Deck 6: Propositional Logic

Question 1

Proposition 2I
Given the following proposition:
[∼(X ∨ B) ≡ (∼ Y ⊃ ∼ X)] ≡ ∼[(A ⊃ ∼ Y) • ∼(∼ X ⊃ ∼ B)]
In Proposition 2I, the main operator is a:

A) Tilde.
B) Wedge.
C) Horseshoe.
D) Triple bar.
E) Dot.

Answer 1

Triple bar.

Question 2

Hertz runs a sale or both Avis buys new cars and Budget lowers rates.

A) H ∨ A • B
B) H ∨ (A • B)
C) (H ∨ A) • B
D) H • (A ∨ B)
E) H • A ∨ B

Answer 2

H ∨ (A • B)

Question 3

Not both Hertz runs a sale and Thrifty lays off workers if either Avis buys new cars or Budget lowers rates.

A) (∼H • ∼T) ⊃ (A ∨ B)
B) (A ∨ B) ⊃ (∼H • ∼T)
C) ∼(H • T) ⊃ (A ∨ B)
D) (A ∨ B) ⊃ ∼(H • T)
E) (A • B) ⊃ ∼(H ∨ T)

Answer 3

(A ∨ B) ⊃ ∼(H • T)

Question 4

National does not improve efficiency or Enterprise does not reorganize, given that Thrifty lays off workers.

A) T ⊃ ∼(N ∨ E)
B) (∼N ∨ ∼E) ⊃ T
C) ∼(N ∨ E) ⊃ T
D) (T ⊃ ∼N) ∨ ∼E
E) T ⊃ (∼N ∨ ∼E)

Question 5

Statement 1I
Given the following statement:
(G • ∼ M) ⊃ (M ∨ G)
Statement 1I is:

A) Consistent.
B) Logically equivalent.
C) Tautologous.
D) Contingent.
E) Self-contradictory.

Question 6

Statement 3I
Given the following statement:
[K • (P ∨ ∼ R)] • [K ⊃ (R • ∼ P)]
The truth table for Statement 3I has how many lines?

A) Six.
B) Nine.
C) Eleven.
D) Eight.
E) Four.

Question 7

National's improving efficiency or Enterprise's reorganizing is a sufficient and necessary condition for Thrifty's laying off workers, provided that Avis buys new cars.

A) (N ∨ E) ≡ (A ⊃ T)
B) A ⊃ [(N ∨ E) ≡ T]
C) [(N ∨ E) ≡ T] ⊃ A
D) A ⊃ [(N ⊃ E) • (E ⊃ N)]
E) A ≡ [(N ∨ E) ⊃ T]

Question 8

Statement 1I
Given the following statement:
(G • ∼ M) ⊃ (M ∨ G)
The truth table for Statement 1I has how many lines?

A) Two.
B) Eight.
C) Four.
D) Six.
E) Nine.

Question 9

Statement 3I
Given the following statement:
[K • (P ∨ ∼ R)] • [K ⊃ (R • ∼ P)]
Statement 3I is:

A) Valid.
B) Self-contradictory.
C) Tautologous.
D) Inconsistent.
E) Contingent.

Question 10

Thrifty lays off workers only if neither National improves efficiency nor Enterprise reorganizes.

A) T ⊃ ∼(N ∨ E)
B) ∼(N ∨ E) ⊃ T
C) T ⊃ (∼N ∨ ∼T)
D) (∼N ∨ ∼E) ⊃ T
E) T ≡ ∼(N ∨ E)

Question 11

Hertz's running a sale and Budget's lowering rates are a necessary condition for either Avis's buying new cars or Thrifty's laying off workers.

A) (A • T) ⊃ (H ∨ B)
B) (H • B) ⊃ (A ∨ T)
C) (H • B) ∨ (A ∨ T)
D) (A ∨ T) ⊃ (H • B)
E) (H • B) ≡ (A ∨ T)

Question 12

If National's improving efficiency implies that Enterprise reorganizes, then Avis buys new cars.

