7.11 (Describing a world) Launch Tarski™s World and choose Hide Labels from the Display menu.

‚ Then, with the labels hidden, open Montague™s World. In this world, each object has a name,

and no object has more than one name. Start a new sentence ¬le where you will describe some

features of this world. Check each of your sentences to see that it is indeed a sentence and that

it is true in this world.

1. Notice that if c is a tetrahedron, then a is not a tetrahedron. (Remember, in this world

each object has exactly one name.) Use your ¬rst sentence to express this fact.

2. However, note that the same is true of b and d. That is, if b is a tetrahedron, then d

isn™t. Use your second sentence to express this.

3. Finally, observe that if b is a tetrahedron, then c isn™t. Express this.

4. Notice that if a is a cube and b is a dodecahedron, then a is to the left of b. Use your

next sentence to express this fact.

5. Use your next sentence to express the fact that if b and c are both cubes, then they

are in the same row but not in the same column.

6. Use your next sentence to express the fact that b is a tetrahedron only if it is small.

[Check this sentence carefully. If your sentence evaluates as false, then you™ve got the

arrow pointing in the wrong direction.]

7. Next, express the fact that if a and d are both cubes, then one is to the left of the

other. [Note: You will need to use a disjunction to express the fact that one is to the

left of the other.]

8. Notice that d is a cube if and only if it is either medium or large. Express this.

9. Observe that if b is neither to the right nor left of d, then one of them is a tetrahedron.

Express this observation.

10. Finally, express the fact that b and c are the same size if and only if one is a tetrahedron

and the other is a dodecahedron.

Save your sentences as Sentences 7.11. Now choose Show Labels from the Display menu.

Verify that all of your sentences are indeed true. When verifying the ¬rst three, pay particular

attention to the truth values of the various constituents. Notice that sometimes the conditional

has a false antecedent and sometimes a true consequent. What it never has is a true antecedent

and a false consequent. In each of these three cases, play the game committed to true. Make

sure you understand why the game proceeds as it does.

7.12 (Translation) Translate the following English sentences into fol. Your translations will use all

‚ of the propositional connectives.

1. If a is a tetrahedron then it is in front of d.

2. a is to the left of or right of d only if it™s a cube.

3. c is between either a and e or a and d.

4. c is to the right of a, provided it (i.e., c) is small.

Chapter 7

Biconditional symbol: ” / 185

5. c is to the right of d only if b is to the right of c and left of e.

6. If e is a tetrahedron, then it™s to the right of b if and only if it is also in front of b.

7. If b is a dodecahedron, then if it isn™t in front of d then it isn™t in back of d either.

8. c is in back of a but in front of e.

9. e is in front of d unless it (i.e., e) is a large tetrahedron.

10. At least one of a, c, and e is a cube.

11. a is a tetrahedron only if it is in front of b.

12. b is larger than both a and e.

13. a and e are both larger than c, but neither is large.

14. d is the same shape as b only if they are the same size.

15. a is large if and only if it™s a cube.

16. b is a cube unless c is a tetrahedron.

17. If e isn™t a cube, either b or d is large.

18. b or d is a cube if either a or c is a tetrahedron.

19. a is large just in case d is small.

20. a is large just in case e is.

Save your list of sentences as Sentences 7.12. Before submitting the ¬le, you should complete

Exercise 7.13.

7.13 (Checking your translations) Open Bolzano™s World. Notice that all the English sentences from

‚ Exercise 7.12 are true in this world. Thus, if your translations are accurate, they will also be

true in this world. Check to see that they are. If you made any mistakes, go back and ¬x them.

Remember that even if one of your sentences comes out true in Bolzano™s World, it does not

mean that it is a proper translation of the corresponding English sentence. If the translation is

correct, it will have the same truth value as the English sentence in every world. So let™s check

your translations in some other worlds.

Open Wittgenstein™s World. Here we see that the English sentences 3, 5, 9, 11, 12, 13, 14,

and 20 are false, while the rest are true. Check to see that the same holds of your translations.

If not, correct your translations (and make sure they are still true in Bolzano™s World).

