Predicate Logic – Discrete Mathematics

The part of a statement that contains a verb that states something about the subject is known as the predicate. It’s more a property. For example, let’s take the statement “Anna is pretty.” In this sentence, “is pretty” is the predicate. And “Anna” is the set that it’s applied to.

Predicate logic vs Propositional logic

Predicate logic is different from proportional logic, but to understand predicate logic, you need to understand proportional logic first.

In a proportional logic, the statement bears a truth value. For example, “Washington, D.C. is the capital of the USA”, so it can be either true or false. And these prepositions are labeled using capital and simple letters. Moreover, they are connected using the following logical connectives.

Logical connectivesLogic
¬  not
∧  and
∨  or
⇒  if and then
⇔  if and only if

Therefore, proportional logic has statements connected using logical connectives. But they do not have quantifiers.

A predicate is just a property of a statement. But a predicate logic is a developed version of proportional logic. Instead of sticking to statements, it uses quantifiers and predicates which are parts of the statements.  

Statement: P: “Einstein is smart.”

Predicate: P(x): “x is smart.”

Statement: Q: Ǝx ∈ D, P(x): “Some people in the x domain are smart.”

As you can see, the predicate is just a fraction of a statement, but it does have much control over the statement. And when we combine predicates and quantifiers, we can build a more advanced statement.

Predicates

A predicate is a part of the statement which describes the verbal part. And it’s sometimes true and sometimes false. Moreover, a predicate is usually represented by capital ‘P’. If the predicate is represented for a variable x, then we can write it as P(x).

Examples,

P(x) = x is a student.

P(classroom) = Classroom is empty.

P(classroom) = Classroom has 25 students

And we can combine quantifiers and predicates,

(Ɐx)P(x) = x passed the exam.

Meaning: All the x passed the exam.

(Ǝx)P(x) = x passed the exam.

Meaning: Some of the x passed the exam. (At least one of the x has passed the exam.)

Universal Quantifier Ɐ

Universal quantifier represents all the data in the set and is represented by upside-down A which is Ɐ.

Ɐx ∈ D, P(x)

Meaning: For all the x values in the domain, the predicate logic P(x) is true.

Example: Every human has a nose.

Ɐ(human) ∈ H, P(human)

Meaning: For all humans, the predicate logic P(human) is true. The predicate is ‘has a nose’, so it means that every people has a nose.

Existential Quantifier Ǝ

Existential Quantifier represents at least some data in the set and is represented by backward E which is Ǝ.

Ǝx ∈ D, P(x)

Meaning: For some of the x values in the domain, the predicate P(x) is true.

Example: Some people have a beard.

Ǝ (human) ∈ H, P(human)

Meaning: For some humans, the predicate logic P(human) is true. The predicate is ‘has a beard’, so it means that some people have a beard. And it does not apply to all the people in the domain.