Question 1

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:
 $$\begin{array} { | c | c | } 
\hline \mathbf { n } & \boldsymbol { \beta } \
\hline 1 & 1 \
\hline 1 & 0 \
\hline 0 & 1 \
\hline 0 & 0 \
\hline
\end{array}$$ 
For exercises with three propositional variables, the standard assignment is:
 $$\begin{array} { | c | c | c | } 
\hline \alpha & \beta & \gamma \
\hline 1 & 1 & 1 \
\hline 1 & 1 & 0 \
\hline 1 & 0 & 1 \
\hline 1 & 0 & 0 \
\hline 0 & 1 & 1 \
\hline 0 & 1 & 0 \
\hline 0& 0 & 1 \
\hline 0 & 0 &0\
\hline
\end{array}$$ 
For exercises with four propositional variables, the standard assignment is:
 $\begin{array}{|c|c|c|c|}
\hline\alpha & \beta & \gamma & \delta \
\hline 1 & 1 & 1 & 1 \
\hline 1 & 1 & 1 & 0 \
\hline 1 & 1 & 0 & 1 \
\hline 1 & 1 & 0 & 0\
\hline 1 & 0 & 1 & 1 \
\hline 1 & 0 & 1 & 0 \
\hline 1 & 0 & 0 & 1 \
\hline 1 & 0 & 0 & 0 \
\hline 0 & 1 & 1 & 1 \
\hline 0 & 1 & 1 & 0 \
\hline 0 & 1 & 0 & 1\
\hline 0 & 1 & 0 & 0 \
\hline 0 & 0 & 1 & 1 \
\hline 0 & 0 & 1 & 0 \
\hline 0 & 0 & 0 & 1 \
\hline 0 & 0 & 0 & 0\\hline
\end{array}$ 
Complete truth tables for each of the following propositions.
-(K ? L) ? (?K ?L)

Accepted Answer

The answer of Note: the solutions to most of the...

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.
-X ⊃ [(∼X $ \lor $ A) ⊃ X]

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.
-∼[(A ⊃ P) $ \lor $ (A ⊃ Q)] • (P • Q)

Accepted Answer

A)  True 
B)  False 
C)  Unknown

Question 4

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 • ∼G) ⊃ (G $ \lor $ ∼G)

Accepted Answer

A)  Tautology 
B)  Contingency 
C)  Contradiction

Question 5

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, specify a counterexample.
-(W $ \lor $ X) ⊃∼(Y • ∼Z)
∼(Y • W) ⊃∼Z
∼(W $ \lor $ X) $ \lor $ Z / (Z $ \lor $ W) ⊃ (X • Y)

Accepted Answer

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

Question 6

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:
 $$\begin{array} { | c | c | } 
\hline \mathbf { n } & \boldsymbol { \beta } \
\hline 1 & 1 \
\hline 1 & 0 \
\hline 0 & 1 \
\hline 0 & 0 \
\hline
\end{array}$$ 
For exercises with three propositional variables, the standard assignment is:
 $$\begin{array} { | c | c | c | } 
\hline \alpha & \beta & \gamma \
\hline 1 & 1 & 1 \
\hline 1 & 1 & 0 \
\hline 1 & 0 & 1 \
\hline 1 & 0 & 0 \
\hline 0 & 1 & 1 \
\hline 0 & 1 & 0 \
\hline 0& 0 & 1 \
\hline 0 & 0 &0\
\hline
\end{array}$$ 
For exercises with four propositional variables, the standard assignment is:
 $\begin{array}{|c|c|c|c|}
\hline\alpha & \beta & \gamma & \delta \
\hline 1 & 1 & 1 & 1 \
\hline 1 & 1 & 1 & 0 \
\hline 1 & 1 & 0 & 1 \
\hline 1 & 1 & 0 & 0\
\hline 1 & 0 & 1 & 1 \
\hline 1 & 0 & 1 & 0 \
\hline 1 & 0 & 0 & 1 \
\hline 1 & 0 & 0 & 0 \
\hline 0 & 1 & 1 & 1 \
\hline 0 & 1 & 1 & 0 \
\hline 0 & 1 & 0 & 1\
\hline 0 & 1 & 0 & 0 \
\hline 0 & 0 & 1 & 1 \
\hline 0 & 0 & 1 & 0 \
\hline 0 & 0 & 0 & 1 \
\hline 0 & 0 & 0 & 0\\hline
\end{array}$ 
Complete truth tables for each of the following propositions.
-[J ⊃ (K $ \lor $ L)] ⊃ [∼L ⊃ (K $ \lor $ J)]

Accepted Answer

The answer of Note: the solutions to most of the...