A) (N ⊃ E ⊃ A)
B) N ⊃ (E ⊃ A)
C) (N ⊃ E) ⊃ A
D) A ⊃ (N ⊃ E)
E) (E ⊃ N) ⊃ A

Question 13

Budget lowers rates unless Hertz and Thrifty do not overhaul operations.

A) B ∨ (∼H • ∼T)
B) B ∨ ∼(H • T)
C) B ⊃ (∼H • ∼T)
D) (∼H • ∼T) ⊃ B
E) (B ∨ ∼H) • ∼T

Question 14

Proposition 1I
Given the following proposition:
[(B ≡ ∼ Y) • ∼(∼ X ⊃ A)] ∨ [∼(X ≡ A) ⊃ (Y • ∼ A)]
In Proposition 1I, the main operator is a:

A) Dot.
B) Tilde.
C) Horseshoe.
D) Triple bar.
E) Wedge.

Question 15

Thrifty's laying off workers is a sufficient condition for National's improving efficiency, if Enterprise reorganizes.

A) E ⊃ (N ⊃ T)
B) (T ⊃ N) ⊃ E
C) E ⊃ (T ⊃ N)
D) T ⊃ (N ⊃ E)
E) E ≡ (T ⊃ N)

Question 16

Proposition 2I
Given the following proposition:
[∼(X ∨ B) ≡ (∼ Y ⊃ ∼ X)] ≡ ∼[(A ⊃ ∼ Y) • ∼(∼ X ⊃ ∼ B)]
Given that A and B are true and X and Y are false, determine the truth value of Proposition 2I.

Question 17

Budget's lowering rates implies that Hertz runs a sale, only if Avis does not buy new cars.

A) (B ⊃ H) ⊃ ∼A
B) ∼A ⊃ (B ⊃ H)
C) (B ⊃ H) ≡ ∼A
D) (H ⊃ B) ⊃ ∼A
E) (B ⊃ H ⊃ A)

Question 18

Proposition 1I
Given the following proposition:
[(B ≡ ∼ Y) • ∼(∼ X ⊃ A)] ∨ [∼(X ≡ A) ⊃ (Y • ∼ A)]
Given that A and B are true and X and Y are false, determine the truth value of Proposition 1I.

Question 19

Given the pair of statements: R ∨ S and S ⊃ ∼ R
These statements are:

A) Inconsistent.
B) Contradictory.
C) Invalid.
D) Logically equivalent.
E) Consistent.

Question 20

Statement 2I
Given the following statement:
(L • ∼ E) ≡ (E • ∼ L)
Statement 2I is:

A) Self-contradictory.
B) Tautologous.
C) Consistent.
D) Contingent.
E) Inconsistent.

Question 21

?Q ? S $\frac { \sim \mathrm { Q } } { \mathrm { S } }$

A) Disjunctive syllogism.
B) Modus ponens.
C) Invalid.
D) Modus tollens.
E) Denying the antecedent.

Question 22

Given the argument: J ⊃ (E • Q) / ∼ Q ⊃ L / (E ∨ M) ⊃ (L ∨ ∼ Q) / M ∨ J // L
This argument is:

A) Valid.
B) Invalid.
C) Sound.
D) Cogent.
E) Uncogent.

Question 23

In the expression 'P • Q', 'P' is called:

A) A conjunction.
B) A disjunct.
C) An antecedent.
D) A conjunct.
E) A disjunction.

Question 24

?F ? L $\frac{\mathrm{F}}{\sim \mathrm{L}}$

A) Denying the antecedent.
B) Modus ponens.
C) Affirming the consequent.
D) Modus tollens.
E) Invalid.

Question 25

K $\frac { \mathrm { G } \supset \sim \mathrm { K } } { \sim \mathrm { G } }$

A) Disjunctive syllogism.
B) Invalid.
C) Denying the antecedent.
D) Modus tollens.
E) Affirming the consequent.

