PHI 2100, Formal Logic I

Traditional Logic Examples of Application

Posted on 1/20/2001

This is not a graded assignment.

The problems worked out in RED were done in class on 1/29/2001.

#6 and #7 are done in green.

9-12 are done in blue.

1,2, 8 are done in black in case we don’t have time to do them in class.

13 was done in class.

Work out the answers to these problems as soon as you can, and then when we’re finished with the section on traditional logic, do them again. Compare the time and difficulty before and after finishing the section. You’ll probably be surprised.

1. John likes to write philosophy papers, but he detests taking literature courses. Not all philosophy papers contain factual information. Literature courses are not philosophy courses.

J))W – (All) John (J) is a person who likes to write philosophy papers (W)

J))L – (All) John (J) is a person who detests taking literature courses (L).

P((F – Some philosophy papers (P) are not things containing factual information (F).

C)(O – No literature course C is a philosophy course (O).

Based on the information given in the passage above, which statement(s) below cannot be true?

A. No non-philosophy papers are papers that John likes to write. Non-W)(Z (Z=papers John likes to write). What you know about John and the papers he likes to write is that he likes to write philosophy papers, but it doesn’t mean he doesn’t like to write anything else. Undetermined.

B. Nobody detests literature courses more than John detests them. You know from the information given in the original (above) that John doesn’t like literature courses, but you don’t have enough information to assume that he hates them more than anyone else hates them.

C. Some papers that do not contain factual information are non-philosophy papers. – non-F()non-P. What you know from the information given above is that P((F. By contraposition, P((F becomes non-F((non-P. Moving from the O form (transformation of the given) to the I form in C, you find that C is undetermined.

D. All philosophy courses are non-literature courses. O))non-C. From the information above, you know that C)(O. That converts to O)(C, which obverts to O))non-C. They are the same. D is true.

E. All courses John likes are non-literature courses. J))L. This is the same as the given information. True.

F. None of the above.

2. There is no job better than teaching; yet some people work at jobs for which they are not suited. Not everyone prefers to teach for a living. There is at least one truck driver who is not suited for the job. There is a truck driver who knows that there is no job better than teaching.

J)(B – No job is better than a teaching job.

P()non-S. Some people are non-suited for their jobs.

P((T – Some person is not one who prefers teaching.

D()non-S. Some truck drivers are non-suited for their jobs.

D()B – Some truck drivers are those who know that there is no job better than a teaching job.

Based on the information given in the passage above, which statement(s) below must be true?

A. Some truck drivers are not suited for their jobs. D((S obv D()non-S (identical to original). True.

B. Non-teaching jobs are all non-truck driving jobs. Non-E (non-teaching jobs) ))non-R (truck driving jobs). You have no information on either of these things, specifically. Undetermined.

C. Some people do not prefer to drive a truck for a living. P((U (U=prefers to drive a truck). You have no information in the given information regarding preference to drive a truck. You have a statement about preference for teaching, but that is irrelevant in this case.

D. All people unsuited for their jobs are non-truck drivers. Non-S))non-D. Compare the statement in the original, D()non-S. That original can be obverted to D((S. The statement being considered is non-S))non-D, which contraposes to D))S. This one says that all truck drivers are suited for their jobs. Your original information says that some truck drivers are not suited. These are contradictory. False.

E. Only non-teachers are unsuited for their jobs. Did you notice that there is no reference to teachers in the original information? There is reference to teaching jobs and there is reference to people (truck drivers, in particular) who know that there’s no job better than teaching, but no term in the given information refers to teachers in particular. Cannot be determined.

F. None of the above.

3. Assume that it is FALSE that “all non-exempt funds are taxable.”

Explanation: If it is false that “all non-E are T,” then it is true that “some non-E are not T.” So start from there. It’s much easier to start with a true given statement than it is with a false one. The idea is this. Whenever you start out with a false given statement, state its contradiction. The given in this case is an A form statement (an A form is a universal affirmative). The contradictory statement is the exact opposite (which is a particular negative). If the A form is false, the O form has to be true. So use the true statement as your given statement. It is non-E((T.

Based on the information given above, which statement(s) below may be true, but are not necessarily true?