Next open Leibniz™s World. Here half the English sentences are true (1, 2, 4, 6, 7, 10, 11, 14,

18, and 20) and half false (3, 5, 8, 9, 12, 13, 15, 16, 17, and 19). Check to see that the same

holds of your translations. If not, correct your translations.

Finally, open Venn™s World. In this world, all of the English sentences are false. Check to

see that the same holds of your translations and correct them if necessary.

There is no need to submit any ¬les for this exercise, but don™t forget to submit Sentences

7.12.

Section 7.2

186 / Conditionals

7.14 (Figuring out sizes and shapes) Open Euler™s Sentences. The nine sentences in this ¬le uniquely

‚ determine the shapes and sizes of blocks a, b, and c. See if you can ¬gure out the solution just

by thinking about what the sentences mean and using the informal methods of proof you™ve

already studied. When you™ve ¬gured it out, submit a world in which all of the sentences are

true.

7.15 (More sizes and shapes) Start a new sentence ¬le and use it to translate the following English

‚ sentences.

1. If a is a tetrahedron, then b is also a tetrahedron.

2. c is a tetrahedron if b is.

3. a and c are both tetrahedra only if at least one of them is large.

4. a is a tetrahedron but c isn™t large.

5. If c is small and d is a dodecahedron, then d is neither large nor small.

6. c is medium only if none of d, e, and f are cubes.

7. d is a small dodecahedron unless a is small.

8. e is large just in case it is a fact that d is large if and only if f is.

9. d and e are the same size.

10. d and e are the same shape.

11. f is either a cube or a dodecahedron, if it is large.

12. c is larger than e only if b is larger than c.

Save these sentences as Sentences 7.15. Then see if you can ¬gure out the sizes and shapes of

a, b, c, d, e, and f . You will ¬nd it helpful to approach this problem systematically, ¬lling in

the following table as you reason about the sentences:

a b c d e f

Shape:

Size:

When you have ¬lled in the table, use it to guide you in building a world in which the twelve

English sentences are true. Verify that your translations are true in this world as well. Submit

both your sentence ¬le and your world ¬le.

7.16 (Name that object) Open Sherlock™s World and Sherlock™s Sentences. You will notice that none

‚ of the objects in this world has a name. Your task is to assign the names a, b, and c in such a

way that all the sentences in the list come out true. Submit the modi¬ed world as World 7.16.

7.17 (Building a world) Open Boolos™ Sentences. Submit a world in which all ¬ve sentences in this

‚ ¬le are true.

Chapter 7

Conversational implicature / 187

7.18 Using the symbols introduced in Table 1.2, page 30, translate the following sentences into fol.

‚ Submit your translations as a sentence ¬le.

1. If Claire gave Folly to Max at 2:03 then Folly belonged to her at 2:00 and to him at

2:05.

2. Max fed Folly at 2:00 pm, but if he gave her to Claire then, Folly was not hungry ¬ve

minutes later.

3. If neither Max nor Claire fed Folly at 2:00, then she was hungry.

4. Max was angry at 2:05 only if Claire fed either Folly or Scru¬y ¬ve minutes before.

5. Max is a student if and only if Claire is not.

7.19 Using Table 1.2 on page 30, translate the following into colloquial English.

1. (Fed(max, folly, 2:00) ∨ Fed(claire, folly, 2:00)) ’ Pet(folly)

2. Fed(max, folly, 2:30) ” Fed(claire, scru¬y, 2:00)

3. ¬Hungry(folly, 2:00) ’ Hungry(scru¬y, 2:00)

4. ¬(Hungry(folly, 2:00) ’ Hungry(scru¬y, 2:00))

7.20 Translate the following into fol as best you can. Explain any predicates and function symbols

you use, and any shortcomings in your ¬rst-order translations.

