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:
 $$\begin{array} { | c | c | } 
\hline \alpha& \boldsymbol { \beta } \
\hline 1 & 1 \
\hline 1 & 0 \
\hline 0 & 1 \
\hline0& 0\
\hline
\end{array}$$ 
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.
-[J ? (K $ \lor $ L)] ? [?L ? (K $ \lor $ J)]
A) 1111 1111
B) 1111 1110
C) 1110 1111
D) 1111 1100
E) 0101 0101

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:
 $$\begin{array} { | c | c | } 
\hline \alpha& \boldsymbol { \beta } \
\hline 1 & 1 \
\hline 1 & 0 \
\hline 0 & 1 \
\hline0& 0\
\hline
\end{array}$$ 
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)]
A) 1111 1111
B) 1100 1100
C) 1111 1100
D) 1100 0000
E) 0000 1100

Accepted Answer

The correct sequence under the main operator for the given proposition, considering the standard assignment of truth values, is 1111 1100. This is because the proposition combines disjunctions and conjunctions in a way that, when evaluated with the standard truth assignments for each variable, results in true outcomes for all cases except the last two when M is false and N is true, regardless of the values of O.

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:
 $$\begin{array} { | c | c | } 
\hline \alpha& \boldsymbol { \beta } \
\hline 1 & 1 \
\hline 1 & 0 \
\hline 0 & 1 \
\hline0& 0\
\hline
\end{array}$$ 
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)]
A) 1111 1111
B) 1110 1111
C) 1110 1110
D) 0000 0011
E) 0000 1100

Accepted Answer

Given the standard assignment of truth values and the logical operators involved, the correct sequence under the main operator for the given proposition is determined by evaluating the proposition with the assigned truth values for each variable. The proposition involves disjunctions (OR) and conjunctions (AND), as well as negations (NOT). Evaluating the proposition step by step for each assignment of truth values to the variables ($\alpha$, $\beta$, $\gamma$, and $\delta$) leads to the sequence 1110 1111, which matches option B.

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:
 $$\begin{array} { | c | c | } 
\hline \alpha& \boldsymbol { \beta } \
\hline 1 & 1 \
\hline 1 & 0 \
\hline 0 & 1 \
\hline0& 0\
\hline
\end{array}$$ 
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.
-(S ? ?T) ? {[V ? (?S • T)] $ \lor $ [V ? (?T • S)]}
A) 0011 1111
B) 0011 0000
C) 0101 1101
D) 1111 0101
E) 1111 1101

Accepted Answer

The correct sequence under the main operator for the given proposition, considering the standard assignment of truth values, is "1111 1101". This is because the proposition involves a combination of logical operations (implication, negation, conjunction, and disjunction) applied to the variables S, T, and V with their respective truth values assigned according to the standard assignment. The operations result in the final sequence of truth values as described.

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:
 $$\begin{array} { | c | c | } 
\hline \alpha& \boldsymbol { \beta } \
\hline 1 & 1 \
\hline 1 & 0 \
\hline 0 & 1 \
\hline0& 0\
\hline
\end{array}$$ 
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)]
A) 1111 1111
B) 1110 1111
C) 1010 1111
D) 0001 1110
E) 0001 0110

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:
 $$\begin{array} { | c | c | } 
\hline \alpha& \boldsymbol { \beta } \
\hline 1 & 1 \
\hline 1 & 0 \
\hline 0 & 1 \
\hline0& 0\
\hline
\end{array}$$ 
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.
-[(E ? ?F) $ \lor $ (G ? ?H)] $ \lor $ [(?E • ?F) • (?G • ?H)]
A) 1001 1111 1111 1001
B) 1001 1111 1111 1000
C) 0110 1111 1111 0110
D) 0110 1111 1111 0111
E) 0110 1111 1111 0000

Accepted Answer

The correct sequence under the main operator for the given proposition, considering the standard assignment of truth values, reflects the behavior of logical OR and AND operations. The sequence "0110 1111 1111 0111" correctly represents the outcomes of these operations, considering all possible truth values for the variables involved.

