# Chapter 5 Connectives

In the previous chapter, we considered symbolizing fairly basic English sentences with sentence letters of TFL. This leaves us wanting to deal with the English expressions ‘and’, ‘or’, ‘not’, and so forth. These are connectives—they can be used to form new sentences out of old ones. In TFL, we will make use of logical connectives to build complex sentences from atomic components. There are five logical connectives in TFL. This table summarizes them, and they are explained throughout this section.

These are not the only connectives of English of interest. Others are, e.g., ‘unless’, ‘neither … nor …’, and ‘because’. We will see that the first two can be expressed by the connectives we will discuss, while the last cannot. ‘Because’, in contrast to the others, is not truth functional.

## 5.1 Negation

Consider how we might symbolize these sentences:

1. 1.

Mary is in Barcelona.

2. 2.

It is not the case that Mary is in Barcelona.

3. 3.

Mary is not in Barcelona.

In order to symbolize sentence 1, we will need a sentence letter. We might offer this symbolization key:

$B$:

Mary is in Barcelona.

Since sentence 2 is obviously related to sentence 1, we will not want to symbolize it with a completely different sentence letter. Roughly, sentence 2 means something like ‘It is not the case that $B$’. In order to symbolize this, we need a symbol for negation. We will use ‘$\neg$’. Now we can symbolize sentence 2 with ‘$\neg B$’.

Sentence 3 also contains the word ‘not’, and it is obviously equivalent to sentence 2. As such, we can also symbolize it with ‘$\neg B$’.

A sentence can be symbolized as $\neg\mathscr{A}$ if it can be paraphrased in English as ‘It is not the case that…’.

It will help to offer a few more examples:

1. 4.

The widget can be replaced.

2. 5.

The widget is irreplaceable.

3. 6.

The widget is not irreplaceable.

Let us use the following representation key:

$R$:

The widget is replaceable

Sentence 4 can now be symbolized by ‘$R$’. Moving on to sentence 5: saying the widget is irreplaceable means that it is not the case that the widget is replaceable. So even though sentence 5 does not contain the word ‘not’, we will symbolize it as follows: ‘$\neg R$’.

Sentence 6 can be paraphrased as ‘It is not the case that the widget is irreplaceable.’ Which can again be paraphrased as ‘It is not the case that it is not the case that the widget is replaceable’. So we might symbolize this English sentence with the TFL sentence ‘$\neg\neg R$’.

But some care is needed when handling negations. Consider:

1. 7.

Jane is happy.

2. 8.

Jane is unhappy.

If we let the TFL-sentence ‘$H$’ symbolize ‘Jane is happy’, then we can symbolize sentence 7 as ‘$H$’. However, it would be a mistake to symbolize sentence 8 with ‘$\neg{H}$’. If Jane is unhappy, then she is not happy; but sentence 8 does not mean the same thing as ‘It is not the case that Jane is happy’. Jane might be neither happy nor unhappy; she might be in a state of blank indifference. In order to symbolize sentence 8, then, we would need a new sentence letter of TFL.

## 5.2 Conjunction

Consider these sentences:

1. 9.

2. 10.

Barbara is athletic.

3. 11.

Adam is athletic, and also Barbara is athletic.

We will need separate sentence letters of TFL to symbolize sentences 9 and 10; perhaps

$A$:

$B$:

Barbara is athletic.

Sentence 9 can now be symbolized as ‘$A$’, and sentence 10 can be symbolized as ‘$B$’. Sentence 11 roughly says ‘A and B’. We need another symbol, to deal with ‘and’. We will use ‘$\wedge$’. Thus we will symbolize it as ‘$(A\wedge B)$’. This connective is called conjunction . We also say that ‘$A$’ and ‘$B$’ are the two conjuncts of the conjunction ‘$(A\wedge B)$’.

Notice that we make no attempt to symbolize the word ‘also’ in sentence 11. Words like ‘both’ and ‘also’ function to draw our attention to the fact that two things are being conjoined. Maybe they affect the emphasis of a sentence, but we will not (and cannot) symbolize such things in TFL.

Some more examples will bring out this point:

1. 12.

Barbara is athletic and energetic.

2. 13.

Barbara and Adam are both athletic.

3. 14.

Although Barbara is energetic, she is not athletic.

4. 15.

