The limitations of propositional logic and predicates covered here are as follows:
- Limitations of Propositional Logic
- Predicate Logic and Predicates
Limitations of Propositional Logic
Limitation: Propositional Logic does not allow us to conclude the truth of ALL or SOME statements.
Hence, some valid arguments cannot be concluded or translated into purely propositional logic.
It is not possible to mention properties that apply to categories of object, or even about relationships between those properties.
Please read the following examples carefully
Examples
Example 1:
- Every computer connected to the college network is functioning properly.
- Computer Science lab is functioning properly
- However, it is not possible to find out the truth about whether Business lab is functioning.
Example 2:
- Adam is ill.
- If Adam is ill, he should not go to school.
- So, Adam should not go to school.
Example 3:
- Computer X has been infected by a virus.
- Computer Y has been infected by a virus.
- However, it is not possible to say the city network has been infected by virus.
Basically, propositional logic is limited to infer statements from general rules.
Predicate Logic
Statements involving variables (e.g. x, y) are neither true nor false when the values of the variables are not specified.
For example:
Predicate Logic allows to make propositions from statements with variables.
A statement with variable has two parts:
- x is greater than 9
- The first part, the variable x, is the subject of the statement.
- The second part—the predicate, "is greater than 9" refers to a property that the subject of the statement can have.
- We denote the statement "x is greater than 3" by P(x), where P denotes the predicate "is greater than 3" and x is the variable.
- The statement P(x) is also said to be the value of the propositional function P at x
- Once a value has been assigned to the variable x, the statement P(x) becomes a proposition and has a truth value.
Predicates Examples
One Variable
Example 1:Let P(x) denote the statement "x > 3."
What are the truth values of P(4) and P(2)?
- P(1), which is the statement "1 > 3," is false.
- P(2), which is the statement "2 > 0," is true.
Example 2: Let A(x) denote the statement "Computer x is infltrated by a virus." Suppose CS and Business are infilitrated by a virus.
What are truth values of A(CS10), A(CS20), and A(Business)?
- A(CS10) is false, as CS1 is not on the list of infiltrated computers.
- A(CS20) and A(Business) are true, because CS20 and Business are on the list of infiltrated computers.
Two Variables
We can also have statements that involve more than one variable.
Example 1: Let Q(x, y) denote the statement "x = y + 9."
What are the truth values of the propositions Q(1, 2) and Q(5, 0)?
- The statement Q(1, 2) is the statement "1 = 2 + 9," which is false.
- The statement Q(5, 0) is the proposition "5 = 0 + 5," which is true.
Example 2: Let Q(x, y) denote the statement "x = y + 3.".
What are the truth values of the propositions Q(3, 2) and Q(7, 0)?
- The statement Q(3, 2) is the statement "3 = 2 + 3," which is false.
- The statement Q(7, 0) is the proposition "7 = 0 + 7," which is true.
n-ary predicate
In general, a statement involving the n variables x1, x2, . . . , xn can be denoted by
P(xi, x2, . . . , xn).
A statement of the form P(x1, x2, . . . , xn) is the value of the propositional function P
at the n-tuple (xi, x2, . . . , xn), and P is also called a n-ary predicate.
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Date of last modification: March 27, 2019