Question 1

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-Q • (∼A • ∼Q)

Accepted Answer

A)  True 
B)  False 
C)  Unknown 
A)  True 
B)  False 
C)  Unknown

Question 2

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-∼[(Z ⊃ B) • (P ⊃ C)] $ \lor $ [(X • Y) ≡ A]

Accepted Answer

The answer of Assume A, B, C are true; X,...

Question 3

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-(C $ \lor $ X) ⊃ (Q $ \lor $ A)

Accepted Answer

The answer of Assume A, B, C are true; X,...

Question 4

Construct a complete truth table for each of the following arguments. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, specify a counterexample.
-M ⊃ (N • ∼O)
N ⊃ (O $ \lor $ M) / O ⊃∼N

Accepted Answer

The answer of Construct a complete truth table for each...

Question 5

Construct a complete truth table for each of the following arguments. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)
-A / A • ∼A

Accepted Answer

A)  Valid 
B)  Invalid. Counterexample when A is true 
C)  Invalid. Counterexample when A is false 
D)  Invalid. Counterexamples when A is either true or false. 
A)  Valid 
B)  Invalid. Counterexample when A is true 
C)  Invalid. Counterexample when A is false 
D)  Invalid. Counterexamples when A is either true or false.

Question 6

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-[(P • Q) $ \lor $ (∼P • Q)] $ \lor $ [(P • ∼Q) $ \lor $ (∼P • ∼Q)]

Accepted Answer

The answer of Assume A, B, C are true; X,...

Question 7

construct a complete truth table for each of the following pairs of propositions. Then, using the truth table, determine whether the statements are logically equivalent or contradictory. If neither, determine whether they are consistent or inconsistent.
-J ⊃ (K ⊃ ∼J) and J • K

Accepted Answer

A)  Logically equivalent 
B)  Contradictory 
C)  Neither logically equivalent nor contradictory, but consistent 
D)  Inconsistent 
A)  Logically equivalent 
B)  Contradictory 
C)  Neither logically equivalent nor contradictory, but consistent 
D)  Inconsistent

Question 8

construct a complete truth table for each of the following pairs of propositions. Then, using the truth table, determine whether the statements are logically equivalent or contradictory. If neither, determine whether they are consistent or inconsistent.
-(D ⊃ E) • (E ⊃ D) and (E $ \lor $ ∼D) • (D $ \lor $ ∼E)

Accepted Answer

A)  Logically equivalent 
B)  Contradictory 
C)  Neither logically equivalent nor contradictory, but consistent 
D)  Inconsistent 
A)  Logically equivalent 
B)  Contradictory 
C)  Neither logically equivalent nor contradictory, but consistent 
D)  Inconsistent

Question 9

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)
-A ⊃ B
C ⊃ ∼B
∼A ⊃ D
E ⊃ ∼F
∼E ⊃ G
∼F ⊃ C / D $ \lor $ G

Accepted Answer

A)  Valid 
B)  Invalid. Counterexample when A, B, C, E, and F are true and D and G are false 
C)  Invalid. Counterexample when A, B, and F are true and C, D, E, and G are false 
D)  Invalid. Counterexample when B and E are true and A, C, D, F, and G are false 
E)  Invalid. Counterexample when C is true and A, B, D, E, F, and G are false 
A)  Valid 
B)  Invalid. Counterexample when A, B, C, E, and F are true and D and G are false 
C)  Invalid. Counterexample when A, B, and F are true and C, D, E, and G are false 
D)  Invalid. Counterexample when B and E are true and A, C, D, F, and G are false 
E)  Invalid. Counterexample when C is true and A, B, D, E, F, and G are false

Question 10

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-X ⊃ [(∼X $ \lor $ A) ⊃ X]

Accepted Answer

A)  True 
B)  False 
C)  Unknown 
A)  True 
B)  False 
C)  Unknown

Question 11

Construct a complete truth table for each of the following arguments. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)
-M ⊃ (N • ∼O)
N ⊃ (O $ \lor $ M) / O ⊃ ∼N

