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Which of the following is a simple statement? A) Moe is a dog but he acts like a human. B) Sasha is hungry because Willa ate Sasha’s dinner. C) Moe is a dog, whereas Sasha is a cat. D) This exam is easy.
D) This exam is easy.

Consider the following: If those roller skates hadn’t been on the stairs, she would not have tripped. What kind of conditional is this? A) material, B) counterfactual/subjunctive, C) logically equivalent, D) indicative
B) counterfactual/subjunctive

Which of the following is most obviously not a truthfunctional statement? A) Frank believes that the Earth is 13.7 billion years old. B) The Earth is 13.7 billion years old. C) The Earth is 13.7 billion years old but grass is not green. D) The Earth is either 13.7 billion years old or grass is green.
A) Frank believes that the Earth is 13.7 billion years old.

Which of the following is not one of the three “Laws of Thought”? A) (p ⊃ p) is always true, B) (p • ~p) is always false, C) (p v ~p) is always true, D) (p ⊃ ~p) is always false
D) (p ⊃ ~p) is always false

Which of the following is NOT true? A: ~(x)Fx⇔(∃x)~Fx. B: (x)Fx⇔~(∃x)~Fx. C: ~(x)~Fx⇔(∃x)Fx. D: (x)~Fx⇔~(∃x)Fx. E: all of the above. F: none of the above.
 D is not true: (x)Fx⇔~(∃x)Fx
 It should read: (x)~Fx⇔~(∃x)Fx

Symbolize the following statement, using capital letters to abbreviate the simple statements involved: Argentina will mobilize only if Brazil protests to the U.N., while Chile will call for a meeting of all of the Latin American states if the Dominican Republic does not call for such a meeting.
 A: Argentina will mobilize.
 B: Brazil will protest to the UN.
 C: Chile will call for a meeting of all Latin American states.
 D: The Dominican Republic will call for a meeting of all Latin American states.
 (A ⊃ B) • (~D ⊃ C)

Symbolize the following statement, using capital letters to abbreviate the simple statements involved: If oil consumption continues to grow, then either oil imports will increase or domestic oil reserves will be depleted. If oil imports increase and domestic oil reserves are depleted, then, unless a new source of income is found, the nation will go bankrupt. Therefore, the nation will go bankrupt if and only if oil consumption continues to grow.
 B: The nation will go bankrupt.
 C: Oil consumption will continue to grow.
 I: Oil imports will increase.
 R: Domestic oil reserves will be depleted.
 N: A new source of income will be found.
 C ⊃ (I v R)
 (I • R) ⊃ (N v B)
 ∴ B ≡ C

The following is a valid argument. State the rule of inference by which its conclusion follows from its premise or premises. 1. (S≡T) v [Q ⊃ (O • R)] 2. ~(S≡T) 3. [Q ⊃ (O • R)]
 Disjunction Syllogism
 p v q
 ~p
 ∴ q

The following is a valid argument. State the rule of inference by which its conclusion follows from its premise or premises. 1. [N ⊃ (O • P)] • [Q ⊃ (O • R)] 2. N v Q 3. (O • P) v (Q • R)
 Constructive Dilemma
 (p ⊃ q) • (r ⊃ s)
 p v r
 ∴ q v s

The following is a valid argument. State the rule of inference by which its conclusion follows from its premise or premises. 1. [T v (U ⊃ S)] • [(W •~V) ⊃ ~T] 2. [T v (U ⊃ S)] • [W ⊃ (~V ⊃ ~T)]
 Exportation (Replacement Rule)
 [(p • q) ⊃ r] ⇔ [p ⊃ (q ⊃ r)]

The following is a valid argument. State the rule of inference by which its conclusion follows from its premise or premises. 1. [(W • Z) ⊃ (Y ⊃ Z)] ≡ (~X v Y) 2. {[(W • Z) ⊃ (Y ⊃ Z)] ≡ (~X v Y)} v B

The following is a valid argument. State the rule of inference by which its conclusion follows from its premise or premises. 1. (~A ⊃ B) ⊃ (~C v ~D) 2. (~A ⊃ B) ⊃~(C • D)
 De Morgan’s Theorems (Replacement Rule)
 ~(p • q) ⇔ (~p v ~q)
 ~(p v q) ⇔ (~p • ~q)

The following is a valid argument. State the rule of inference by which its conclusion follows from its premise or premises. 1. (~F v G) • (H ≡ I) 2. (F ⊃ G) • (H ≡ I)
 Material Implication (Replacement Rule)
 (p ⊃ q) ⇔ (~p v q)

The following is a valid argument. State the rule of inference by which its conclusion follows from its premise or premises. 1. (C v D) ⊃ [(J v K) ⊃ (J • K)] 2. ~[(J v K) ⊃ (J • K)] 3. ~(C v D)

The following is a correct proof. Justify each line that is not a premise with the rule of inference and the line from which it came. 1. (A ⊃ B), 2. (C ⊃~B), ∴ (A ⊃ ~C), 3. (~~B ⊃~C), 4. (B ⊃~C), 5. (A ⊃~C)
 1. (A ⊃ B)
 2. (C ⊃~B)
 ∴ (A ⊃ ~C)
 3. (~~B ⊃~C) 2, transposition
 4. (B ⊃~C) 3, double negation
 5. (A ⊃~C) 1,4 hypothetical syllogism

Use natural deduction to prove the following: 1. (~A ⊃ A), ∴ A
 1. (~A ⊃ A)
 ∴ A
 2. (~~A v A) 1, material implication
 3. (A v A) 2, double negation
 4. A 3, tautology

