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INSTRUCTIONS: Select the correct translation for each problem.
1. Either Safeco reduces premiums and Geico cuts costs or Farmers hires agents.
a. S • (G ∨ F)
b. (S • G) ⊃ F
c. (S • G) ∨ F
d. (S ∨ G) • F
e. S ∨ (G • F)
2. Not both Travelers and Conseco run an ad if Liberty opens new offices.
a. L ⊃ ∼(T • C)
b. ∼[C • (L ⊃ C)]
c. L ⊃ (∼T • ∼C)
d. ∼(T • C) ⊃ L
e. (∼T • ∼C) ⊃ L
3. Progressive increases its territory if neither Safeco cuts costs nor Travelers runs an ad.
a. ∼(S ∨ T) ≡ P
b. P ⊃ (∼S ∨ ∼T)
c. (∼S ∨ ∼T) ⊃ P
d. ∼(S ∨ T) ⊃ P
e. P ⊃ ∼(S ∨ T)
4. If either Nationwide or Geico does not open new offices, then Metropolitan does not hire agents.
a. (∼N ∨ ∼G) ⊃ ∼M
b. (∼N • ∼G) ⊃ ∼M
c. ∼(N ∨ G) ⊃ ∼M
d. ∼N ∨ (∼G ⊃ ∼M)
e. ∼[(N ∨ G) ⊃ M]
5. Progressive expands coverage if and only if both Liberty and Conseco do not cut costs.
a. (∼L • ∼C) ⊃ P
b. P ≡ (∼L • ∼C)
c. P ≡ ∼(L • C)
d. P ≡ (∼L ∨ ∼C)
e. P ⊃ (∼L • ∼C)
6. Metropolitan’s reorganizing implies that Allstate hires agents, provided that Liberty reduces premiums.
a. M ⊃ (L ⊃ A)
b. L ⊃ (A ⊃ M)
c. (L ⊃ M) ⊃ A
d. M ⊃ (A ⊃ L)
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e. L ⊃ (M ⊃ A)
7. If either Farmers runs an ad or Nationwide cuts costs, then if Safeco expands coverage then Geico pays a dividend.
a. (F ∨ N) ⊃ (G ⊃ S)
b. [F ⊃ (S ⊃ G)] ∨ [N ⊃ (S ⊃ G)]
c. [(F ∨ N) ⊃ S] ⊃ G
d. (F ∨ N) ⊃ (S ⊃ G)
e. F ∨ [N ⊃ (S ⊃ G)]
8. Allstate’s increasing its territory is a sufficient condition for Liberty’s restructuring only if Progressive’s cutting costs is
a necessary condition for Conseco’s hiring agents.
a. (C ⊃ P) ⊃ (A ⊃ L)
b. (P ⊃ C) ⊃ (L ⊃ A)
c. (A ⊃ L) ⊃ (C ⊃ P)
d. (A ⊃ L) ≡ (C ⊃ P)
e. (L ⊃ A) ⊃ (P ⊃ C)
9. If either Nationwide’s running an ad or Progressive’s paying a dividend implies that Allstate hires agents, then Safeco’s
opening new offices is a sufficient and necessary condition for Liberty’s reducing premiums.
a. [A ⊃ (N ∨ P)] ⊃ (S ≡ L)
b. (S ≡ L) ⊃ [(N ∨ P) ⊃ A]
c. (N ∨ P) ⊃ [A ⊃ (S ≡ L)]
d. [(N ∨ P) ⊃ A] ⊃ (S ⊃ L)
e. [(N ∨ P) ⊃ A] ⊃ (S ≡ L)
10. Travelers and Farmers cut costs given that Safeco opens new offices, but Metropolitan restructures if and only if either
Nationwide expands coverage or Conseco reduces premiums.
a. [(T • F) ⊃ S] • [(N ∨ C) ⊃ M]
b. [(T • F) ⊃ S] ⊃ [M ≡ (N ∨C)]
c. [(T • F) ⊃ S] • [M ≡ (N ∨ C)]
d. [S ⊃ (T • F)] • [M ≡ (N ∨ C)]
e. [S ⊃ (T • F)] • [M ⊃ (N ∨ C)]
Proposition 1F
Given the following proposition:
∼ [∼ (X ∨ ∼ Y) ≡ (∼ X ⊃ ∼A)] ∨ ∼ (B • ∼ A)
11. Given that A and B are true and X and Y are false, determine the truth value of Proposition 1F.
a. False.
b. True.
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12. In Proposition 1F, the main operator is a:
a. Dot.
b. Horseshoe.
c. Wedge.
d. Tilde.
Proposition 2F
Given the following proposition:
[(X ⊃ ∼ Y) • ∼ (X ∨ ∼ B)] ⊃ [∼ (B ⊃ ∼ Y) ≡ (A • ∼ Y)]
13. Given that A and B are true and X and Y are false, determine the truth value of Proposition 2F.
a. False.
b. True.
14. In Proposition 2F, the main operator is a:
a. Triple bar.
b. Dot.
c. Tilde.
d. Wedge.
e. Horseshoe.
INSTRUCTIONS: Use an ordinary truth table to answer the following problems. Construct the truth table as per the
instructions in the textbook.
Statement 1F
Given the following statement:
∼ [K ⊃ (N ⊃ K)]
15. Statement 1F is:
a. Self-contradictory.
b. Contingent.
c. Consistent.
d. Tautologous.
e. Logically equivalent.
