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.
-[E $ \lor $ ?(E • ?F)] ? [F ? ?(E $ \lor $ ?F)]

Accepted Answer

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

Question 2

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

Accepted Answer

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

Question 3

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 4

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.
-[M $ \lor $ (N • O)] $ \lor $ [(M • N) $ \lor $ ∼(M • N)]

Accepted Answer

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

Question 5

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

Accepted Answer

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

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.
-(S ⊃ ∼T) ⊃ {[V ⊃ (∼S ⊃ T)] $ \lor $ [V ⊃ (∼T ⊃ S)]}

Accepted Answer

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

Question 7

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 8

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

Accepted Answer

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

Question 9

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.
-(I ⊃ ∼I) ⊃ [I ⊃ (I ⊃ ∼I)]

Accepted Answer