Arguments and their Validity

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A compound proposition is a tautology if it is always true for all possible combination of the truth values of its components.

If a compound proposition which is always false for all possible combination of the truth values, of its components then it is called a contradiction.

Consider two compound statements P and Q. If the conditional PQ is a tautology then we say that "Q logically follows from P or Q is a valid conclusion (consequence) of the premise P. (P;Q) is called the argument.

In general, if P₁, P₂, P₃ ….P_n are n compound statements and Q is a compound statement which follows from the other statements P₁, P₂, P₃.........P_n, then we say the conclusion Q logically follows from the set of premises P₁, P₂, P₃.......P_n. These sequences of premises ending with the conclusion is called an argument.

(P₁, P₂, P₃.......P_n;Q) is called an argument where P₁, P₂, P₃.......P_n are premises and Q is a conclusion or consequence. Premises are also known as Hypotheses.

Validity of the Argument

The argument (P₁, P₂, P₃.....P_n;Q) is said to be valid if Q is true whenever all P₁, P₂, P₃.....P_n are true.

We discuss two methods to determine the validity of the argument.

Method 1:

Construct the truth table of the conditional

P₁ P₂ P₃ .... P_m Q and find out whether it is a tautology or not. If it is a tautology then the argument is valid otherwise it is said to be invalid.

Example:

Test the validity of the following argument.

P₁: p q

P₂: q p, Q:pq

To test the validity of the argument, we have to show

P₁ P₂ Q is a tautology.

Since the truth value of the last column are not all T,

Method 2:

Steps to test the validity of an argument.

Step 1:

Construct the truth table for truth values of all the hypotheses and the conclusion.

Step 2:

Find the rows in which all the hypotheses have truth value T. Such a row is called critical row.

Step 3:

If in each critical row, the truth value of the conclusion is also true, then the argument is valid.

If there is at least one critical row in which the conclusion is false, then the argument is invalid.

Examine the validity of the following argument

We select the critical rows and wherever the premises (hypotheses) have truth value T.

Since the corresponding conclusion Q also has Truth value T in the critical rows, the given argument is valid.