Question 26

(?W ? L) • (N ? ?R) $\frac { \mathrm { N } \vee \sim \mathrm { W } } { \mathrm { L } \vee \sim \mathrm { R } }$

A) Invalid.
B) Pure hypothetical syllogism.
C) Destructive dilemma.
D) Denying the antecedent.
E) Constructive dilemma.

Question 27

Given the argument: S ⊃ E / N ⊃ (∼ S ⊃ E) // N ⊃ E
This argument is:

A) Invalid; fails in 5th line.
B) Invalid; fails in 6th line.
C) Valid.
D) Invalid; fails in 2nd line.
E) Invalid; fails in 3rd line.

Question 28

Given the argument: K ∨ B / K ≡ (D ∨ ∼ B) // K ⊃ D
This argument is:

A) Invalid; fails in 1st line.
B) Invalid; fails in 2nd line.
C) Valid.
D) Invalid; fails in 4th line.
E) Invalid; fails in 3rd line.

Question 29

E ? ?T $\frac { \sim \mathrm { N } \supset \mathrm { E } } { \sim \mathrm { N } \supset \sim \mathrm { T } }$

A) Modus ponens.
B) Constructive dilemma.
C) Pure hypothetical syllogism.
D) Invalid.
E) Disjunctive syllogism.

Question 30

(M ? ?T) • (?H ? T) $\frac { \mathrm { T } \vee \sim \mathrm { T } } { \sim \mathrm { M } \vee \mathrm { H } }$

A) Invalid.
B) Destructive dilemma.
C) Constructive dilemma
D) Modus tollens.
E) Affirming the consequent.

Question 31

Given the pair of statements: (H • J) ∨ (N • J) and (N ∨ H) ⊃ ∼ J
These statements are:

A) Contradictory.
B) Valid.
C) Logically equivalent.
D) Consistent.
E) Inconsistent.

Question 32

Given the statements: ∼ S ⊃ D / K ⊃ S / D ⊃ K / R ≡ ∼(S • C) / R ≡ C
These statements are:

A) Contradictory.
B) Tautologous.
C) Consistent.
D) Inconsistent.
E) Valid.

Question 33

F ? ?H $\frac { \mathrm { H } \supset \sim \mathrm { T } } { \mathrm { F } \supset \mathrm { T } }$

A) Pure hypothetical syllogism.
B) Constructive dilemma.
C) Invalid.
D) Affirming the consequent.
E) Disjunctive syllogism.

Question 34

R $\frac{\mathrm{R} \supset \sim \mathrm{D}}{\sim \mathrm{D}}$

A) Invalid.
B) Affirming the consequent.
C) Disjunctive syllogism.
D) Modus tollens.
E) Modus ponens.

Question 35

Given the statements: W ⊃ (E ∨ ∼ G) / A ⊃ (G ∨ ∼ E) / H ⊃ (A • W) / H • E
These statements are:

A) Inconsistent.
B) Tautologous.
C) Invalid.
D) Logically equivalent.
E) Consistent.

Question 36

?G ? ?B $\underline { B }$
$\sim \mathrm { G }$

A) Disjunctive syllogism.
B) Invalid.
C) Denying the antecedent.
D) Affirming the consequent.
E) Modus tollens.

Question 37

Given the argument: G ⊃ (N • E) / H ⊃ N / K ⊃ ∼E / (H • K) ≡ G
This argument is:

A) Valid.
B) Invalid.
C) Uncogent.
D) Cogent.
E) Sound.

Question 38

According to De Morgan's rule, ∼(P • Q) is logically equivalent to:

A) P ∨ Q
B) ∼P • ∼Q
C) ∼P ∨ ∼Q
D) P • Q
E) P ≡ Q

Question 39

S ? ?C $\frac{\sim \mathrm{C}}{\mathrm{S}}$

A) Modus tollens.
B) Affirming the consequent.
C) Denying the antecedent.
D) Modus ponens.
E) Invalid.

