Description

Problem Set #2

1. Translate the following argument into SL, then use a truth table to determine whether it is valid or invalid. If it is invalid, specify a counterexample.

(10 pts. for translation, 10 pts. for truth table, 5 pts. for verdict)

Although most Americans approve of gun control, it is neither wise

nor moral. For gun control is wise if and only if it prevents criminals

from obtaining weapons. And gun control is moral if and only if it

preserves our liberty. But it is not the case that gun control both

preserves our liberty and prevents criminals from obtaining weapons.

2. Using only NOR (↓), construct a formula that is equivalent to α → β. (5

pts.)

3. Construct a proof for the following argument: ¬E, F → (D∨E), ¬D ∴ ¬F .

(10 pts.)

4. Prove that the following formula is a theorem: (P &¬Q) → ¬(P ↔ Q).

(10 pts.)

1

Problem Set #1

1. Suppose α is a contradiction. (5 pts. each)

(a) Is it possible for α ∨ β to be a tautology? If yes, give an example. If

no, explain why.

(b) Is it possible for α → β to be contingent? If yes, give an example. If

no, explain why.

2. Suppose the set {α, β, γ} is inconsistent. (5 pts. each)

(a) What can you determine about the formula ¬α ∨ ¬β ∨ ¬γ? Explain

your answer.

(b) What can you determine about the argument α ∴ ¬(β&γ)? Explain

your answer.

3. You are on the island of knights and knaves, where (a) every local is either

a knight or a knave, (b) knights always tell the truth, and (c) knaves always

lie. You meet three locals: Peggy, Joe, and Zoey. Peggy says, “I’m a knave

only if Joe is.” Joe says, “Peggy is a knight but Zoey is a knave.” Zoey says,

“Joe and I are different.” Using a truth table, can you determine who

is a knight and who is a knave? (10 pts. for translation and symbolization

key, 10 pts. for truth table, 10 pts. for interpreting truth table)

1

Problem Set #1 Answer Key

1. Suppose α is a contradiction.

(a) Yes. α ∨ β is a tautology when β is a tautology.

(b) No. Conditionals with a false antecedent are true; contradictions are

always false; so α → β is always true when α is a contradiction.

2. Suppose the set {α, β, γ} is inconsistent.

(a) The formula ¬α ∨ ¬β ∨ ¬γ is a tautology. To say that {α, β, γ} is

inconsistent is to say that its members can’t all be true at the same

time. This means that in every case at least one of them must be

false, which means that in every case at least one of the disjuncts of

¬α ∨ ¬β ∨ ¬γ must be true. Since a disjunction is true when at least

one of its disjuncts is true, this makes ¬α ∨ ¬β ∨ ¬γ a tautology.

(b) The argument α ∴ ¬(β&γ) is valid. By the inconsistency of {α, β, γ},

there’s no case where α is true and β&γ is true. Thus there is no case

where α is true and ¬(β&γ) is false, i.e., there are no counterexamples

to the argument.

3. P = Peggy is a knight; J = Joe is a knight; Z = Zoey is a knight.

P

T

T

T

T

F

F

F

F

J

T

T

F

F

T

T

F

F

Z

T

F

T

F

T

F

T

F

P ↔ (¬P → ¬J)

T

T

T

T

T

T

F

F

J ↔ (P &¬Z)

F

T

T

F

F

F

T

T

Peggy is a knight. Joe is a knave. Zoey is a knight.

1

Z ↔ ¬(J ↔ Z)

F

F

T

T

F

F

T

T

1. Translate the following argument into SL, then use a truth table to deter-

mine whether it is valid or invalid. If it is invalid, specify a counterexample.

(10 pts. for translation, 10 pts. for truth table, 5 pts. for verdict)

Although most Americans approve of gun control, it is neither wise

nor moral. For gun control is wise if and only if it prevents criminals

from obtaining weapons. And gun control is moral if and only if it

preserves our liberty. But it is not the case that gun control both

preserves our liberty and prevents criminals from obtaining weapons.

2. Using only NOR (1), construct a formula that is equivalent to a + B. (5

pts.)

3. Construct a proof for the following argument: -E, F + (DVE), -D ::-F.

(10 pts.)

4. Prove that the following formula is a theorem: (P&-Q) +-(P HQ).

(10 pts.)

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