16. The truth table for Statement 1F has how many lines?
a. Six.
b. Eight.
c. Two.
d. Four.
e. Nine.
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INSTRUCTIONS: Use an ordinary truth table to answer the following problems. Construct the truth table as per the
instructions in the textbook.
Statement 2F
Given the following statement:
(Q ⊃ ∼ C) • (R ⊃ ∼ Q)
17. Statement 2F is:
a. Tautologous.
b. Inconsistent.
c. Contingent.
d. Self-contradictory.
e. Consistent.
18. The truth table for Statement 2F has how many lines?
a. Twelve.
b. Nine.
c. Four.
d. Six.
INSTRUCTIONS: Use an ordinary truth table to answer the following problems. Construct the truth table as per the
instructions in the textbook.
19. Given the statement:
∼ (S • ∼ D) ∨ (D ⊃ S)
This statement is:
a. Valid.
b. Tautologous.
c. Inconsistent.
d. Contingent.
e. Self-contradictory.
20. Given the pair of statements:
K ⊃ ∼ (E • R) and (E • K) ⊃ ∼ R
These statements are:
a. Inconsistent.
b. Contradictory.
c. Consistent.
d. Logically equivalent.
e. Invalid.
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21. Given the pair of statements:
M • ∼ D and D • ∼ M
These statements are:
a. Logically equivalent.
b. Contradictory.
c. Inconsistent.
d. Consistent.
e. Valid.
22. Given the pair of statements:
Q ≡ ∼ A and A • ∼ Q
These statements are:
a. Consistent.
b. Inconsistent.
c. Valid.
d. Contradictory.
e. Logically equivalent.
23. Given the argument:
K ⊃ Q / Q ⊃ ∼ K // K ≡ Q
This argument is:
a. Invalid; fails in 2nd line.
b. Invalid; fails in 1st line.
c. Valid.
d. Invalid; fails in 4th line.
e. Invalid; fails in 3rd line.
24. Given the argument:
H ⊃ G / G ⊃ H / H ∨ G // G • H
This argument is:
a. Invalid; fails in 1st line.
b. Invalid; fails in 6th line.
c. Invalid; fails in 3rd line.
d. Valid.
e. Invalid; fails in 4th line.
INSTRUCTIONS: Use indirect truth tables to answer the following problems.
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25. Given the argument:
R ⊃ D / D ⊃ (N ∨ S) / S ⊃ (P ∨ N) / N ⊃ R // R ≡ N
This argument is:
a. Uncogent.
b. Invalid.
c. Sound.
d. Cogent.
e. Valid.
26. Given the argument:
Q ⊃ (A ∨ M) / A ⊃ (W ∨ C) / M ⊃ (E ∨ C) / ∼(W ∨ E) // Q ⊃ C
This argument is:
a. Invalid.
b. Uncogent.
c. Cogent.
d. Valid.
e. Sound.
27. Given the statements:
J ⊃ B / B ⊃ (N ∨ C) / N ⊃ C / J • ∼C
These statements are:
a. Consistent.
b. Invalid.
c. Inconsistent.
d. Logically equivalent.
e. Tautologous.
28. Given the statements:
R ⊃ (G ∨ Q) / G ⊃ (H • P) / Q ⊃ H / R ⊃ ∼ H
These statements are:
a. Consistent.
b. Contradictory.
c. Tautologous.
d. Inconsistent.
INSTRUCTIONS: Determine whether the following symbolized arguments are valid or invalid by identifying the form
of each. In some cases the argument must be rewritten using double negation or commutativity before it has a named
form. Those arguments without a specific name are invalid.
29. ∼N ∨ W
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∼W
∼N
a. MP—valid.
b. DA—invalid.
c. MT—valid.
d. DS—valid.
e. Invalid.
30. M
H ⊃ ∼M
∼H
a. Invalid.
b. MT—valid.
c. MP—valid.
d. DS—valid.
e. DA—invalid.
31. ∼N ⊃ H
H
∼N
a. DA—invalid.
b. DS—valid.
c. MP—valid.
d. MT—valid.
e. AC—invalid.
32. (∼M ⊃ E) • (K ⊃ ∼C)
K ∨ ∼M
E ∨ ∼C
a. HS—valid.
b. CD—valid.
c. DD—valid.
d. MP—valid.
e. Invalid.
33. ∼T ⊃ F
∼F ⊃ G
∼T ⊃ G
a. Invalid.
b. CD—valid.
c. HS—valid.
d. CD—invalid.
e. DA—invalid.
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34. ∼A ⊃ ∼F
C ⊃ ∼A
C ⊃ ∼F
a. MT—valid.
b. CD—valid.
c. Invalid.
d. DD—invalid.
e. HS—valid.
35. (Q ⊃ E) • (D ⊃ ∼E)
E ∨ ∼E
∼Q ∨ ∼D
a. CD—valid.
b. Invalid.
c. DD—valid.
d. MT—valid.
e. CD—invalid.
36. ∼G ⊃ K
∼G
K
a. DS—invalid.
b. DA—valid.
c. MT—valid.
d. MP—valid.
e. HS—valid.
37. (H ∨ ∼S) • (G ∨ ∼A)
H ∨ G
∼S ∨ ∼A
a. MP—valid.
b. CD—valid.
c. DD—valid.
d. CD—invalid.
e. Invalid
38. ∼D
D ⊃ ∼L
L
a. AC—invalid.
b. MP—valid.
c. DA—invalid.
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d. MT—invalid.
e. HS—valid.