A. No taxable funds are non-exempt.

B. Almost all exempt funds are non-taxable.

C. All non-taxable funds are exempt.

D. All non-exempt funds are charitable contributions.

E. Some non-taxable funds are exempt.

F. None of the above.

Explanation:

A. T)(non-E: The given statement is non-E((T, so you need to transform either the original given statement, or the one in (A) so that the subject and predicate terms “match” each other. That is, your original has non-E first, then T. The one you’re working on has T first and non-E second. The easiest thing to do is transform the one in (A) and change it to non-E)(T by conversion. Now the given and the one in (A) have the same subject and predicate terms, and you can determine the truth value of (A) by the square of opposition. The original is an O form and it is true, so comparing that to non-E)(T (which is an E form) tells you that the statement (A) is undetermined. So, this is one of the ones that may be true, but it’s not necessarily true.

B. This one we translated in class simply as E()non-T, but you could have done it as E((non-T. The phrase “almost all” is an ‘exceptive indicator’ which tells you that it is permissible to translate the statement in (B) as either an I form or an O form, or both. Some logic texts indicate that you ought to translate both ways, but either one is sufficient for our purposes. So in this case, we’ll choose the I form translation as the one to be determined.

The original, given statement is, again non-E((T. The one you’re working on now is E()non-T. So, basically, you have a problem. The problem is that the given statement has non-E as the subject and T as the predicate where the one in (B) has E as the subject and non-T as the predicate. You have to do something to determine the value of this statement. You can try to remove complementary classes, or add some. It’s up to you how to proceed. (And of course you’ve already seen that there are quite a few ways to approach these problems. You may begin one way, another person begins another way, and yet the two of you reach the same answer.) But to get moving with this, go ahead and look at the one you’re working with in (B). You can obvert it to E((T. But now you have no complementary class in it. But that’s not a problem, since you can use contraposition on it and get non-T((non-E. Of course, now you have TWO complementary classes, and the given statement had only one. You can fix that. Look at the given statement. It is non-E((T. You can obvert it to non-E()non-T. Now this one is in the wrong order. You need non-T first and non-E second. The rule that simply moves terms from one position to the other is conversion. The statement you wish to transform is an I form, and it becomes non-T()non-E. Your original, given statement is an O form with non-T as the subject and non-E as the predicate. The one in (B) has the same subject and predicate and is an I form. On the square of opposition, when you move from an I form to an O form, or from an O form to an I form, you get “undetermined.” So that’s the answer. And since “undetermined” means it may be true, this is one of those that may be true, but is not necessarily true.

C. All non-T are E = non-T))E. Remember that the given is non-E((T and you know it is true. So what do you do next? Don’t give up…. Take non-T))E and use contraposition. That gets non-E))T. Now the statement in C and the original have the same subject and predicate. The original is an O form and the one in C is an A form. When you move from O to A on the square, you get a false. So this one is false, and so it is not one that may be true (because it is false).

D. All non-E are C. Now, all you have to do is look at this one and compare the original given. You see that the original, given proposition had to do with E and T, not with E and C. So you can’t determine the value of (D) because it is about something different. Given the information you have in the original, you can’t determine the value of the one in D, so it is undetermined. This, then, is one of those that may be true, but is not necessarily true.

E. Some non-T are E. = non-T()E. Your original given proposition is non-E((T. Again, you have to decide something about which statement to transform. The original can be transformed with contraposition to get non-T((E. Now the original and the one in (E) have the same subject and predicate. The original, given is transformed to non-T((E and the one in (E) is non-T()E. When you move from a true O form to determine the corresponding I form, you get “undetermined.” So, (E) is one of those that may be true, but is not necessarily true.

4. Assume that it is TRUE that “dogs winning the competition are awarded a blue ribbon.”

Explanation: This one, and all subsequent problems, will be done with symbols only.

Given: W))B is true – obverse = W)(non-B. Contrapositive of original = non-B))non-W. Obverse of this is non-B)(W, which you could also get by converting the obverse of the original.

Based on the information given above, which statement(s) below must be false?

A. Some animals not awarded a blue ribbon are not dogs winning the competition.

Non-B((W (Note: you could also say Non-B()non-W.) Non-B((W by contrap = non-W((B; obverse = non-W()non-B; convert this to non-B()non-W and compare non-B))non-W as a transformation of the original given. Moving from A to I is TRUE. This one is NOT one of those that must be false because it is TRUE.