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:
 $$\begin{array} { | c | c | } 
\hline \alpha& \boldsymbol { \beta } \
\hline 1 & 1 \
\hline 1 & 0 \
\hline 0 & 1 \
\hline0& 0\
\hline
\end{array}$$ 
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.
-[(A ? B) ? (?C ? D)] ? {[A • (B $ \lor $ D)] ? [D • (B $ \lor $ ?D)]}
A) 1111 1010 0000 0000
B) 1110 0001 1110 1110
C) 1011 1111 0101 1111
D) 1100 0100 1100 0100
E) 1110 1111 0011 1010

Accepted Answer

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

Question 8

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.
-(I ⊃ ∼I) ⊃ [I ⊃ (I ⊃ ∼I)]
A) Tautology
B) Contingency
C) Contradiction

Accepted Answer

The truth table for (I ⊃ ∼I) is:

I | ∼I | I ⊃ ∼I
--|----|--------
T |  F |   F 
F |  T |   T

The truth table for (I ⊃ (I ⊃ ∼I)) is:

I | (I ⊃ ∼I) | I ⊃ (I ⊃ ∼I)
--|----------|--------------
T |    F     |       F 
F |    T     |       T

Finally, the truth table for (I ⊃ ∼I) ⊃ [I ⊃ (I ⊃ ∼I)] is:

I | (I ⊃ ∼I) | I ⊃ (I ⊃ ∼I) | (I ⊃ ∼I) ⊃ [I ⊃ (I ⊃ ∼I)]
--|----------|-------------|--------------------------------
T |    F     |      F      |                T
F |    T     |      T      |                T

As we can see, the final column is true for all truth values of I. Therefore, the proposition is a tautology.

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.
-(G • ∼G) ⊃ (G $ \lor $ ∼G)
A) Tautology
B) Contingency
C) Contradiction

Accepted Answer

The truth table for (G • ∼G) is:

G   ∼G   G • ∼G 
T   F     F 
F   T     F

The truth table for (G $ \lor $ ∼G) is:

G   ∼G   G $ \lor $ ∼G 
T   F     T 
F   T     T

The truth table for the entire proposition is:

G   ∼G   G • ∼G   G $ \lor $ ∼G   (G • ∼G) ⊃ (G $ \lor $ ∼G) 
T   F     F         T            T 
F   T     F         T            T

Since the proposition is true for every possible truth value of G and ∼G, it is a tautology.

Question 10

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

Accepted Answer

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

Question 11

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)
A) Tautology
B) Contingency
C) Contradiction

Accepted Answer

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

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

Accepted Answer

(A ≡ ∼B) ⊃ ∼(B • A) is a tautology because it is true for all possible combinations of truth values for A and B.

Question 13

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.
-(C ⊃ ∼D) $ \lor $ (∼D ⊃ C)
A) Tautology
B) Contingency
C) Contradiction

Accepted Answer

This proposition is a tautology because, for all possible truth values of C and D, the proposition is always true. The structure of the proposition ensures that at least one of the implications will always be true, regardless of the truth values of C and D.

Question 14

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)]
A) Tautology
B) Contingency
C) Contradiction

Accepted Answer

A contingency is a proposition that is neither always true (tautology) nor always false (contradiction). The given proposition, (G ≡ H) ⊃ ∼[(G • H) ∨ (∼G • ∼H)], is true in some cases and false in others, depending on the truth values of G and H. The truth table for this proposition shows that it is not always true nor always false, hence it is a contingency.

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.
-[N ⊃ (O ⊃ P)] ⊃ (N ⊃ O)
A) Tautology
B) Contingency
C) Contradiction

Accepted Answer

To construct the truth table:
- Use four columns for each variable: N, O, P, and the entire proposition.
- Fill in the N, O, and P columns with all possible combinations of T and F (there are 8 total).
- Use the given logical operators to determine the truth value of each part of the proposition and then the entire proposition for each row.
- If the proposition is true for all rows, it is a tautology. If it is false for all rows, it is a contradiction. If it is true for some rows and false for others, it is a contingency.

Using this method, we get the following truth table:

N | O | P | (N ⊃ (O ⊃ P)) ⊃ (N ⊃ O)
---------------------------------------
T | T | T | T
T | T | F | T
T | F | T | T
T | F | F | T
F | T | T | T
F | T | F | F
F | F | T | T
F | F | F | T

Since the entire proposition is false for one row (row 6) and true for all others, it is a contingency. Therefore, the answer is B.

Question 16

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)
A) Tautology
B) Contingency
C) Contradiction

Accepted Answer

To construct the truth table:
- We start with the variables Q, R, and S and list all possible combinations of their truth values (2^3 = 8 rows).
- Then we evaluate each part of the proposition based on those truth values, using the logical operators.
- Finally, we determine the truth value of the entire proposition by comparing the truth values of each side of the biconditional (≡) sign.

