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The Predicate Calculus
Predicate Calculus deals with predicates, which are propositions containing variables.
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 following statement.
Ram is a student.
Now consider the above statement in terms of Predicate calculus.
Here "is a student" is a predicate and Ram is subject.
Let's denote "Ram" as x and "is a student" as a predicate P then we can write the above statement as P(x).
Generally a statement expressed by Predicate must have at least one object associated with Predicate. In our case, Ram is the required object with associated with predicate P.
Statement Function
Earlier we denoted "Ram" as x and "is a student" as predicate P then we have statement as P(x). Here P(x) is a statement function where if we replace x with a Subject say Sunil then we'll be having a statement "Sunil is a student."
Thus a statement function is an expression having Predicate Symbol and one or multiple variables. This statement function gives a statement when we replaced the variables with objects. This replacement is called substitution instance of statement function.
Quantifiers
The variable of predicates is quantified by quantifiers. There are two types of quantifier in predicate logic − Universal Quantifier and Existential Quantifier.
Universal Quantifier
Universal quantifier states that the statements within its scope are true for every value of the specific variable. It is denoted by the symbol ∀.
∀ x P(x) is read as for every value of x, P(x) is true.
Example − "Man is mortal" can be transformed into the propositional form ∀ x P(x) where P(x) is the predicate which denotes x is mortal and ∀ x represents all men.
Existential Quantifier
Existential quantifier states that the statements within its scope are true for some values of the specific variable. It is denoted by the symbol ∃.
∃ x P(x) is read as for some values of x, P(x) is true.
Example − "Some people are dishonest" can be transformed into the propositional form ∃ x P(x) where P(x) is the predicate which denotes x is dishonest and ∃ x represents some dishonest men.
Predicate Formulas
Consider a Predicate P with n variables as P(x1, x2, x3, ..., xn). Here P is n-place predicate and x1, x2, x3, ..., xn are n individuals variables. This n-place predicate is known as atomic formula of predicate calculus. For Example: P(), Q(x, y), R(x,y,z)
Well Formed Formula
Well Formed Formula (wff) is a predicate holding any of the following −
All propositional constants and propositional variables are wffs
If x is a variable and Y is a wff, ∀ x Y and ∀ x Y are also wff
Truth value and false values are wffs
Each atomic formula is a wff
All connectives connecting wffs are wffs
Free and Bound variables
Consider a Predicate formula having a part in form of (∃ x) P(x) of (x)P(x), then such part is called x-bound part of the formula. Any occurrence of x in x-bound part is termed as bound occurrence and any occurrence of x which is not x-bound is termed as free occurrence. See the examples below -
(∃ x) (P(x) ∧ Q(x))
(∃ x) P(x) ∧ Q(x)
In first example, scope of (∃ x) is (P(x) ∧ Q(x)) and all occurrences of x are bound occurrences. Whereas in second example, scope of (∃ x) is P(x) and last occurrence of x in Q(x) is a free occurrence.
Universe of Discourse
We can limit the class of individuals/objects used in a statment. Here limiting means confining the input variable to a set of particular individuals/objects. Such a restricted class is termed as Universe of Discourse/domain of individual or universe. See the example below:
Some cats are black.
C(x) : x is a cat.
B(x) : x is black.
(∃ x)(C(x) ∧ B(x))
If Universe of discourse is E = { Katy, Mille } where katy and Mille are white cats then our third statement is false when we replace x with either Katy or Mille where as if Universe of discourse is E = { Jene, Jackie } where Jene and Jackie black cats then our third statement stands true for Universe of Discourse F.
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