1. If Abe can fool Stephen, surely he can fool Ulysses.

2. If you scratch my back, I™ll scratch yours.

3. France will sign the treaty only if Germany does.

4. If Tweedledee gets a party, so will Tweedledum, and vice versa.

5. If John and Mary went to the concert together, they must like each other.

7.21 (The monkey principle) One of the stranger uses of if. . . then. . . in English is as a roundabout

‚| way to express negation. Suppose a friend of yours says If Keanu Reeves is a great actor, then

I™m a monkey™s uncle. This is simply a way of denying the antecedent of the conditional, in

this case that Keanu Reeves is a great actor. Explain why this works. Your explanation should

appeal to the truth table for ’, but it will have to go beyond that. Turn in your explanation

and also submit a Boole table showing that A ’ ⊥ is equivalent to ¬A.

Section 7.3

Conversational implicature

In translating from English to fol, there are many problematic cases. For

example, many students resist translating a sentence like Max is home unless

Claire is at the library as:

¬Library(claire) ’ Home(max)

Section 7.3

188 / Conditionals

These students usually think that the meaning of this English sentence would

be more accurately captured by the biconditional claim:

¬Library(claire) ” Home(max)

The reason the latter seems natural is that when we assert the English sen-

tence, there is some suggestion that if Claire is at at the library, then Max is

not at home.

To resolve problematic cases like this, it is often useful to distinguish be-

tween the truth conditions of a sentence, and other things that in some sense

follow from the assertion of the sentence. To take an obvious case, suppose

someone asserts the sentence It is a lovely day. One thing you may conclude

from this is that the speaker understands English. This is not part of what

the speaker said, however, but part of what can be inferred from his saying it.

The truth or falsity of the claim has nothing to do with the speaker™s linguistic

abilities.

The philosopher H. P. Grice developed a theory of what he called con-

versational implicature to help sort out the genuine truth conditions of a

sentence from other conclusions we may draw from its assertion. These other

conclusions are what Grice called implicatures. We won™t go into this theory

conversational

implicatures in detail, but knowing a little bit about it can be a great aid in translation,

so we present an introduction to Grice™s theory.

Suppose we have an English sentence S that someone asserts, and we

are trying to decide whether a particular conclusion we draw is part of the

meaning of S or, instead, one of its implicatures. Grice pointed out that if

the conclusion is part of the meaning, then it cannot be “cancelled” by some

cancelling implicatures

further elaboration by the speaker. Thus, for example, the conclusion that

Max is home is part of the meaning of an assertion of Max and Claire are

home, so we can™t cancel this conclusion by saying Max and Claire are home,

but Max isn™t home. We would simply be contradicting ourselves.

Contrast this with the speaker who said It is a lovely day. Suppose he

had gone on to say, perhaps reading haltingly from a phrase book: Do you

speak any French? In that case, the suggestion that the speaker understands

English is e¬ectively cancelled.

A more illuminating use of Grice™s cancellability test concerns the expres-

sion either. . . or. . . . Recall that we claimed that this should be translated into

fol as an inclusive disjunction, using ∨. We can now see that the suggestion

that this phrase expresses exclusive disjunction is generally just a conversa-

tional implicature. For example, if the waiter says You can have either soup or

salad, there is a strong suggestion that you cannot have both. But it is clear

that this is just an implicature, since the waiter could, without contradicting

Chapter 7

Conversational implicature / 189

himself, go on to say And you can have both, if you want. Had the original

either. . . or. . . expressed the exclusive disjunction, this would be like saying

You can have soup or salad but not both, and you can have both, if you want.

Let™s go back now to the sentence Max is at home unless Claire is at the

library. Earlier we denied that the correct translation was

¬Library(claire) ” Home(max)

which is equivalent to the conjunction of the correct translation

¬Library(claire) ’ Home(max)

with the additional claim

Library(claire) ’ ¬Home(max)

Is this second claim part of the meaning of the English sentence, or is it

simply a conversational implicature? Grice™s cancellability test shows that it

is just an implicature. After all, it makes perfectly good sense for the speaker

to go on to say On the other hand, if Claire is at the library, I have no idea

where Max is. This elaboration takes away the suggestion that if Claire is at

the library, then Max isn™t at home.

Another common implicature arises with the phrase only if, which people

often construe as the stronger if and only if. For example, suppose a father

tells his son, You can have dessert only if you eat all your lima beans. We™ve

seen that this is not a guarantee that if the child does eat his lima beans he

will get dessert, since only if introduces a necessary, not su¬cient, condition.