The complete truth table is:

| Q | R | S | Q ⊃ R | R ⊃ S | (Q ⊃ R) ⊃ (R ⊃ S) | ∼(∼Q \/ S) | ((Q ⊃ R) ⊃ (R ⊃ S)) ≡ ∼(∼Q \/ S) |
|---|---|---|-------|-------|----------------------|-------------|------------------------------------|
| T | T | T |   T   |   T   |          T           |      F      |                  F                 |
| T | T | F |   T   |   F   |          F           |      F      |                  T                 |
| T | F | T |   F   |   T   |          T           |      F      |                  F                 |
| T | F | F |   F   |   T   |          T           |      T      |                  F                 |
| F | T | T |   T   |   T   |          T           |      F      |                  F                 |
| F | T | F |   T   |   F   |          F           |      F      |                  T                 |
| F | F | T |   T   |   T   |          T           |      T      |                  F                 |
| F | F | F |   T   |   T   |          T           |      T      |                  F                 |

Notice that the proposition is not always true (T) or always false (F), since there are cases where it is true (rows 2 and 6) and cases where it is false (rows 1, 3, 4, 5, 7, and 8). Thus, it is not a tautology or a contradiction. Therefore, the best choice is B, contingency.

Question 17

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.
-[(P $ \lor $ Q) • (R • S)] ⊃ [(P • R) $ \lor $ (Q • S)]
A) Tautology
B) Contingency
C) Contradiction

Accepted Answer

The proposition is a tautology because it is true for all possible combinations of truth values for P, Q, R, and S.

Question 18

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.
-A ⊃ ∼A and ∼A ⊃ A
A) Logically equivalent
B) Contradictory
C) Neither logically equivalent nor contradictory, but consistent
D) Inconsistent

Accepted Answer

A truth table for A ⊃ ∼A and ∼A ⊃ A shows that both statements are true when A is false and both are false when A is true, making them contradictory because they cannot both be true in the same scenario.

Question 19

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 and ∼(E ⊃ ∼D)
A) Logically equivalent
B) Contradictory
C) Neither logically equivalent nor contradictory, but consistent
D) Inconsistent

Accepted Answer

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

Question 20

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.
-G ≡ ∼H and (H • ∼G) $ \lor $ (G • ∼H)
A) Logically equivalent
B) Contradictory
C) Neither logically equivalent nor contradictory, but consistent
D) Inconsistent

Accepted Answer

To construct the truth table, we first need to create columns for G, H, ∼H, ∼G, G ≡ ∼H, and (H • ∼G) $ \lor $ (G • ∼H):

G| H| ∼H| ∼G| G ≡ ∼H| (H • ∼G) $ \lor $ (G • ∼H)
T| T| F | F | F        | F
T| F| T | F | F        | T
F| T| F | T | F        | T
F| F| T | T | T        | T

Since the column for G ≡ ∼H matches the column for (H • ∼G) $ \lor $ (G • ∼H), we can see that the two statements are logically equivalent.

Quiz 2: Propositional Logic: Syntax and Semantic

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. -(I ⊃ ∼I) ⊃ [I ⊃ (I ⊃ ∼I) ]

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)

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 ⊃ A)

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)

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 • A)

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. -(C ⊃ ∼D) $\lor$ (∼D ⊃ C)

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) ]

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. -[N ⊃ (O ⊃ P) ] ⊃ (N ⊃ O)

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)

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

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 and ∼(E ⊃ ∼D)

Quiz 2: Propositional Logic: Syntax and Semantic

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. -(I ⊃ ∼I) ⊃ [I ⊃ (I ⊃ ∼I) ]

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)

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 ⊃ A)

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)

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 • A)

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. -(C ⊃ ∼D) ∨ \lor ∨ (∼D ⊃ C)

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) ]

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. -[N ⊃ (O ⊃ P) ] ⊃ (N ⊃ O)

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)

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. -[(P ∨ \lor ∨ Q) • (R • S) ] ⊃ [(P • R) ∨ \lor ∨ (Q • 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. -A ⊃ ∼A and ∼A ⊃ A

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 and ∼(E ⊃ ∼D)

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)

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)

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. -(C ⊃ ∼D) $\lor$ (∼D ⊃ C)

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) ]

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)

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