11.7: Exercises

Logical Equivalence

■ 1. Is every statement logically equivalent to itself?

■ 2. Does the definition of logical equivalence ever permit false sentences to be logically equivalent to each other?

3. Let A = "x is 4," let B = "x is an even number," and let C = "x is 8/2." Then are A and B logically equivalent?

4. Let A = "x is 4," let B = "x is an even number," and let C = "x is 8/2." Then are A and C logically equivalent?

5. If no items from column C are nondeductible, can we infer with certainty that all items for column C are deductible? How about vice versa?

The Logic of Not, And, Or, and If-Then

■ 1. Show, by appeal to its logical form in sentential logic, why the following argument is valid or is invalid, and be sure to say which it is.

If politicians are corrupt, their friends are also corrupt. Thus, if politicians are not corrupt, they don't have peculiar friends because they have peculiar friends if their friends are corrupt.

■ 2. The following argument is

a. deductively valid
b. deductively invalid.

I don't know whether polyvinyls are corrosive or not, but I do know that if they are, then ferrophenyls are also corrosive. Therefore, if polyvinyls are not corrosive, they don't have picoferrous properties because they have picoferrous properties if ferrophenyls are corrosive.

Defend your answer by appeal to logical form.

3. The following argument is

a. deductively valid
b. deductively invalid

I don't know whether polyvinyls are corrosive or not, but I do know that if they are corrosive, then ferrophenyls are also corrosive. Isn't it reasonable, then, to suppose that, if polyvinyls are corrosive, they have picoferrous properties because, as you’ve already said, they have picoferrous properties if ferrophenyls are corrosive?

Defend your answer by appeal to logical form.

4. Consider this amusing argument: "If God had wanted people to fly, He would not have given us bicycles." Here is its implicit premise: But He has given them to us; so it's clear what God wants.

a. State the implicit conclusion.
b. State the logical form of the argument in sentential logic.
c. Is the argument deductively valid? Why or why not?
d. Is the arguer assuming that God is not evil?

5. The following passage contains one or more deductive sub-arguments, all of whose premises are stated explicitly. For each one, (a) identify the sub-argument by rewriting it in standard form, (b) give its logical form, and (c) say whether it is valid.

You’ve got to give up this sort of behavior. God frowns on homosexuality. Besides, all the community standards oppose it, and this is hurting your father's business. He has to serve the public, remember. If your homosexuality is illegal, you should give it up, and your homosexuality is illegal, as you well know. You say it's OK because it feels right, but there is more to this world than your feelings. I love you, but you must quit this nonsense.

■ 6. Valid or invalid?

■ 7. Is this a deductively valid argument pattern?

■ 8. Is the following argument form deductively valid in sentential logic?

9. Here is the logical form of an argument in sentential logic. Is it valid or invalid?

10. Which statement patterns below would be inconsistent with the pattern "P or Q"?

a. not-P
b. not-Q
c. not-P and not-Q
d. not-P or not-Q
e. If P, then not-Q

11. Which of the following statement forms has the logical form “If A, then B”?

a. A, from which it follows that B.
b. A, which follows from B.

■ 12. Is this argument deductively valid?

The report writing was not difficult. Since report writing is either difficult or pleasant, the report writing must have been pleasant.

13. Identify the lowercase letter preceding any passage below that contains an argument or a sub-argument that has the following logical form:

not -B
A implies B
not-A

a. X is the family of all open-closed intervals together with the null set. If X is the family of all open-closed intervals together with the null set, then X is closed under the operation of intersection. Consequently, X is not closed under the operation of intersection.
b. X is not the family of all open-closed intervals together with the null set. If X is the family of all open-closed intervals together with the null set, then X is closed under the operation of intersection. Consequently, X is not closed under the operation of intersection.
c. X is not the family of all open-closed intervals together with the null set. But X is the family of all open-closed intervals together with the null set if X is closed under the operation of intersection. Consequently, X is not closed under the operation of intersection.
d. If X is the family of all open-closed intervals together with the null set, then X is closed under the operation of intersection. X is the family of all open-closed intervals together with the null set. Consequently, X is closed under the operation of intersection.
e. X is the family of all open-closed intervals together with the null set. If X is the family of all open-closed intervals together with the null set, then X is closed under the operation of intersection. Consequently, X is closed under the operation of intersection.
f. If X is closed under the operation of intersection, then X is the family of all open-closed intervals together with the null set. X is not the family of all open-closed intervals together with the null set. Consequently, X is not closed under the operation of intersection.

