Question 1

use indirect truth tables to determine, for each given set of propositions, whether it is consistent. If the set is consistent, choose an option with a consistent valuation. (There may be other consistent valuations.)
-F • (A ⊃ D)
E $ \lor $ ∼B
∼[C ⊃ (D $ \lor $ F)]
A $ \lor $ (B • D)
E ⊃ A
A) Inconsistent
B) Consistent. Consistent valuation when F and D are true and A, B, C, and E are false
C) Consistent. Consistent valuation when A, B, D, E, and F are true and C is false
D) Consistent. Consistent valuation when A, B, E, and F are true and C and D are false
E) Consistent. Consistent valuation when B and C are true and A, D, E, and F are false

Accepted Answer

The answer of use indirect truth tables to determine, for...

Question 2

use indirect truth tables to determine, for each given set of propositions, whether it is consistent. If the set is consistent, choose an option with a consistent valuation. (There may be other consistent valuations.)
-(L • N) $ \lor $ I
L ≡ ∼K
K ⊃ (I ≡ ∼M)
(J $ \lor $ K) • ∼N
A) Inconsistent
B) Consistent. Consistent valuation when I, J, and K are true and L, M, and N are false
C) Consistent. Consistent valuation when J, K, and M are true and I, L, and N are false
D) Consistent. Consistent valuation when J, L, M, and N are true and I and L are false
E) Consistent. Consistent valuation when I, J, L, and N are true and K and M are false

Accepted Answer

Step 1: Set up the truth table with all the propositions and connectives.
Step 2: Fill in the truth values for the propositions that are given in the problem.
Step 3: Fill in the truth values for the propositions that are implied by the connectives in the problem.
Step 4: Check for any rows where all the propositions are false. If there are none, the set of propositions is consistent.

| I | J | K | L | M | N | (L • N) $ \lor $ I | L ≡ ~K | K ⊃ (I ≡ ~M) | (J $ \lor $ K) • ~N |
|---|---|---|---|---|---|------------------------|----------|------------------------|------------------------|
| T | T | T | F | F | F |           T            |     T    |           F            |           F            |
| T | T | F | F | F | F |           T            |     T    |           T            |           T            |
| T | F | T | F | F | F |           T            |     T    |           F            |           T            |
| T | F | F | F | F | F |           T            |     T    |           T            |           T            |
| F | T | T | F | F | F |           F            |     T    |           F            |           F            |
| F | T | F | F | F | F |           F            |     T    |           T            |           T            |
| F | F | T | T | T | T |           F            |     F    |           F            |           F            |
| F | F | F | T | T | T |           T            |     F    |           T            |           F            |

There are no rows where all the propositions are false, so the set of propositions is consistent. The consistent valuation is when I, J, and K are true and L, M, and N are false.

Question 3

use indirect truth tables to determine, for each given set of propositions, whether it is consistent. If the set is consistent, choose an option with a consistent valuation. (There may be other consistent valuations.)
-P ≡ (Q • ∼R)
P ⊃ ∼Q
(∼P • Q) ⊃ S
A) Inconsistent
B) Consistent. Consistent valuation when P, Q, and S are true and R is false
C) Consistent. Consistent valuation when P and Q are true and S and R are false
D) Consistent. Consistent valuation when Q, R and S are true and P is false
E) Consistent. Consistent valuation when Q and R are false and P and S are true

Accepted Answer

The answer of use indirect truth tables to determine, for...

Question 4

use indirect truth tables to determine, for each given set of propositions, whether it is consistent. If the set is consistent, choose an option with a consistent valuation. (There may be other consistent valuations.)
-A $ \lor $ B
(C • D) ≡ (E • F)
F ⊃ ∼A
D ⊃ B
A) Inconsistent
B) Consistent. Consistent valuation when A, C, and E are true and B, D, and F are false
C) Consistent. Consistent valuation when A, D, and E are true and B, C, and F are false
D) Consistent. Consistent valuation when B, C, and F are true and A, D, and E are false
E) Consistent. Consistent valuation when B, D, and E are true and A, C, and F are false

Accepted Answer

The set is consistent with the valuation where B, D, and E are true, and A, C, and F are false. This satisfies all the given propositions.

Question 5

use indirect truth tables to determine, for each given set of propositions, whether it is consistent. If the set is consistent, choose an option with a consistent valuation. (There may be other consistent valuations.)
-A ⊃ (B • C)
D ⊃ (E $ \lor $ F)
E ⊃ ∼B
F ⊃ ∼C
A $ \lor $ D
A) Inconsistent
B) Consistent. Consistent valuation when A, B, C, and D are true and E and F are false
C) Consistent. Consistent valuation when A, B, C, and F are true and D and E are false
D) Consistent. Consistent valuation when A, B, C, and E are true and D and F are false
E) Consistent. Consistent valuation when D, E, and F are true and A, B, and C are false

Accepted Answer

E) is correct because it provides a consistent valuation that does not violate any of the given propositions. If D, E, and F are true, and A, B, and C are false, all the given propositions can be satisfied. Specifically, since A is false, the first proposition A ⊃ (B • C) is satisfied regardless of the truth values of B and C due to the way material implication works (if the antecedent is false, the implication is true regardless of the consequent). D being true and leading to E ∨ F being true is satisfied by having either E or F true, which is the case here. E being true and leading to ∼B being true is consistent with B being false. F being true and leading to ∼C being true is consistent with C being false. Lastly, A ∨ D being true is satisfied by D being true. This valuation does not lead to any contradictions among the propositions, making the set consistent.

Question 6

use indirect truth tables to determine, for each given set of propositions, whether it is consistent. If the set is consistent, choose an option with a consistent valuation. (There may be other consistent valuations.)
-P ⊃ Q
R ⊃ Q
∼S ⊃ (P $ \lor $ R)
(Q • S) ⊃ T
∼(U ⊃ T)
∼W ≡ S
A) Inconsistent
B) Consistent. Consistent valuation when P, Q, S, U, and W are true and R and T are false
C) Consistent. Consistent valuation when Q, R, and U are true and P, S, T, and W are false
D) Consistent. Consistent valuation when P, Q, U, and W are true and R, S, and T are false
E) Consistent. Consistent valuation when S and U are true and P, Q, R, T, and W are false

Accepted Answer

The given set of propositions is consistent with the valuation where P, Q, U, and W are true, and R, S, and T are false. This satisfies all the propositions without contradiction.

Question 7

use indirect truth tables to determine, for each given set of propositions, whether it is consistent. If the set is consistent, choose an option with a consistent valuation. (There may be other consistent valuations.)
-D ≡ (A • B)
D $ \lor $ (∼E • F)
∼E ⊃ A
F ⊃ B
A $ \lor $ B
∼A $ \lor $ ∼B
A) Inconsistent
B) Consistent. Consistent valuation when A and F are true and B, D, and E are false
C) Consistent. Consistent valuation when A, D, and E are true and B and F are false
D) Consistent. Consistent valuation when B, D, and E are true and A and F are false
E) Consistent. Consistent valuation when B and F are true and A, D, and E are false

Accepted Answer

The answer of use indirect truth tables to determine, for...

Question 8

use the following key to translate each of the given arguments into symbols of PL.
B: Brouwer is an intuitionist. 
F: Frege is a logicist.
G: Gödel is a platonist. 
H: Hilbert is a formalist.
-It is not the case that either Frege is a logicist or Brouwer is an intuitionist. Gödel being a platonist is necessary and sufficient for Brouwer being an intuitionist. Hilbert is a formalist. So, Gödel is not a platonist; however, Hilbert is a formalist.

Accepted Answer