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.)
-X ≡ Y
Y ≡ ∼Z
Z ≡ ∼X

Accepted Answer

A)  Inconsistent 
B)  Consistent. Consistent valuation when X, Y, and Z are true 
C)  Consistent. Consistent valuation when X and Y are true and Z is false 
D)  Consistent. Consistent valuation when X and Z are true and Y is false 
E)  Consistent. Consistent valuation when Y and Z are true and X is false

Question 8

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, specify a counterexample.
-(U • ∼V) $ \lor $ W
W ≡ ∼V / ∼U

Accepted Answer

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

Question 9

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.
-[(R • S) ⊃ U] ⊃ {{∼U ⊃ [R ⊃ (S ⊃ T)]}

Accepted Answer

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

Question 10

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.
-Hilbert is a formalist if, and only if, Gödel is a platonist. Hilbert is not a formalist and Brouwer is an intuitionist. Hilbert is a formalist if Frege is a logicist. Therefore, Frege is not a logicist and Gödel is not a platonist.

Accepted Answer

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

Question 11

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.)
-(P $ \lor $ Q) ⊃ R
(S $ \lor $ ∼U) ⊃ (∼R • ∼W)
S ⊃ (P • T) / ∼S

Accepted Answer

A)  Valid 
B)  Invalid. Counterexample when P, Q, R, S, T, and U are true and W is false 
C)  Invalid, Counterexample when P, S, T and W are true and Q, R, and U are false 
D)  Invalid . Counterexample when S, T, and U are true and P, Q, R, and W are false 
E)  Invalid. Counterexample when S, T, and U are true and P, Q, R, and W are false

Question 12

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.
-[P ? (?R ? Q)] $ \lor $ [(Q $ \lor $ R) $ \lor $ (?P • R)]

Accepted Answer

A)  1111 1111 
B)  1110 1111 
C)  1110 1110 
D)  0000 0011 
E)  0000 1100

Question 13

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

Accepted Answer

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

Question 14

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.
-P ⊃ (Q ⊃ R) ⊃ [(P • ∼R) ⊃ ∼Q]

Accepted Answer

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

Question 15

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.
-(V ? W) • (V ? ?W)

Accepted Answer

A)  1111 
B)  1001 
C)  0110 
D)  1000 
E)  0000

Question 16

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:
 $$\begin{array} { | c | c | } 
\hline \mathbf { n } & \boldsymbol { \beta } \
\hline 1 & 1 \
\hline 1 & 0 \
\hline 0 & 1 \
\hline 0 & 0 \
\hline
\end{array}$$ 
For exercises with three propositional variables, the standard assignment is:
 $$\begin{array} { | c | c | c | } 
\hline \alpha & \beta & \gamma \
\hline 1 & 1 & 1 \
\hline 1 & 1 & 0 \
\hline 1 & 0 & 1 \
\hline 1 & 0 & 0 \
\hline 0 & 1 & 1 \
\hline 0 & 1 & 0 \
\hline 0& 0 & 1 \
\hline 0 & 0 &0\
\hline
\end{array}$$ 
For exercises with four propositional variables, the standard assignment is:
 $\begin{array}{|c|c|c|c|}
\hline\alpha & \beta & \gamma & \delta \
\hline 1 & 1 & 1 & 1 \
\hline 1 & 1 & 1 & 0 \
\hline 1 & 1 & 0 & 1 \
\hline 1 & 1 & 0 & 0\
\hline 1 & 0 & 1 & 1 \
\hline 1 & 0 & 1 & 0 \
\hline 1 & 0 & 0 & 1 \
\hline 1 & 0 & 0 & 0 \
\hline 0 & 1 & 1 & 1 \
\hline 0 & 1 & 1 & 0 \
\hline 0 & 1 & 0 & 1\
\hline 0 & 1 & 0 & 0 \
\hline 0 & 0 & 1 & 1 \
\hline 0 & 0 & 1 & 0 \
\hline 0 & 0 & 0 & 1 \
\hline 0 & 0 & 0 & 0\\hline
\end{array}$ 
Complete truth tables for each of the following propositions.
-{[X ≡ ∼(Y $ \lor $ Z)] • [Y ≡ ∼(X $ \lor $ Z)]} ⊃ [(X • Z) $ \lor $ ∼(X • ∼Z)]