Accepted Answer

A)  Valid 
B)  Invalid. Counterexample when M, N, and O is true 
C)  Invalid. Counterexample when M and N are true and O is false 
D)  Invalid. Counterexample when M is false and N and O are true 
E)  Invalid. Counterexample when M and N are false and O is true 
A)  Valid 
B)  Invalid. Counterexample when M, N, and O is true 
C)  Invalid. Counterexample when M and N are true and O is false 
D)  Invalid. Counterexample when M is false and N and O are true 
E)  Invalid. Counterexample when M and N are false and O is true

Question 12

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction.
-[(Q ⊃ R) ⊃ (R ⊃ S)] ≡ ∼(∼Q $ \lor $ S)

Accepted Answer

A)  Tautology 
B)  Contingency 
C)  Contradiction 
A)  Tautology 
B)  Contingency 
C)  Contradiction

Question 13

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the α assignment; the second variable in the formula read left to right (if any) gets the β assignment; the third variable in the formula read left to right (if any) gets the γ assignment; and the fourth variable in the formula read left to right (if any) gets the δ assignment.
For exercises with only one propositional variable, the standard assignment is:
 $\begin{array} { |l|  }
 \hline \alpha\ \hline \text { 1}\ \hline \text {0 }\\hline
\end{array}$
 
For exercises with two propositional variables, the standard assignment is:
 $$\begin{array} { | c | c | } 
\hline \alpha& \boldsymbol { \beta } \
\hline 1 & 1 \
\hline 1 & 0 \
\hline 0 & 1 \
\hline0& 0\
\hline
\end{array}$$ 
For exercises with three propositional variables, the standard assignment is:
 $$\begin{array} { | c | c | c | } 
\hline \alpha & \beta & \gamma \
\hline 1 & 1 & 1 \
\hline 1 & 1 & 0 \
\hline 1 & 0 & 1 \
\hline 1 & 0& 0e \
\hline 0 & 1 & 1 \
\hline 0 & 1 & 0 \
\hline 0 &0& 1 \
\hline 0 & 0& 0 \
\hline
\end{array}$$ 
For exercises with four propositional variables, the standard assignment is:
 $$\begin{array} { | l | l | l | l | } 
\hline \alpha & \boldsymbol { \beta } &\gamma& \delta\ \
\hline 1 & 1 & 1 & 1 \
\hline 1 & 1 & 1 & 0 \
\hline 1 & 1 & 0 & 1 \
\hline 1 & 1 & 0 & 0 \
\hline 1 & 0 & 1 & 1 \
\hline 1 & 0 & 1 & 0\
\hline 1 & 0 & 0 & 1 \
\hline 1 & 0& 0 & 0 \
\hline 0 & 1 & 1 & 1 \
\hline 0 & 1 & 1 &0 \
\hline
\end{array}$$ 
 $$\begin{array} { | c | c | c | c | } 
\hline 0 & 1 & 0 & 1 \
\hline 0 & 1 & 0 & 0 \
\hline 0& 0 & 1 & 1 \
\hline 0 & 0 & 1 & 0 \
\hline 0 & 0 & 0 & 1 \
\hline 0 & 0 & 0 & 0 \
\hline
\end{array}$$ 
 $$\begin{array} { |l | l | l | l |} 
\hline 0 & 1 & 0 & 1 \
\hline
\end{array}$$ 
For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions.
-[M $ \lor $ (N • O)] $ \lor $ [(M • N) $ \lor $ (?M • N)]

Accepted Answer

A)  1111 1111 
B)  1100 1100 
C)  1111 1100 
D)  1100 0000 
E)  0000 1100 
A)  1111 1111 
B)  1100 1100 
C)  1111 1100 
D)  1100 0000 
E)  0000 1100

Question 14

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.
-If Gödel is a platonist, then Frege is a logicist. If Frege is a logicist, Brouwer being an intuitionist is a sufficient condition for Hilbert being a formalist. Gödel is a platonist. Gödel is a platonist if Hilbert is a formalist. Therefore, Gödel is a platonist if Brouwer is an intuitionist.

