Symbolic Logic 5E: 2.3, II

“Use truth tables to determine the validity or invalidity of each of the following argument forms”

Note: Lines which contain true premises and a true conclusion are in green, while lines with true premises and a false conclusion are in red.

1. Valid.

P Q P ∧ Q

T

T

T

V

T

F

F

F

F

F

F

T

F

2. Invalid.

P Q P ∧ Q

T

T

T

V

T

F

F

X

F

F

F

F

T

F

3. Invalid.

P Q P ∨ Q

T

T

T

V

T

F

T

V

F

T

T

X

F

T

F

4. Valid.

P P Q Q

T

T

T

V

F

T

T

T

T

F

V

F

F

F

5. Invalid.

P Q P→Q

T

T

T

V

T

F

F

X

F

T

T

F

F

T

6. Valid.

P Q Q→P

T

T

T

V

T

F

T

V

F

T

F

F

F

T

7. Valid.

P Q

¬Q

¬P

P→Q

¬Q→¬P

T

T

F

F

T

T

V

T

F

T

F

F

F

F

T

F

T

T

T

V

F

F

T

T

T

T

V

8. Invalid.

P Q

¬Q

¬P

P→Q

¬P→¬Q

T

T

F

F

T

T

V

T

F

T

F

F

T

F

T

F

T

T

F

X

F

F

T

T

T

T

V

9. Valid.

P Q R P→(Q ∧ R) ¬(Q ∧ R)→ ¬P Q ∧ R ¬(Q ∧ R) ¬P

T

T

T

T

T

T

F

F

V

T

T

F

F

F

F

T

F

T

F

F

F

F

F

T

F

T

F

T

T

T

F

T

T

V

F

F

F

T

T

F

T

T

V

F

F

T

T

T

T

F

T

V

T

F

T

F

F

F

T

F

F

T

F

T

T

F

T

T

10. Invalid.

P Q ¬Q P ∨ Q

T

T

F

T

X

T

F

T

T

V

F

T

F

T

F

F

T

F

11. Valid.

P Q P ∧ Q

T

T

T

V

T

F

F

F

T

F

F

F

F

12. Invalid.

P Q P→Q Q→P P v Q

T

T

T

T

T

V

T

F

F

T

T

F

T

T

F

T

F

F

T

T

F

X

13. Valid.

P Q P→Q Q v P

T

T

T

T

V

T

F

F

T

F

T

T

T

V

F

F

T

F

14. Valid.

P Q R P→Q Q→R P→(Q→R) P→R

T

T

T

T

T

T

T

V

T

T

F

T

F

F

F

T

F

F

F

T

T

F

F

F

F

T

T

T

T

V

F

F

T

T

T

T

T

V

F

T

T

T

T

T

T

V

F

T

F

T

F

T

T

V

T

F

T

F

T

T

T

15. Valid.

P Q R P→Q P→R Q v R (P→Q) ∧ (P→R)

T

T

T

T

T

T

T

V

T

T

F

T

F

T

F

T

F

F

F

F

F

F

F

F

F

T

T

F

T

F

F

T

T

T

T

T

F

T

T

T

T

T

T

F

T

F

T

T

T

T

T

F

T

F

T

T

F

16. Invalid.

P Q R Q v R P→(Q vR) P→¬Q P v R ¬Q

T

T

T

T

T

F

T

F

T

T

F

T

T

T

T

F

V

T

F

F

F

T

T

T

T

V

F

F

F

F

T

T

F

T

X

F

F

T

T

T

T

T

T

V

F

T

T

T

T

T

T

F

V

F

T

F

T

T

T

F

F

X

T

F

T

T

T

T

T

T

V

17. Valid.

P Q R S P→Q R→S (P→Q) ∧ (R→S) P v R Q v S

T

T

T

T

T

T

T

T

T

V

T

T

T

F

T

F

F

T

T

T

T

F

F

T

T

T

T

T

V

T

F

F

F

T

T

T

T

T

T

T

F

T

T

T

T

T

T

V

T

F

T

T

F

T

F

T

T

T

F

F

T

F

T

F

T

T

T

F

T

F

F

F

F

T

F

F

F

F

F

T

T

T

F

F

F

F

F

T

T

T

T

F

T

F

F

T

T

T

T

T

T

T

V

F

T

T

T

T

T

T

T

T

V

F

F

T

F

T

F

F

T

F

F

T

F

F

T

T

T

F

T

F

T

T

F

T

F

F

T

T

F

T

F

T

T

T

T

F

T

18. Valid.

P Q R S P→Q R→S (P→Q) ∧ (R→S) ¬P v ¬R ¬Q v ¬S

¬P

¬Q

¬R

¬S

T

T

T

T

T

T

T

F

F

F

F

F

F

T

T

T

F

T

F

F

F

T

F

F

F

T

T

T

F

F

T

T

T

T

T

F

F

T

T

V

T

F

F

F

T

T

T

T

F

F

T

T

T

T

T

F

T

T

T

T

T

F

F

F

T

F

T

F

T

T

F

T

F

F

T

F

T

F

F

T

F

F

T

F

T

F

T

T

F

T

T

F

T

F

T

F

F

F

F

F

T

F

T

F

T

F

F

F

F

T

T

T

T

T

T

T

T

T

V

F

F

F

T

T

T

T

T

T

T

T

T

F

V

F

F

T

T

T

T

T

T

T

T

T

F

F

V

F

T

T

T

T

T

T

T

F

T

F

F

F

F

F

T

F

T

F

F

T

T

T

T

F

T

F

T

F

F

T

T

T

T

T

T

F

T

T

V

F

T

T

F

T

F

F

T

T

T

F

F

T

F

T

F

T

T

T

T

T

F

T

F

T

F

19. Valid.

P Q P v Q P ∧ Q (P v Q)→(P ∧ Q)

T

T

T

T

T

V

T

F

T

F

F

F

T

T

F

F

F

F

F

F

T

20. Valid.

P Q ¬P (Q ∧ ¬P) P v (Q ∧ ¬P) ¬(Q ∧ ¬P)

T

T

F

F

T

T

V

T

F

F

F

T

T

V

F

T

T

T

T

F

F

F

T

F

F

T

21. Valid.

P Q P v Q P ∧ Q (P v Q)→(P ∧ Q) ¬(P v Q) ¬(P ∧ Q)

T

T

T

T

T

F

F

T

F

T

F

F

F

T

F

T

T

F

F

F

T

F

F

F

F

T

T

T

V

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