Accepted Answer

The answer of Note: the solutions to most of the...

Question 17

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.
-(B ? B) $ \lor $ ?B

Accepted Answer

A)  11 
B)  10 
C)  01 
D)  00

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.
-X $ \lor $ [A • (B ⊃ Y)]

Accepted Answer

A)  True 
B)  False 
C)  Unknown

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.
-(C $ \lor $ X) ⊃ (Q $ \lor $ A)

Accepted Answer

A)  True 
B)  False 
C)  Unknown

Question 20

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.
-(?C ? ?D) ? [D ? (C ? D)]

Accepted Answer

A)  1111 
B)  1101 
C)  1011 
D)  0011 
E)  0101

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 ⊃ [(∼X $\lor$ A) ⊃ 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. -∼[(A ⊃ P) $\lor$ (A ⊃ Q) ] • (P • Q)

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 • ∼G) ⊃ (G $\lor$ ∼G)

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, specify a counterexample. -(W $\lor$ X) ⊃∼(Y • ∼Z) ∼(Y • W) ⊃∼Z ∼(W $\lor$ X) $\lor$ Z / (Z $\lor$ W) ⊃ (X • Y)

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.) -X ≡ Y Y ≡ ∼Z Z ≡ ∼X

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, specify a counterexample. -(U • ∼V) $\lor$ W W ≡ ∼V / ∼U

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. -[(R • S) ⊃ U] ⊃ {{∼U ⊃ [R ⊃ (S ⊃ T)]}

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.) -(P $\lor$ Q) ⊃ R (S $\lor$ ∼U) ⊃ (∼R • ∼W) S ⊃ (P • T) / ∼S

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

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. -P ⊃ (Q ⊃ R) ⊃ [(P • ∼R) ⊃ ∼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. -X $\lor$ [A • (B ⊃ Y) ]

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)

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 ⊃ [(∼X ∨ \lor ∨ A) ⊃ 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. -∼[(A ⊃ P) ∨ \lor ∨ (A ⊃ Q) ] • (P • Q)

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 • ∼G) ⊃ (G ∨ \lor ∨ ∼G)

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, specify a counterexample. -(W ∨ \lor ∨ X) ⊃∼(Y • ∼Z) ∼(Y • W) ⊃∼Z ∼(W ∨ \lor ∨ X) ∨ \lor ∨ Z / (Z ∨ \lor ∨ W) ⊃ (X • Y)

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.) -X ≡ Y Y ≡ ∼Z Z ≡ ∼X

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, specify a counterexample. -(U • ∼V) ∨ \lor ∨ W W ≡ ∼V / ∼U

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. -[(R • S) ⊃ U] ⊃ {{∼U ⊃ [R ⊃ (S ⊃ T)]}

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

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. -P ⊃ (Q ⊃ R) ⊃ [(P • ∼R) ⊃ ∼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. -X ∨ \lor ∨ [A • (B ⊃ Y) ]

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)

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]

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. -∼[(A ⊃ P) $\lor$ (A ⊃ Q) ] • (P • Q)

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 • ∼G) ⊃ (G $\lor$ ∼G)

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, specify a counterexample. -(W $\lor$ X) ⊃∼(Y • ∼Z) ∼(Y • W) ⊃∼Z ∼(W $\lor$ X) $\lor$ Z / (Z $\lor$ W) ⊃ (X • Y)

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, specify a counterexample. -(U • ∼V) $\lor$ W W ≡ ∼V / ∼U

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

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 $\lor$ [A • (B ⊃ Y) ]

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)