Adam is athletic, but Barbara is more athletic than him.

Sentence 12 is obviously a conjunction. The sentence says two things (about Barbara). In English, it is permissible to refer to Barbara only once. It might be tempting to think that we need to symbolize sentence 12 with something along the lines of ‘$B$ and energetic’. This would be a mistake. Once we symbolize part of a sentence as ‘$B$’, any further structure is lost, as ‘$B$’ is a sentence letter of TFL. Conversely, ‘energetic’ is not an English sentence at all. What we are aiming for is something like ‘$B$ and Barbara is energetic’. So we need to add another sentence letter to the symbolization key. Let ‘$E$’ symbolize ‘Barbara is energetic’. Now the entire sentence can be symbolized as ‘$(B\wedge E)$’.

Sentence 13 says one thing about two different subjects. It says of both Barbara and Adam that they are athletic, even though in English we use the word ‘athletic’ only once. The sentence can be paraphrased as ‘Barbara is athletic, and Adam is athletic’. We can symbolize this in TFL as ‘$(B\wedge A)$’, using the same symbolization key that we have been using.

Sentence 14 is slightly more complicated. The word ‘although’ sets up a contrast between the first part of the sentence and the second part. Nevertheless, the sentence tells us both that Barbara is energetic and that she is not athletic. In order to make each of the conjuncts a sentence letter, we need to replace ‘she’ with ‘Barbara’. So we can paraphrase sentence 14 as, ‘Both Barbara is energetic, and Barbara is not athletic’. The second conjunct contains a negation, so we paraphrase further: ‘Both Barbara is energetic and it is not the case that Barbara is athletic’. Now we can symbolize this with the TFL sentence ‘$(E\wedge\neg B)$’. Note that we have lost all sorts of nuance in this symbolization. There is a distinct difference in tone between sentence 14 and ‘Both Barbara is energetic and it is not the case that Barbara is athletic’. TFL does not (and cannot) preserve these nuances.

Sentence 15 raises similar issues. The word ‘but’ suggests a contrast or difference, but this is not something that TFL can deal with. All we can do is paraphrase the sentence as ‘Both Adam is athletic, and Barbara is more athletic than Adam’. (Notice that we once again replace the pronoun ‘him’ with ‘Adam’.) How should we deal with the second conjunct? We already have the sentence letter ‘$A$’, which is being used to symbolize ‘Adam is athletic’, and the sentence ‘$B$’ which is being used to symbolize ‘Barbara is athletic’; but neither of these concerns their relative athleticity. So, to symbolize the entire sentence, we need a new sentence letter. Let the TFL sentence ‘$R$’ symbolize the English sentence ‘Barbara is more athletic than Adam’. Now we can symbolize sentence 15 by ‘$(A\wedge R)$’.

A sentence can be symbolized as $(\mathscr{A}\wedge\mathscr{B})$ if it can be paraphrased in English as ‘Both…, and…’, or as ‘…, but …’, or as ‘although …, …’.

We noted above that the contrast suggested by ‘but’ cannot be captured in TFL, and that we simply ignore it. A phenomenon that cannot simply be ignored is temporal order. E.g., consider:

1. 16.

Harry stood up and objected to the proposal.

2. 17.

Harry objected to the proposal and stood up.

If Harry stood up after he objected, sentence 17 is true but sentence 16 is false—the use of ‘and’ here is asymmetric. The symbol ‘$\wedge$’ of TFL, however, is always symmetric (or “commutative” as logicians say). TFL cannot deal with asymmetric ‘and’. We’ll assume for all our examples and exercises that ‘and’ is symmetric.11 1 On symmetric and asymmetric conjunction in linguistics, see, e.g., Robin Lakoff, “If’s, and’s, and but’s about conjunction”, in: C. J. Fillmore and D. T. Langendoen (eds.), Studies in Linguistic Semantics, Holt, Rinehart & Winston, 1971, and Susan Schmerling, “Asymmetric conjunction and rules of conversation”, in: P. Cole and J. L. Morgan (eds.), Speech Acts, Brill, 1975, pp. 211–31.

You might be wondering why we put brackets around the conjunctions. The reason can be brought out by thinking about how negation interacts with conjunction. Consider:

1. 18.

It’s not the case that you will get both soup and salad.

2. 19.

You will not get soup but you will get salad.

