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)]

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

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

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)

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

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)]

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

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

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

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)]

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

Exam 2: Propositional Logic: Syntax and Semantic

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: \alpha 1 0 For exercises with two propositional variables, the standard assignment is: 1 1 1 0 0 1 0 0 For exercises with three propositional variables, the standard assignment is: \alpha \beta \gamma 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 For exercises with four propositional variables, the standard assignment is: \alpha \beta \gamma \delta 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 Complete truth tables for each of the following propositions. -(K ? L) ? (?K ?L)

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Question 201

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]

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Question 202

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)

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Question 203

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)

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Question 204

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)

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Question 205

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: \alpha 1 0 For exercises with two propositional variables, the standard assignment is: 1 1 1 0 0 1 0 0 For exercises with three propositional variables, the standard assignment is: \alpha \beta \gamma 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 For exercises with four propositional variables, the standard assignment is: \alpha \beta \gamma \delta 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 Complete truth tables for each of the following propositions. -[J ⊃ (K $\lor$ L)] ⊃ [∼L ⊃ (K $\lor$ J)]

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Question 206

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

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Question 207

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

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Question 208

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)]}

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Question 209

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.

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Question 210

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

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Question 211

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: \alpha 1 0 For exercises with two propositional variables, the standard assignment is: \alpha 1 1 1 0 0 1 0 0 For exercises with three propositional variables, the standard assignment is: \alpha \beta \gamma 1 1 1 1 1 0 1 0 1 1 0 0e 0 1 1 0 1 0 0 0 1 0 0 0 For exercises with four propositional variables, the standard assignment is: \alpha \gamma \delta\ 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 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)]

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Question 212

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

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Question 213

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]

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Question 214

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: \alpha 1 0 For exercises with two propositional variables, the standard assignment is: \alpha 1 1 1 0 0 1 0 0 For exercises with three propositional variables, the standard assignment is: \alpha \beta \gamma 1 1 1 1 1 0 1 0 1 1 0 0e 0 1 1 0 1 0 0 0 1 0 0 0 For exercises with four propositional variables, the standard assignment is: \alpha \gamma \delta\ 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 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)

(Multiple Choice)

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Question 215

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

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Question 216

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: \alpha 1 0 For exercises with two propositional variables, the standard assignment is: \alpha 1 1 1 0 0 1 0 0 For exercises with three propositional variables, the standard assignment is: \alpha \beta \gamma 1 1 1 1 1 0 1 0 1 1 0 0e 0 1 1 0 1 0 0 0 1 0 0 0 For exercises with four propositional variables, the standard assignment is: \alpha \gamma \delta\ 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 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

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Question 217

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)]

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Question 218

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)

(Multiple Choice)

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Question 219

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: \alpha 1 0 For exercises with two propositional variables, the standard assignment is: \alpha 1 1 1 0 0 1 0 0 For exercises with three propositional variables, the standard assignment is: \alpha \beta \gamma 1 1 1 1 1 0 1 0 1 1 0 0e 0 1 1 0 1 0 0 0 1 0 0 0 For exercises with four propositional variables, the standard assignment is: \alpha \gamma \delta\ 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 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)]

(Multiple Choice)

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Question 220

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)

Introducing Logic

Inference in Propositional Logic

Monadic Predicate Logic

Full First-Order Logic

Filters

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)

Introducing Logic

Inference in Propositional Logic

Monadic Predicate Logic

Full First-Order Logic

Filters

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)