Accepted Answer

The answer of use the following key to translate each...

Question 15

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction.
Justify your answers by appeal to the meanings of those terms.
-[(P $ \lor $ Q) • (R • S)] ⊃ [(P • R) $ \lor $ (Q • S)]

Accepted Answer

The answer of construct a complete truth table for each...

Question 16

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-∼Q $ \lor $ (∼X $ \lor $ Q)

Accepted Answer

A)  True 
B)  False 
C)  Unknown 
A)  True 
B)  False 
C)  Unknown

Question 17

For each of the following questions, determine whether the given formula is a wff or not. If it is a wff, indicate its main operator.
-∼[(R • S) ⊃ U] ⊃ {{∼U ⊃ [R ⊃ (S ⊃ T)]}

Accepted Answer

A)  It's a wff. The main operator is the first ⊃, reading left to right. 
B)  It's a wff. The main operator is the first ∼, reading left to right. 
C)  It's a wff. The main operator is the second ⊃, reading left to right. 
D)  It's a wff. The main operator is the third ⊃, reading left to right. 
E)  Not a wff 
A)  It's a wff. The main operator is the first ⊃, reading left to right. 
B)  It's a wff. The main operator is the first ∼, reading left to right. 
C)  It's a wff. The main operator is the second ⊃, reading left to right. 
D)  It's a wff. The main operator is the third ⊃, reading left to right. 
E)  Not a wff

Question 18

Instructions: For questions 11-20, construct complete a truth table for each of the following pairs of propositions. Then, using the truth table, determine whether the statements are logically equivalent or contradictory. If neither, determine whether they are consistent or inconsistent. Justify your answers.
-A ⊃ ∼A and ∼A⊃ A

Accepted Answer

The answer of Instructions: For questions 11-20, construct complete a...

Question 19

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-(∼P ⊃ P) $ \lor $ (A ⊃ P)

Accepted Answer

A)  True 
B)  False 
C)  Unknown 
A)  True 
B)  False 
C)  Unknown

Question 20

For each of the following questions, determine whether the given formula is a wff or not. If it is a wff, indicate its main operator.
-I • E ⊃ ∼F

Accepted Answer

A)  It's a wff. The main operator is the •. 
B)  It's a wff. The main operator is the ⊃. 
C)  It's a wff. The main operator is the ∼. 
D)  Not a wff 
A)  It's a wff. The main operator is the •. 
B)  It's a wff. The main operator is the ⊃. 
C)  It's a wff. The main operator is the ∼. 
D)  Not a wff

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -Q • (∼A • ∼Q)

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -∼[(Z ⊃ B) • (P ⊃ C)] $\lor$ [(X • Y) ≡ A]

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -(C $\lor$ X) ⊃ (Q $\lor$ A)

Construct a complete truth table for each of the following arguments. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, specify a counterexample. -M ⊃ (N • ∼O) N ⊃ (O $\lor$ M) / O ⊃∼N

Construct a complete truth table for each of the following arguments. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.) -A / A • ∼A

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -[(P • Q) $\lor$ (∼P • Q)] $\lor$ [(P • ∼Q) $\lor$ (∼P • ∼Q)]

construct a complete truth table for each of the following pairs of propositions. Then, using the truth table, determine whether the statements are logically equivalent or contradictory. If neither, determine whether they are consistent or inconsistent. -J ⊃ (K ⊃ ∼J) and J • K

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.) -A ⊃ B C ⊃ ∼B ∼A ⊃ D E ⊃ ∼F ∼E ⊃ G ∼F ⊃ C / D $\lor$ G

