The Predicate Calculus

Predicate Calculus (also called first-order logic) extends propositional logic by dealing with predicates − statements that contain variables. While propositional logic works with fixed true/false statements, predicate calculus allows us to express properties of objects and relationships between them.

Predicate

A predicate is an expression of one or more variables defined on some specific domain. A predicate with variables can be made a proposition by either assigning a value to the variable or by quantifying the variable.

Consider the statement − "Ram is a student."

• "is a student" is the predicate (P), and "Ram" is the subject (x).
• We write this as P(x), where P denotes "is a student" and x denotes "Ram".
• A predicate must have at least one object associated with it. Here, Ram is the required object.

Statement Function

A statement function is an expression having a predicate symbol and one or more variables. It becomes a statement when we replace the variables with specific objects. This replacement is called a substitution instance.

For example, if P(x) means "x is a student", then −

P(Ram)   ? "Ram is a student"   (a proposition - can be true or false)
P(Sunil) ? "Sunil is a student" (a proposition - can be true or false)
P(x)     ? "x is a student"     (a statement function - not yet a proposition)

Quantifiers

The variables of predicates are quantified by quantifiers. There are two types −

Universal Quantifier (∀)

The universal quantifier states that the statement is true for every value of the variable. It is denoted by ∀.

∀x P(x) is read as "for every value of x, P(x) is true."

Example − "Man is mortal" can be written as ∀x P(x) where P(x) denotes "x is mortal" and ∀x represents all men.

Existential Quantifier (∃)

The existential quantifier states that the statement is true for at least one value of the variable. It is denoted by ∃.

∃x P(x) is read as "there exists some x for which P(x) is true."

Example − "Some people are dishonest" can be written as ∃x P(x) where P(x) denotes "x is dishonest."

Predicate Formulas

A predicate P with n variables is written as P(x₁, x₂, ..., x_n). This is called an n-place predicate and is known as an atomic formula of predicate calculus. For example: P(), Q(x, y), R(x, y, z).

Well Formed Formula (wff)

A Well Formed Formula is a syntactically valid expression satisfying any of the following −

• All propositional constants and propositional variables are wffs.
• If x is a variable and Y is a wff, then ∀x Y and ∃x Y are also wffs.
• Truth values (true and false) are wffs.
• Each atomic formula is a wff.
• Connectives (∧, ∨, ¬, →, ⇔) connecting wffs produce wffs.

Free and Bound Variables

When a variable appears within the scope of a quantifier (∀x or ∃x), it is a bound occurrence. Any occurrence outside the scope of a quantifier is a free occurrence −

(?x)(P(x) ? Q(x))   ? Scope of ?x covers both P(x) and Q(x)
                        All x are bound.

(?x) P(x) ? Q(x)    ? Scope of ?x covers only P(x)
                        x in P(x) is bound, x in Q(x) is free.

Universe of Discourse

The universe of discourse (or domain) restricts the set of objects that variables can represent. The truth of a predicate formula depends on the chosen universe.

Example − "Some cats are black" can be expressed as −

C(x) : x is a cat
B(x) : x is black
Formula: (?x)(C(x) ? B(x))

If Universe = {Katy, Millie} (both white cats)
    ? Formula is FALSE

If Universe = {Jene, Jackie} (both black cats)
    ? Formula is TRUE

The same formula can be true or false depending on the universe of discourse chosen.

Conclusion

Predicate calculus extends propositional logic by introducing variables, predicates, and quantifiers. Universal (∀) and existential (∃) quantifiers specify whether a predicate holds for all or some values, and the truth of a predicate depends on the universe of discourse.