Question

(4) Determine if the following logical statement is valid. First convert each statement to a symbolic logic, and then create a truth table. Don’t forget you need to state if the statement is valid or not and state why.

Prop A. Steve can exclusively either Study or Sleep.

Prop B. If Steve studies, then he’ll Pass his Exam.

Conclusion: If Steve Sleeps, then he’ll NOT Pass his Exam.

Answer 1

Let,

A = Study B = Sleep P = Pass ~P = not Pass

1st Proposition:   Steve can exclusively either Study or Sleep. So we can denote it as A V B, which we shorten into S

2nd Proposition:   If Steve studies, then he’ll Pass his Exam. So we can denote it as A --> P, shortened into T

The two propositions will work together to reach the conclusion :

Conclusion is : If Steve Sleeps, then he’ll NOT Pass his Exam. Which we can denote as B --> ~P, shortened into U

So, we have to find tautology for (S AND T) --> U, Where we shortened (S AND T) as V. So, V --> U will be final expression

Truth Table

A	B	P	~P	S =A V B	T = A --> P	U = B --> ~P	V = S AND T	V --> U
0	0	0	1	0	1	1	0	1
0	0	1	0	0	1	1	0	1
0	1	0	1	1	1	1	1	1
0	1	1	0	1	1	0	1	0
1	0	0	1	1	0	1	0	1
1	0	1	0	1	1	1	1	1
1	1	0	1	1	0	1	0	1
1	1	1	0	1	1	0	1	0

As we see, we can not find all truth values for the last column, we come to the opinion that "Conclusion: If Steve Sleeps, then he’ll NOT Pass his Exam." is NOT VALID