Limitation and Propositional Logic and PredicatesIn this section, we will learn about the limitations of Propositional logic and predicates. For this, we will cover the following topics:
Limitations of Propositional LogicAs 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 logicThere are various examples of propositional logic, and some of them are shown below: Example 1:
Example 2:
Example 3:
So, in order to infer the statements, we use propositional logic from general rules. Predicate LogicSuppose 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".
Examples of PredicateA 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:
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".
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".
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".
Hence, the truth value of Q(7, 4) is false, and the truth value of Q(0, 5) is true. n-ary PredicateIn 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. |