14. In regard to the previous question, identify the letters of the passages that are deductively invalid.

15. Is this argument deductively valid? Defend your answer by appeal to sentential logic.

If state senators are corrupt, their staff members are corrupt. The staff members of state senators are indeed corrupt, so state senators are corrupt.

■ 16. Is this argument deductively valid? Defend your answer by appeal to sentential logic.

If Einstein were alive today, the physics department at Princeton University in New Jersey would be affected by his presence. So, if you look at the department, you’ll see he's one dead duck.

■ 17. Is this a valid sequent in Sentential Logic?

A → B, ~A ⊢ ~B

That is, if we obey the rules of the truth tables, can we be confident that there is no way to assign truth-values to the simple statement letters that will produce a counterexample?

The Logic of Only, Only If, and Unless

■ 1. You can usually get from the bottom floor to the top floor of a building that has an elevator _________ you walk up the stairs.

a. only if
b. if and only if
c. just when
d. unless
e. none of the above

2. You are president of the United States _______________ you are a U.S. citizen.

a. only if
b. if and only if
c. provided that
d. if e. unless

■ 3. You are president of the United States ________________ you are not president of the

United States.

a. only if
b. if and only if
c. just when
d. if
e. unless

4. For any whole number x, x is even ____________ x is odd. (Which ones cannot be used to fill in the blank and still leave a true statement?)

a. only if
b. if and only if
c. provided that
d. if
e. unless

■ 5. For any whole number x, x is even ____________ x is not odd. (Which ones cannot be used to fill in the blank and still leave a true statement?)

a. only if
b. if and only if
c. provided that
d. if
e. unless

6. If it were the case that only people favoring cost-cutting techniques in the administration are advocates of decreasing the number of administrative positions, would it follow that you have got to be an advocate of decreasing the number of administrative positions to be a person favoring cost-cutting techniques in the administration?

7. A sign says, "Only adults may view this film." Does it follow with certainty that if you're an adult, you may view this film?

8. Joseph will not graduate in cosmetology unless he passes either the developmental cosmetology course or the course in experimental design. So, if Joseph passes experimental design, he will graduate in cosmetology.

a. deductively valid
b. not deductively valid

9. Carlucci calls us only if the war room is in condition orange, but the war room is in condition orange. So, Carlucci will call.

In analyzing this argument, let the word Orange stand for the statement "The war room is in condition orange," and let the word Call stand for "Carlucci calls us." Rewriting the first premise as a conditional and then generalizing to the argument pattern yields which one of the following?

a. If Orange, then Call.
Orange.
Call.
b. If Call, then Orange.
Orange.
Call.
c. Call only if Orange.
Orange.
Call.

10. The argument in the previous question commits the fallacy of denying the antecedent when the first premise is rewritten as a logically equivalent conditional statement and then the argument is translated into its form in sentential logic.

a. true
b. false

11. Consider this memo from an employer:

Employees must be given the opportunity to give or withhold their consent before the private aspects of their lives are investigated. The firm is justified in inquiring into the employee's life only if the employee has a clear understanding that the inquiry is being made. The means used to gain this information are also important; extraordinary methods would include hidden microphones, lie detector tests, spies, and personality inventory tests.