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -X ⊃ [(∼X $\lor$ A) ⊃ X]

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction. -[(Q ⊃ R) ⊃ (R ⊃ S)] ≡ ∼(∼Q $\lor$ S)

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -∼Q $\lor$ (∼X $\lor$ Q)

For each of the following questions, determine whether the given formula is a wff or not. If it is a wff, indicate its main operator. -∼[(R • S) ⊃ U] ⊃ {{∼U ⊃ [R ⊃ (S ⊃ T)]}

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -(∼P ⊃ P) $\lor$ (A ⊃ P)

For each of the following questions, determine whether the given formula is a wff or not. If it is a wff, indicate its main operator. -I • E ⊃ ∼F

Introducing Logic

Inference in Propositional Logic

Monadic Predicate Logic

Full First-Order Logic

Filters

Exam 2: Propositional Logic: Syntax and Semantic

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -Q • (∼A • ∼Q)

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -∼[(Z ⊃ B) • (P ⊃ C)] ∨ \lor ∨ [(X • Y) ≡ A]

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -(C ∨ \lor ∨ X) ⊃ (Q ∨ \lor ∨ A)

Construct a complete truth table for each of the following arguments. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, specify a counterexample. -M ⊃ (N • ∼O) N ⊃ (O ∨ \lor ∨ M) / O ⊃∼N

Construct a complete truth table for each of the following arguments. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.) -A / A • ∼A

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -[(P • Q) ∨ \lor ∨ (∼P • Q)] ∨ \lor ∨ [(P • ∼Q) ∨ \lor ∨ (∼P • ∼Q)]

construct a complete truth table for each of the following pairs of propositions. Then, using the truth table, determine whether the statements are logically equivalent or contradictory. If neither, determine whether they are consistent or inconsistent. -J ⊃ (K ⊃ ∼J) and J • K

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.) -A ⊃ B C ⊃ ∼B ∼A ⊃ D E ⊃ ∼F ∼E ⊃ G ∼F ⊃ C / D ∨ \lor ∨ G

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -X ⊃ [(∼X ∨ \lor ∨ A) ⊃ X]

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction. -[(Q ⊃ R) ⊃ (R ⊃ S)] ≡ ∼(∼Q ∨ \lor ∨ S)

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -∼Q ∨ \lor ∨ (∼X ∨ \lor ∨ Q)

For each of the following questions, determine whether the given formula is a wff or not. If it is a wff, indicate its main operator. -∼[(R • S) ⊃ U] ⊃ {{∼U ⊃ [R ⊃ (S ⊃ T)]}

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -(∼P ⊃ P) ∨ \lor ∨ (A ⊃ P)

For each of the following questions, determine whether the given formula is a wff or not. If it is a wff, indicate its main operator. -I • E ⊃ ∼F

Introducing Logic

Inference in Propositional Logic

Monadic Predicate Logic

Full First-Order Logic

Filters

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -∼[(Z ⊃ B) • (P ⊃ C)] $\lor$ [(X • Y) ≡ A]

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -(C $\lor$ X) ⊃ (Q $\lor$ A)

Construct a complete truth table for each of the following arguments. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, specify a counterexample. -M ⊃ (N • ∼O) N ⊃ (O $\lor$ M) / O ⊃∼N

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -[(P • Q) $\lor$ (∼P • Q)] $\lor$ [(P • ∼Q) $\lor$ (∼P • ∼Q)]

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.) -A ⊃ B C ⊃ ∼B ∼A ⊃ D E ⊃ ∼F ∼E ⊃ G ∼F ⊃ C / D $\lor$ G

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -X ⊃ [(∼X $\lor$ A) ⊃ X]

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction. -[(Q ⊃ R) ⊃ (R ⊃ S)] ≡ ∼(∼Q $\lor$ S)

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -∼Q $\lor$ (∼X $\lor$ Q)

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -(∼P ⊃ P) $\lor$ (A ⊃ P)