Question

QUESTION 3 Symbolize the following argument using the variables p, q, and r. Then construct a complete truth table to show whether or not the argument is valid. Use 1 for T(true) and 0 for F(false). Valid or Invalid? Why? Prove. Explain what your truth table shows. 10 points Total: 3 points for correct symbolic form, 4 points for valid/invalid and reason, 3 points for correct truth table. If Max studies hard, then Max gets an 'A' or Max gets a good grade. Max got an 'A'. .. Max studied hard. IP q

Answer 1

The argument is symbolized as follows :

p : Max studies hard.

q : Max gets an 'A'.

r : Max gets a good grade.

So, the premises are :

• p -> ( q v r )
• q

The conclusion to be proved is :

p.

The truth table is :

p	q	r	( q v r )	p -> ( q v r )	( p -> ( q v r ) ) ∧ q	( ( p -> ( q v r ) ) ∧ q ) -> p
0	0	0	0	1	0	1
0	0	1	1	1	0	1
0	1	0	1	1	1	0
0	1	1	1	1	1	0
1	0	0	0	0	0	1
1	0	1	1	1	0	1
1	1	0	1	1	1	1
1	1	1	1	1	1	1

The given argument is invalid.

The reason is that the resulting final expression ( ( p -> ( q v r ) ) ∧ q ) -> p is not a tautology but a contingency. In order to have the argument to be valid, the resulting expression must be a tautology but the given truth table proves that the final expression is a contingency that is all values are not 1.