■ i. If the quotation is correct, then if the employee has a clear understanding that the inquiry is being made, the firm is justified in inquiring into the employee's life.

a. follows
b. does not follow

ii. If the quotation is correct, the firm is justified in inquiring into the employee's life if the employee has a clear understanding that the inquiry is being made.

a. follows
b. does not follow

iii. If the quotation is correct, then if the firm is justified in inquiring into the employee's life, the employee has a clear understanding that the inquiry is being made.

a. follows
b. does not follow

12. Suppose x = 4 if and only if y < 22. From this fact, which follow with certainty? (There might be more than one.)

a. x = 4 provided that y < 22.
b. x = 4 unless y < 22.
c. y < 22 if x = 4.
d. x = 4 or y < 22.
e. (y = 22 or y > 22) just when x is not equal to 4.

13. Are these two arguments logically analogous? Is either of them deductively valid?

Carlucci will call us only if the war room is in condition orange, but the war room is in condition orange. So, Carlucci will call.

Carlucci will call us only if he is alive. Carlucci is alive, so he will call.

■ 14. Are these two sentence forms logically equivalent?

Not-A unless B.

A only if B.

Sentential Logic

1. Create a truth table for this argument in (ordinary Sentential Logic): B & (~C → ~B), so C, then say how you can look at the table and tell whether the argument is valid. Is it valid?

2. Create a truth table for this argument: B & (C v ~B), so C, then say how you can look at the table and tell whether the argument is valid. Is it valid?

3. (a) Is this sentence a tautology: (C & B) v (~C v ~B) ?
(b) Show its truth table.
(c) Say how you can look at the truth table and tell whether the sentence is a tautology.
(d) Is this sentence a tautology in 3-valued logic?
(e) Show its truth table in 3-valued logic.
(f) Say how you can look at the 3-valued truth table and tell whether the sentence is a tautology.

4. Do logicians invent logic or discover it?

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Solutions

Logical Equivalence

1 Yes.

2 Here is an example: “Churchill was the first prime minister” and “The first prime minister was Churchill.”

The Logic of Not, And, Or, And if-Ten

1 The argument can be treated in sentential logic using the following definitions of the sentences (clauses):

PC = politicians are corrupt
FC = the friends of politicians are corrupt
PF = politicians have peculiar friends

Here is the logical form (noticing that the conclusion is not the last statement in English):

If PC, then FC.
If FC, then PF.
-----------------------------
If not-PC, Then not-PF.

This form is invalid. If you can’t tell whether this form is valid, maybe the invalidity is easier to see by using a logical analogy. Here is an analogous argument that is also invalid:

If it's a house cat, then it's a feline.
If it's a feline, then it's a mammal.
So, if it's not a house cat, then it's not a mammal.

2 It is invalid. Here is the form: If PVC then FC. If FC then PF. So, If not-PVC then not-PF.

6 Valid.

7 Yes.

8 The form is valid, and any specific argument with that form is also valid.

12 This is an example of valid reasoning, and it remains valid even if you were to learn that one of the premises is false.

16 Valid because its sentential form is modus tollens. The argument is superficially invalid but is actually valid when the principle of charity is used in these two ways: (1) to say that the conclusion is logically equivalent to "Einstein is not alive," and (2) to add the implicit premise that the physics department at Princeton University in New Jersey is not affected by the presence of Einstein.

17 No. This argument has the invalid form called denying the antecedent. In a situation where A is false and B is true, we have a counterexample because then the argument has true premises and a false conclusion.

The Logic of Only, Only If, and Unless

1 Answer (e).

3 Answer (e).

5 Answer (e). It would be true to say “x is even unless x is odd." Adding the not makes (e) not fit in the blank.

11(i) Answer (b). The phrase only if works like if-then in the sense that "Inquiry is justified only if employee has understanding" is logically equivalent to "If inquiry is justified, then employee has understanding." Note that statement (i) is the converse of this—namely, "If employee has understanding, then inquiry is justified." Consequently, (i) does not follow from the statement containing the only if, which is why the answer is (b).

14 Yes. The first is equivalent to "not-A or B." The second is equivalent to "A implies B." These two are equivalent to each other.