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.
-{X $ \lor $ [C • (Y ⊃ B)]} ⊃ {Z ⊃ [Z ⊃ (Z ⊃ Z)]}

Accepted Answer

A)  True 
B)  False 
C)  Unknown

Question 2

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

Accepted Answer

The answer of For each of the following questions, determine...

Question 3

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.
-F • (∼G $ \lor $ H) and F ≡ (∼H • G)

Accepted Answer

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

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, choose an option which presents a counterexample. (There may be other counterexamples as well.)
-(G $ \lor $ ∼H) $ \lor $ ∼∼I / G $ \lor $ ∼H

Accepted Answer

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

Question 5

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.
-(G ? H) ? ? [(G • H) $ \lor $ (?G • ?H)]

Accepted Answer

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

Question 6

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.
-S ⊃ (T $ \lor $ U)
∼T
T $ \lor $ S / U • S

Accepted Answer

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

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.
-L $ \lor $ (M ⊃ N) and ∼L • (M • ∼N)

Accepted Answer

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

Question 8

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.
-G ≡ ∼H and (H • ∼G) $ \lor $ (G • ∼H)

Accepted Answer

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

Question 9

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.
-∼A ≡ B
(∼A $ \lor $ B) ⊃ C / C

Accepted Answer

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

Question 10

For each of the following questions, determine whether the given formulas is a wff or not. If it is a wff, indicate its main operator.
-[A $ \lor $ B • C] ⊃ (A $ \lor $ C)

Accepted Answer

The answer of For each of the following questions, determine...

Question 11

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, specify a counterexample.
-(M • N) ? (O • P)
(M • O) ? (Q $ \lor $?P)
R ? (Q ? N) / (?M • ?N) $ \lor $ (O • P)

Accepted Answer

The answer of use indirect truth tables to determine whether...

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.
-(A • ∼B) • (B $ \lor $ ∼A)

Accepted Answer

A)  Tautology 
B)  Contingency 
C)  Contradiction

Question 13

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.
-[(W ⊃ X) • (Y $ \lor $ ∼X)] ≡ [∼(Z $ \lor $ Y) ⊃ ∼W]

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

Question 14

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 ⊃ C) $ \lor $ (Z • ∼X)

Accepted Answer

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

Question 15

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 Frege is a logicist, then Brouwer is an intuitionist. If Brouwer is an intuitionist, then Gödel is a platonist only if Hilbert is a formalist. Gödel is a platonist. Frege is a logicist. So, Hilbert is a formalist.

Accepted Answer

The answer of use the following key to translate 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.
-∼Y ⊃ [A ≡ (Y • B)]

Accepted Answer

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.
-[(P ≡ Q) • ∼Q] ⊃ (P ⊃ R)

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)  It's a wff. The main operator is the first ⊃, reading left to right. 
E)  Not a wff

Question 18

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

Accepted Answer

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

Question 19

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.
-{[X ? ?(Y $ \lor $ Z)] • [Y ? ?(X $ \lor $ Z)]} ? [(X • Z) $ \lor $ ?(X • ?Z)]

Accepted Answer

A)  1111 1111 
B)  1110 1111 
C)  1010 1111 
D)  0001 1110 
E)  0001 0110

Question 20

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.
-(G ≡ H) ⊃ ∼[(G • H) $ \lor $ (∼G • ∼H)]

Accepted Answer

A)  Tautology 
B)  Contingency 
C)  Contradiction

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. -{X $\lor$ [C • (Y ⊃ B) ]} ⊃ {Z ⊃ [Z ⊃ (Z ⊃ Z) ]}

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

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. -F • (∼G $\lor$ H) and F ≡ (∼H • G)

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. -(G ? H) ? ? [(G • H) $\lor$ (?G • ?H)]

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. -S ⊃ (T $\lor$ U) ∼T T $\lor$ S / U • S

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. -L $\lor$ (M ⊃ N) and ∼L • (M • ∼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, specify a counterexample. -∼A ≡ B (∼A $\lor$ B) ⊃ C / C

For each of the following questions, determine whether the given formulas is a wff or not. If it is a wff, indicate its main operator. -[A $\lor$ B • C] ⊃ (A $\lor$ C)

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, specify a counterexample. -(M • N) ? (O • P) (M • O) ? (Q $\lor$ ?P) R ? (Q ? N) / (?M • ?N) $\lor$ (O • P)

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. -(A • ∼B) • (B $\lor$ ∼A)

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. -[(W ⊃ X) • (Y $\lor$ ∼X) ] ≡ [∼(Z $\lor$ Y) ⊃ ∼W]

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 ⊃ C) $\lor$ (Z • ∼X)

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. -∼Y ⊃ [A ≡ (Y • B) ]

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. -[(P ≡ Q) • ∼Q] ⊃ (P ⊃ R)

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

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. -(G ≡ H) ⊃ ∼[(G • H) $\lor$ (∼G • ∼H) ]

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. -{X ∨ \lor ∨ [C • (Y ⊃ B) ]} ⊃ {Z ⊃ [Z ⊃ (Z ⊃ Z) ]}

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

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. -(G ? H) ? ? [(G • H) ∨ \lor ∨ (?G • ?H)]

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. -S ⊃ (T ∨ \lor ∨ U) ∼T T ∨ \lor ∨ S / U • S

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. -∼A ≡ B (∼A ∨ \lor ∨ B) ⊃ C / C

For each of the following questions, determine whether the given formulas is a wff or not. If it is a wff, indicate its main operator. -[A ∨ \lor ∨ B • C] ⊃ (A ∨ \lor ∨ C)

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, specify a counterexample. -(M • N) ? (O • P) (M • O) ? (Q ∨ \lor ∨ ?P) R ? (Q ? N) / (?M • ?N) ∨ \lor ∨ (O • P)

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. -(A • ∼B) • (B ∨ \lor ∨ ∼A)

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. -[(W ⊃ X) • (Y ∨ \lor ∨ ∼X) ] ≡ [∼(Z ∨ \lor ∨ Y) ⊃ ∼W]

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 ⊃ C) ∨ \lor ∨ (Z • ∼X)