Question 40

If a group of statements are consistent, this means:

A) At least one of them is true.
B) It is possible for all of them to be true.
C) At least one of them is false.
D) All of them are true.
E) It is possible for all of them to be false.

Question 41

Proposition 2H
Given the following proposition:
∼[(B • ∼X) ⊃ ∼(Y ≡ ∼B)] ⊃ [∼(X ⊃ A) ∨ (B ⊃ ∼Y)]
In Proposition 2H, the main operator is a:

A) Dot.
B) Wedge.
C) Triple bar.
D) Tilde.
E) Horseshoe.

Question 42

American's hiring pilots is a sufficient condition for both Delta's advertising a sale and Southwest's expanding routes.

A) (D • S) ∨ A
B) (D ⊃ A) • (S ⊃ A)
C) (D • S) ⊃ A
D) A ⊃ (D • S)
E) A ≡ (D • S)

Question 43

JetBlue buys planes if and only if both Delta does not advertise a sale and Southwest does not expand routes.

A) J ≡ ∼(D • S)
B) J ⊃ (∼D • ∼S)
C) J ≡ (∼D • ∼S)
D) ∼(D ∨ S) ⊃ J
E) J ∨ (∼D • ∼S)

Question 44

Proposition 1H
Given the following proposition:
[(A ≡ ∼B) ∨ (X ⊃ Y)] • [(Y ⊃ A) ≡ ∼(X ∨ B)]
Given that A and B are true and X and Y are false, determine the truth value of Proposition 1H.

Question 45

Southwest expands routes if either American does not hire pilots or United does not lower fares.

A) ∼(A ∨ U) ⊃ S
B) (∼A • ∼U) ⊃ S
C) S ⊃ (∼A ∨ ∼U)
D) (S ⊃ ∼A ∨ ∼U)
E) (∼A ∨ ∼U) ⊃ S

Question 46

Statement 1H
Given the following statement:
(S ⊃ ∼ R) ⊃ (∼ S • R)
Statement 1H is:

A) Consistent.
B) Logically equivalent.
C) Tautologous.
D) Contingent.
E) Self-contradictory.

Question 47

JetBlue buys planes if Delta advertises a sale, and Frontier improves service only if Southwest expands routes.

A) (D ⊃ J) • (F ⊃ S)
B) (D ≡ J) • (F ≡ S)
C) (J ⊃ D) • (S ⊃ F)
D) (D • J) ⊃ (S • F)
E) (J ⊃ D) • (F ⊃ S)

Question 48

The wedge operator is used to translate:

A) "Nevertheless."
B) "Moreover."
C) "Unless."
D) "Implies."
E) "Provided that."

Question 49

United's lowering fares is a sufficient and necessary condition for American's hiring pilots, given that Frontier does not improve service.

A) (U ≡ A) ⊃ ∼F
B) ∼F ⊃ (U ≡ A)
C) (∼F ⊃ U) • (∼F ⊃ A)
D) (∼F ≡ U) • (∼F ≡ A)
E) ∼F ≡ (U ⊃ A)

Question 50

Proposition 1H
Given the following proposition:
[(A ≡ ∼B) ∨ (X ⊃ Y)] • [(Y ⊃ A) ≡ ∼(X ∨ B)]
In Proposition 1H, the main operator is a:

A) Horseshoe.
B) Tilde.
C) Dot.
D) Triple bar.
E) Wedge.

Question 51

Delta advertises a sale and either United lowers fares or American hires pilots.

A) D • (U ∨ A)
B) D ∨ (U • A)
C) D • U ∨ A
D) (D • U) ∨ A
E) D ∨ (U • A) .

Question 52

Statement 2H
Given the following statement:
[N ∨ ∼ (K ∨ G)] ≡ [(G ⊃ N) • (K ⊃ N)]
Statement 2H is:

A) Inconsistent.
B) Contingent.
C) Consistent.
D) Self-contradictory.
E) Tautologous.

