Write a short note on check the validity of the following argument using direct and indirect truth table techniques : P   Q  Q  R  P  R

To assess the validity of the argument:

P → ¬Q
¬Q → R
Therefore, P → R

We'll utilize both direct and indirect truth table techniques.

Direct Truth Table Technique:

1. Create a truth table with columns for P, Q, R, ¬Q, P → ¬Q, ¬Q → R, and P → R.
2. Assign truth values (T or F) to P, Q, and R systematically.
3. Use the truth values of P, Q, and R to determine the truth values of ¬Q, P → ¬Q, ¬Q → R, and P → R.
4. Check if the final column for P → R is always true when the preceding premises are true.

Indirect Truth Table Technique:

1. Begin by assuming the premises are true and the conclusion is false.
2. Construct a truth table including columns for P, Q, R, ¬Q, P → ¬Q, ¬Q → R, and P → R.
3. Determine the possible combinations of truth values that make the premises true and the conclusion false.
4. If there's at least one row where the premises are true and the conclusion is false, the argument is invalid; otherwise, it's valid.

Upon completing the truth tables using both direct and indirect techniques, we can conclude the validity of the argument. If the final column for P → R is always true under both methods, the argument is valid; otherwise, it's invalid.