B. Some dogs winning the competition are awarded red ribbons. The given statement didn’t say anything about red ribbons, so this one is automatically undetermined. It may be true, so it is not the case that it must be false.

C. All animals not awarded a blue ribbon are non-winning dogs. Non-B))non-W. Compare transformed original as non-B))non-W. They are the same statement. If the original is true, so is this one. It is NOT one of those that must be false.

D. Some dogs not awarded a blue ribbon are non-winners. Non-B()non-W. Again, compare the transformed original as non-B))non-W. Moving from a true A form to an I form yields TRUE.

E. None of the above.

5. Assume that it is FALSE that “some professional athletes are not adequately trained.”

Explanation: If it is false that P((A, then it must be true that P))A. So start with that. You know that P))A is true. You also know that by contraposition of this statement, non-A))non-P is true. Obverting that, non-A)(P is true. Converting that, P)(non-A is true.

Based on the information given above, which statement(s) below must be true?

A. Some non-adequately trained people are non-professional athletes. Non-A()non-P. Compare the contrapositive of the original given yielding non-A))non-P. Moving from A to I is TRUE.

B. All adequately trained people are non-professional athletes. A))non-P. Obvert this: A)(P. Moving from A to E, where A is true, yields a FALSE.

C. All professional athletes are adequately trained. P))A. That is exactly what the original statement says. So it is true.

D. Some professional athletes are adequately trained. P()A. The original was P))A. That’s an A form. The one in D is an I form. The one in D is TRUE.

E. Only those who are professional athletes are adequately trained. A))P. Convert by limitation to P()A. Compare the original, P))A is true. So, P()A must be true, too.

6. If no non-voters are citizens, which of the following statements must be true? Which must be false? Which, if any, are undetermined? Non-V)(C is TRUE. By conversion, C)(non-V. Obverting that, C))V. Using the contrapositive, non-V))non-C is true.

A. All citizens are voters. C))V. This is identical to one of the transformations of the original. So it is true.

B. Some voters are citizens. V()C. Convert this and get C()V. One of the original transformations is C))V. So C()V is true.

C. Some non-voters are non-citizens. Non-V()non-C. Compare the transformed original as non-V))non-C. Moving from A to I yields TRUE.

D. Every non-voter is a non-citizen. Non-V))non-C. The contrapositive of this is C))V. Compare the transformed original as C))V. When you have the same thing, it must be true, too.

E. All voters are citizens. V))C. Convert by limitation to C()V. Compare the original transformed to C))V. You get TRUE.

7. If some non-carnivores are herbivores, which of the following statements cannot be true?

Non-C()H. Convert = H()non-C. Obvert that, get H((C. The contrapositive of that is non-C((non-H.

A. Some herbivores are carnivores. H()C. Undetermined.

B. Every herbivore is a carnivore. H))C. False.

C. At least one herbivore is a non-carnivore. H()non-C. Obvert = H((C. True.

D. Some non-herbivores are carnivores. Non-H()C. Convert = C()non-H. Obvert that, C((H. Contrapositive of that, non-H((non-C. None of the transformations yields the same subject and predicate as the original, so this is UNDETERMINED.

E. Some herbivores are not carnivores. H((C. True.

The only one of these that can’t be true is B.

8. Some fish in that fish tank are expensive, and all fish in the pet store are expensive. John’s fish are all expensive. John’s fish were purchased at the pet store.

F()E = Some fish in that fish tank are expensive things.

P))E = All pet store fish are expensive things.

J))E = All fish belonging to John are expensive.

J))S = All fish belong to John were purchased at the pet store.

If you know that all the statements above are true, which statements below may not be true?

A. All the fish in the tank are John’s. T))J – there is no reference in the original information regarding whose fish are in the tank. They could be anyone’s fish. Simply because they are expensive, it doesn’t imply that they are John’s even if all the fish John has are expensive. Undetermined, so it may not be true.

B. Some expensive things are not in that fish tank. E((T – The reference here may not even be to “fish” at all. It says that there are expensive things that AREN’T in the tank. There’s no reference to fish. There could be expensive pump equipment in the tank. Undetermined, so it may not be true.

C. At least some things possessed by John are non-expensive fish. J()non-E. You already know (above) that J))E. So J()non-E, obverted to show J((E, contradicts the original information, so J((E is false.