Question 53

Proposition 2H
Given the following proposition:
∼[(B • ∼X) ⊃ ∼(Y ≡ ∼B)] ⊃ [∼(X ⊃ A) ∨ (B ⊃ ∼Y)]
Given that A and B are true and X and Y are false, determine the truth value of Proposition 2H.

Question 54

If an indirect truth table requiring more than one line is constructed for either an argument or a series of statements and a contradiction is derived on the first line, then:

A) The second line must be completed.
B) The argument is valid and the statements are consistent.
C) The argument is valid and the statements are inconsistent.
D) The argument is invalid and the statements are consistent.
E) The argument is invalid and the statements are inconsistent.

Question 55

United lowers fares if and only if both JetBlue buys planes and Delta advertises a sale, provided that American hires pilots.

A) (U ≡ A) • (J ≡ D)
B) A ⊃ [U ≡ (J • D)]
C) [U ≡ (J • D)] ⊃ A
D) A ⊃ [(U ⊃ J) • (D ⊃ U]
E) [U ⊃ (J • D)] ⊃ A

Question 56

JetBlue buys planes unless neither Frontier improves service nor United lowers fares.

A) J ∨ ∼(F • U)
B) J ∨ ∼(F ∨ U)
C) J ∨ (∼F ∨ ∼U)
D) J ⊃ ∼(F ∨ U)
E) ∼(F ∨ U) ⊃ J

Question 57

Frontier's improving service implies that not both JetBlue buys planes and Southwest expands routes.

A) F ⊃ ∼(J ∨ S)
B) F ⊃ (∼J • ∼S)
C) F ⊃ ∼(J • S)
D) (F ⊃ ∼J) • S
E) F ≡ ∼(J • S)

Question 58

Statement 1H
Given the following statement:
(S ⊃ ∼ R) ⊃ (∼ S • R)
The truth table for Statement 1H has how many lines?

A) Two.
B) Eight.
C) Four.
D) Six.
E) Nine.

Question 59

Frontier's improving service is a necessary condition for either JetBlue's buying planes or United's lowering fares.

A) F ≡ (J ∨ U)
B) F ⊃ (J ∨ U)
C) (F ⊃ J) • (F ⊃ U)
D) (J ∨ U) ⊃ F
E) (J ⊃F) ∨ (F ⊃ U)

Question 60

Statement 2H
Given the following statement:
[N ∨ ∼ (K ∨ G)] ≡ [(G ⊃ N) • (K ⊃ N)]
The truth table for Statement 2H has how many lines?

A) Five.
B) Nine.
C) Eight.
D) Six.
E) Four.

Question 61

Given the pair of statements: G ⊃ (W • ∼ Q) and (W ∨ ∼ G) • ∼(Q • G)
These statements are:

A) Inconsistent.
B) Contradictory.
C) Invalid.
D) Logically equivalent.
E) Consistent.

Question 62

S ? ?R $\frac { \mathrm { N } \supset \sim \mathrm { R } } { \mathrm { S } \supset \mathrm { N } }$

A) Disjunctive syllogism.
B) Constructive dilemma.
C) Pure hypothetical syllogism.
D) Affirming the consequent.
E) Invalid.

Question 63

(?N ? R) • (F ? ?G) $\frac{F \vee \sim N}{R \vee \sim G}$

A) Constructive dilemma.
B) Invalid.
C) Denying the antecedent.
D) Modus tollens.
E) Destructive dilemma.

Question 64

?L ? F $\underline { L }$
$\sim \mathrm { F }$

A) Disjunctive syllogism.
B) Denying the antecedent.
C) Modus tollens.
D) Modus ponens.
E) Invalid.

Question 65

Given the argument: Q ⊃ A / ∼ (A • Q) / A ≡ Q
This argument is:

A) Invalid; fails in 1^st line.
B) Valid.
C) Invalid; fails in 3^rd line.
D) Invalid; fails in 2^nd line.
E) Invalid; fails in 4^th line.

