Question 1

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)

Accepted Answer

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

Question 2

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.)
-I ⊃ J
∼(J • I) / J ⊃ I

Accepted Answer

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

Question 3

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

Accepted Answer

Logically equivalent

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.)
-B $ \lor $ ∼C
∼∼C / ∼B

Accepted Answer

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

Question 5

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

Accepted Answer

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

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.
-(I ? J) $ \lor $ (J ? I)

Accepted Answer

A)   1111
B)   1100
C)   1010
D)   0000

Question 7

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, specify a counterexample.
-D ⊃ (E $ \lor $ F)
D ⊃ (G $ \lor $ F)
∼(F $ \lor $ H) / D ⊃ (E • G)

Accepted Answer

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

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.
-J ⊃ (K ⊃ ∼J) and J • K

Accepted Answer

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

Question 9

use the following key to translate each of the arguments of PL into natural, English arguments.
A: Peirce studied logic.
B: James was a pluralist.
C: Dewey wrote about thirdness.
D: Dewey denigrated the quest for certainty. E: Peirce emphasized education.
-1. ∼C ⊃ (A ≡ B)
2. D $ \lor $ ∼E
3. ∼D • (∼E ⊃ ∼C) / B ⊃ A

Accepted Answer

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

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.
-It is not the case that either Frege is a logicist or Brouwer is an intuitionist. Gödel being a platonist is necessary and sufficient for Brouwer being an intuitionist. Hilbert is a formalist. So, Gödel is not a platonist; however, Hilbert is a formalist.

Accepted Answer

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

Question 11

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.
-A ? [(?B $ \lor $ A) ? ?A]

Accepted Answer

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

Question 12

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.)
-E ≡ F
F ⊃ ∼E / E

Accepted Answer

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

Question 13

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.
-I ⊃ J
∼(J • I) / J ⊃ I

Accepted Answer

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

Question 14

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)]}

Accepted Answer

A)   0011 1111
B)   0011 0000
C)   0101 1101
D)   1111 0101
E)   1111 1101

Question 15

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.)
-P ≡ (Q • ∼R)
P ⊃ ∼Q
(∼P • Q) ⊃ S

Accepted Answer

A)   Inconsistent
B)   Consistent. Consistent valuation when P, Q, and S are true and R is false
C)   Consistent. Consistent valuation when P and Q are true and S and R are false
D)   Consistent. Consistent valuation when Q, R and S are true and P is false
E)   Consistent. Consistent valuation when Q and R are false and P and S are true

Question 16

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.
-B $ \lor $ ∼C
∼∼C / ∼B

Accepted Answer

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

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.
Justify your answers by appeal to the meanings of those terms.
-(A • ∼B) • (B $ \lor $ ∼A)

Accepted Answer

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

Question 18

use the following key to determine which English sentence best represents the given formula of PL.
A: Peirce studied logic.
B: James was a pluralist.
C: Dewey wrote about thirdness.
D: Dewey denigrated the quest for certainty. 
E: Peirce emphasized education.
-∼B ⊃ (∼D • ∼E)

Accepted Answer

A)   If James was not a pluralist, then neither Dewey denigrated the quest for certainty nor Peirce emphasized education.
B)   Dewey did not denigrate the quest for certainty and Peirce did not emphasize education given that James was not a pluralist.
C)   James not being a pluralist entails neither Dewey denigrated the quest for certainty nor Peirce did not emphasize education.
D)   James was not a pluralist provided that Dewey did not denigrate the quest for certainty and Peirce did not emphasize education.
E)   If James was not a pluralist, then either Dewey did not denigrate the quest for certainty or Peirce did not emphasize education.

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.
-∼Y ⊃ [A ≡ (Y • B)]

Accepted Answer

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

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:
 $$\begin{array} { | c | c | } 
\hline \mathbf { n } & \boldsymbol { \beta } \
\hline 1 & 1 \
\hline 1 & 0 \
\hline 0 & 1 \
\hline 0 & 0 \
\hline
\end{array}$$ 
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...

Exam 2: Propositional Logic: Syntax and Semantic

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, specify a counterexample. -D ⊃ (E $\lor$ F) D ⊃ (G $\lor$ F) ∼(F $\lor$ H) / D ⊃ (E • G)

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.) -E ≡ F F ⊃ ∼E / E

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. -I ⊃ J ∼(J • I) / J ⊃ I

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.) -P ≡ (Q • ∼R) P ⊃ ∼Q (∼P • Q) ⊃ 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. -B $\lor$ ∼C ∼∼C / ∼B

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. -(A • ∼B) • (B $\lor$ ∼A)

use the following key to determine which English sentence best represents the given formula of PL. A: Peirce studied logic. B: James was a pluralist. C: Dewey wrote about thirdness. D: Dewey denigrated the quest for certainty. E: Peirce emphasized education. -∼B ⊃ (∼D • ∼E)

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)]