D. John’s fish tank was purchased from the pet store. Note that there is nothing in the original information telling you whose tank it is. You don’t know whether it is John’s, the pet store’s tank, or whether it belongs to someone else. This may be false.

E. No fish in the pet store are inexpensive. P)(non-E. Probably the closest translation, assuming that fish in the pet store are pet store fish, is that no pet store fish are non-expensive. That means (obversion), that P))E, which is identical to the original information. True.

9. Not every computer is reliable. If you know that this statement is true, which statements below must also be true?

C((R

A. Some computers are reliable. C()R. Undetermined.

B. Some non-reliable things are computers. Non-R()C. Convert: C()non-R. Obvert: C((R. True.

C. Some non-reliable things are not non-computers. Non-R((non-C. Contrap: C((R. Same as given. True.

D. Some computers are not reliable. C((R. Same as given. True.

E. No computer is reliable. C)(R. Undetermined.

10. All snakes are reptiles and almost all reptiles have poisonous bites. No insects are reptiles, and some of them do not have poisonous bites. If you know that all these statements are true, which of the following must be true?

Here are the propositions given in the problem:

1. S))R

2. R()P

3. I)(R

4. I((P

A. No snakes are insects. Combining propositions 1 and 3, the conclusion follows that S)(I.

I)(R

S))R

S)(I.

This conclusion follows with absolute certainty.

B. Some insects are not reptiles. See proposition 3. If no insects are reptiles, then some insects are not reptiles (square of opposition).

C. Some non-reptiles are non-insects. Non-R()non-I. Compare statement 3. It is an E form. E forms contrapose by limitation to O forms: non-R((non-I. Compare C. Undetermined.

D. All snakes have poisonous bites. S))P. Combine propositions 1 and 2. This conclusion does not follow. It may be true, but is not necessarily true. What follows from 1 and 2 is S()P.

E. All insects are non-snakes. I))non-S. Obverse: I)(S. See propositions 1 and 3.

S))R

I)(R

It follows that I)(S. This conclusion follows.

11. Any person who cheats on an exam is dishonest. If this statement is true, which of the following must be false?

C))D (Note: You may also translate ‘dishonest’ as ‘non-honest.’

A. Some people who cheat on exams are not dishonest. C((D. False.

B. Some dishonest people are not those who cheat on exams. D((C. Does not convert equivalently. The original converts by limitation to D()C. Undetermined.

C. No person who cheats on an exam is honest. C)(non-D (note: If you use “D” for “dishonest,” then ‘honest’ must be ‘non-dishonest.’ Obvert C))D. Same as original. True.

D. Some honest people are not those who cheat on exams. Non-D((C. Obvert the original, above, yielding C)(non-D. Convert this to: non-D)(C. Non-D((C is true.

E. No dishonest person cheats on an exam. D)(C. Convert: C)(D. False.

12. Which of the conclusions in I-III below can be deduced from these two statements?

A. Some deltas are not phis. D((P

B. No rhos are non-deltas. R)(non-D. Obvert: R))D

Actually, nothing can be deduced from the two statements together because, if they are both premises of an argument, and you try to derive a conclusion, what you’ll find is that the middle term is not distributed in either occurrence. But, you can figure out I, II and III individually in the following way:

I. Some deltas are not rhos. D((R. You already know that R))D from (B). R))D becomes (by limitation, conversion) D()R. From D()R, D((R is undetermined.

II. No phis are deltas. P)(D. You know that D((P from (A). P)(D converts to D)(P. The given is D((P (O form), II is P)(D (E form). Undetermined.

III. All rhos are phis. R))P. Cannot be determined exactly from the information given since the two given propositions are not appropriately related to reach a definite conclusion (again, the fallacy committed is UM).

A. I only

B. II only

C. I and II only

D. I and III only

E. I, II, and III

F. NONE OF THE ABOVE.

13. Assume that the information below is true. If it is true, which of the statements (A-E) must also be true?

All SUVs are non-fuel efficient.

Bicycles are fuel efficient.

Whenever I drive my SUV on the highway, it rains.

It is raining.

A. Bicycles do not travel on the highway.

B. Bicycles travel on the highway only if it is raining.

C. If my car is not an SUV, then it is not raining.

D. I am not now driving on the highway.

E. My SUV is fuel efficient.