sentences which can’t be represented by Propositional logic.

  • • Some Humans are Intelligent.
  • • All Birds can Fly.

To represent these sentences Propositional Logic is not sufficient so we need to use some more powerful logic such as First-Order Logic.

First-Order Logic:

First Order Logic is called Predicate Logic or First-Order Predicate Logic is an extension to the Propositional Logic for knowledge representation. It is sufficient to express natural language statements in a concise way. First-Order Logic is a powerful language to develop information about the objects in an easier way and also able to express the relationships among objects. First-Order Logic does not only assume that the world contains facts like propositional logic but also assumes the following 3 things in the world, these are: -

1. Objects: X, Y, Birds, People, Numbers etc.

2. Relations: Any relations such as Sister, Brother, has Color, Comes Between etc.

3. Functions: Father of, end of, Left Leg of, square root etc.

As a natural language First-Order Logic also has two main parts:

  • a. Syntax
  • b. Semantics

Syntax of First-Order Logic:

basic syntactic elements of first-order logic are symbols. We write statements in short-hand notation in First-Order-Logic.

Basic elements of First-Order-Logic

Basic elements Example
Constant 2, 3, A, B, C, John, cat, dog
variables x, y, z, a, b
predicates Brother, Father, >, ......
Function sqrt, LeftLegOf, ....
Connectives ∧, ∨, ¬, ⟹, ⟺
Equality ==
Quantifier ∀, ∃

Some examples of First Order Logic are given below:

• Chiki is a Cat -> Cat (Chiki).

• Ravi and Chandra are Brothers -> Brothers (Ravi, Chandra).

• X is always greater than 5 -> ∀X>5

• Every chair is a Physical Object -> ∀x (Chair (x)) → Physical Object (x)

• No table is a Human ->∀x (table (x)) → ¬(Human(x))

• There is a house -> ∃x → house (x)

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