Question 66

(K ? ?A) • (S ? ?D) $\begin{array} { l } \mathrm { A } V \mathrm { D } \\\sim \mathrm { K } V \sim \mathrm { S }\end{array}$

A) Invalid.
B) Modus tollens.
C) Affirming the consequent.
D) Denying the antecedent.
E) Destructive dilemma.

Question 67

Given the argument: S ⊃ W / C ⊃ L / (M • ∼L) ∨ (D • ∼W) / C ∨ S // D • M
This argument is:

A) Invalid.
B) Cogent.
C) Sound.
D) Valid.
E) Uncogent.

Question 68

?G $\frac { \sim \mathrm { G } \supset \mathrm { A } } { \mathrm { A } }$

A) Constructive dilemma.
B) Disjunctive syllogism.
C) Modus ponens.
D) Denying the antecedent.
E) Invalid.

Question 69

G ? ?T $\frac { \sim \mathrm { T } } { \mathrm { G } }$

A) Invalid.
B) Disjunctive syllogism.
C) Modus ponens.
D) Affirming the consequent.
E) Modus tollens.

Question 70

?P ? M $\frac { \mathrm { P } } { \mathrm { M } }$

A) Modus tollens.
B) Denying the antecedent.
C) Pure hypothetical syllogism.
D) Disjunctive syllogism.
E) Invalid.

Question 71

Given the pair of statements: P • ∼ Z and Z ≡ P
These statements are:

A) Inconsistent.
B) Valid.
C) Contradictory.
D) Consistent.
E) Logically equivalent.

Question 72

Given the argument: R ⊃ (H • ∼N) / (H • ∼B) ⊃ (N ∨ ∼P) / K ⊃ (G • P) // (K • R) ⊃ B
This argument is:

A) Invalid.
B) Valid.
C) Uncogent.
D) Cogent.
E) Sound.

Question 73

In the expression 'P ∨ Q', 'P' is called:

A) A disjunction.
B) A conjunct.
C) A disjunct.
D) A conjunction.
E) An antecedent.

Question 74

D ? ?N $\frac { \sim \mathrm { N } } { \mathrm { D } }$

A) Affirming the consequent.
B) Invalid.
C) Denying the antecedent.
D) Modus ponens.
E) Modus tollens.

Question 75

Given the statements: H ⊃ E / M ⊃ R / (M • E) ⊃ G / (R • G) ⊃ E / M ∨ E
These statements are:

A) Invalid.
B) Tautologous.
C) Consistent.
D) Logically equivalent.
E) Inconsistent.

Question 76

Given the argument: M ⊃ J / ∼ J ⊃ R // (M ∨ ∼ R) ⊃ J
This argument is:

A) Invalid; fails in 3rd line.
B) Invalid; fails in 2nd line.
C) Invalid; fails in 4th line.
D) Invalid; fails in 1st line.
E) Valid.

Question 77

Given the statements: C ⊃ L / K ⊃ ∼H / L ⊃ (N ⊃ K) / C ⊃ (H ⊃ N) / C • H
These statements are:

A) Contradictory.
B) Tautologous.
C) Consistent.
D) Inconsistent.
E) Valid.

Question 78

?H ? ?B $\frac { \mathrm { K } \supset \sim \mathrm { H } } { \mathrm { K } \supset \sim \mathrm { B } }$

A) Denying the antecedent.
B) Invalid.
C) Pure hypothetical syllogism.
D) Constructive dilemma.
E) Destructive dilemma.

Question 79

B ⊃ ∼Q Q
∼B

A) Modus ponens.
B) Modus tollens.
C) Disjunctive syllogism.
D) Invalid.
E) Denying the antecedent.

Question 80

Statement 2H
Given the following statement:
[N ∨ ∼ (K ∨ G)] ≡ [(G ⊃ N) • (K ⊃ N)]
Given the statement: (E ∨ ∼ H) • (H • ∼ E)
This statement is:

A) Valid.
B) Self-contradictory.
C) Tautologous.
D) Inconsistent.
E) Contingent.