Limitation and Propositional Logic and Predicates

In this section, we will learn about the limitations of Propositional logic and predicates. For this, we will cover the following topics:

• Limitation of Propositional logic
• Predicate Logic and Predicates

Limitations of Propositional Logic

As we know that the propositional logic contains the statements. In case of propositional logic, we are not allowed to conclude the truth of some or ALL statements. Hence, it is not possible to translate or conclude some valid arguments of the propositional logic into purely propositional logic. In case of propositional logic, there is no possibility to describe properties that apply to the object's category. It is also impossible to describe the relationship between those properties.

Examples of Propositional logic

There are various examples of propositional logic, and some of them are shown below:

Example 1:

• All the chemicals and equipment of the chemistry lab are functioning properly.
• Chemistry lab of my college is functioning properly.
• However, we are not able to determine the truth related to whether the business lab is functioning.

Example 2:

• Harry is playing.
• If Harry is playing, then she will not watch the movie.
• So, Harry will not watch the movie.

Example 3:

• A virus is used to infiltrate computer system A.
• A virus is used to infiltrate the computer system B.
• However, a virus has been used by someone to infiltrate the city network of the organization.

So, in order to infer the statements, we use propositional logic from general rules.

Predicate Logic

Suppose there is a statement that contains variables a and b. If there is a variable that is not specified by any value, then that type of statement will neither be true nor false.

Propositions can be made with the help of predicate logic from statements that have variables. If there is a statement that has a variable, then it will have two parts, which are described as follows:

Suppose there is a statement "a is equal to 5".

• The first part of this statement is "the variable a", which is used to indicate the subject of the statement.
• The second part of this statement is "is equal to 5", which is used to indicate the property that the subject of the statement can have.
• With the help of symbol P(a), we can indicate the statement "a is equal to 5", where P is used to indicate the predicate "is equal to 5", and a is used to indicate the variable.
• Once the variable x is assigned, in this case, statement P(a) becomes the proposition and truth table.

Examples of Predicate

A statement can have one variable or more than one variable. Now we will explain the one variable statement and two variable statements one by one with the help of their examples, which are shown below:

One variable

Here we will explain those types of statements that have only one variable. The examples of statements with one variable are described as follows:

Example 1: Suppose there is a statement P(x) = x>3. Now we have to determine the truth values of p(4) and p(2).

Solution: From the question, we have a statement P(x) = x>3

When we put 2 in place of x, then we will get the following:

• P(2) has a statement "2>;3". This statement is false.
• P(4) has a statement "4>3". This statement is true.

Hence, the truth value of P(2) is false, and the truth value of P(4) is true.

Example 2: Suppose there is a statement P(x) = "A virus is used to infiltrate our computer network". Suppose a virus is used to infiltrate the CS20 and Business. Now we have to determine the truth values of A(CS10), A(CS20), and A(Business).

Solution: From the question, we have a statement:

P(x) = "A virus is used to infiltrate our computer network".

• As we can see that CS10 is not on the infiltrate list. So we can say that A(CS10) will be false.
• The CS20 and Business are on the infiltrate list. So we can say that A(CS20) and A(Business) will be true.

Hence, the truth value of A(CS10) is false, and the truth value of A(CS20) and A(Business) are true.

Two Variables

There can be those types of statements that are related to more than one variable. The examples of statements with two variables are described as follows:

Example 1: Suppose we have a proposition Q(a, b) that has a statement "a = b+6". Now we have to determine the truth value of Q(3, 6) and Q(6, 0).

Solution: From the question, we have a statement

Q(a, b) = "a = b+6".

• Q(3, 6) has a statement "3 = 6 + 6". This statement is false because 3 is not equal to 12.
• Q(6, 0) has a statement "6 = 0 + 6". This statement is true because 6 = 6.

Hence, the truth value of Q(3, 6) is false, and the truth value of Q(6, 0) is true.

Example 2: Suppose we have a proposition Q(a, b) that has a statement "a = b-5". Now we have to determine the truth value of Q(7, 4) and Q(0, 5).

Solution: From the question, we have a statement

Q(a, b) = "a = b-5".

• Q(7, 4) has a statement "7 = 4 - 5". This statement is false because 7 is not equal to -1.
• Q(0, 5) has a statement "0 = 5 - 5". This statement is true because 0 = 0.

Hence, the truth value of Q(7, 4) is false, and the truth value of Q(0, 5) is true.

n-ary Predicate

In general, if there is a statement that is used to contain the n number of variables x1, x2, x3, ...., xn, in this case, it will be denoted in the following way:

P(x1, x2, x3, ...., xn).

Suppose there is a statement in the form P(x1, x2, x3, ...., xn). This statement is used to indicate the value of P at the n-tuple (x1, x2, x3, ..., xn). Here P is used to indicate the propositional function, and P can also be known as the n-ary predicate.