Sentence 18 can be paraphrased as ‘It is not the case that: both you will get soup and you will get salad’. Using this symbolization key:

$S_{1}$:

You will get soup.

$S_{2}$:

we would symbolize ‘both you will get soup and you will get salad’ as ‘$(S_{1}\wedge S_{2})$’. To symbolize sentence 18, then, we simply negate the whole sentence, thus: ‘$\neg(S_{1}\wedge S_{2})$’.

Sentence 19 is a conjunction: you will not get soup, and you will get salad. ‘You will not get soup’ is symbolized by ‘$\neg S_{1}$’. So to symbolize sentence 19 itself, we offer ‘$(\neg S_{1}\wedge S_{2})$’.

These English sentences are very different, and their symbolizations differ accordingly. In one of them, the entire conjunction is negated. In the other, just one conjunct is negated. Brackets help us to keep track of things like the scope of the negation.

## 5.3 Disjunction

Consider these sentences:

1. 20.

Either Fatima will play videogames, or she will watch movies.

2. 21.

Either Fatima or Omar will play videogames.

For these sentences we can use this symbolization key:

$F$:

Fatima will play videogames

$O$:

Omar will play videogames

$M$:

Fatima will watch movies

However, we will again need to introduce a new symbol. Sentence 20 is symbolized by ‘$(F\vee M)$’. The connective is called disjunction . We also say that ‘$F$’ and ‘$M$’ are the disjuncts of the disjunction ‘$(F\vee M)$’.

Sentence 21 is only slightly more complicated. There are two subjects, but the English sentence only gives the verb once. However, we can paraphrase sentence 21 as ‘Either Fatima will play videogames, or Omar will play videogames’. Now we can obviously symbolize it by ‘$(F\vee O)$’ again.

A sentence can be symbolized as $(\mathscr{A}\vee\mathscr{B})$ if it can be paraphrased in English as ‘Either…, or….’

Sometimes in English, the word ‘or’ is used in a way that excludes the possibility that both disjuncts are true. This is called an exclusive or . An exclusive or is clearly intended when it says, on a restaurant menu, ‘Entrees come with either soup or salad’: you may have soup; you may have salad; but, if you want both soup and salad, then you have to pay extra.

At other times, the word ‘or’ allows for the possibility that both disjuncts might be true. This is probably the case with sentence 21, above. Fatima might play videogames alone, Omar might play videogames alone, or they might both play. Sentence 21 merely says that at least one of them plays videogames. This is an inclusive or . The TFL symbol ‘$\vee$’ always symbolizes an inclusive or.

It will also help to see how negation interacts with disjunction. Consider:

1. 22.

Either you will not have soup, or you will not have salad.

2. 23.

You will have neither soup nor salad.

3. 24.

You get either soup or salad, but not both.

Using the same symbolization key as before, sentence 22 can be paraphrased in this way: ‘Either it is not the case that you get soup, or it is not the case that you get salad’. To symbolize this in TFL, we need both disjunction and negation. ‘It is not the case that you get soup’ is symbolized by ‘$\neg S_{1}$’. ‘It is not the case that you get salad’ is symbolized by ‘$\neg S_{2}$’. So sentence 22 itself is symbolized by ‘$(\neg S_{1}\vee\neg S_{2})$’.

Sentence 23 also requires negation. It can be paraphrased as, ‘It is not the case that: either you get soup or you get salad’. Since this negates the entire disjunction, we symbolize sentence 23 with ‘$\neg(S_{1}\vee S_{2})$’.

Sentence 24 is an exclusive or. We can break the sentence into two parts. The first part says that you get one or the other. We symbolize this as ‘$(S_{1}\vee S_{2})$’. The second part says that you do not get both. We can paraphrase this as: ‘It is not the case both that you get soup and that you get salad’. Using both negation and conjunction, we symbolize this with ‘$\neg(S_{1}\wedge S_{2})$’. Now we just need to put the two parts together. As we saw above, ‘but’ can usually be symbolized with ‘$\wedge$’. So sentence 24 can be symbolized as ‘$((S_{1}\vee S_{2})\wedge\neg(S_{1}\wedge S_{2}))$’.

This last example shows something important. Although the TFL symbol ‘$\vee$’ always symbolizes inclusive or, we can symbolize an exclusive or in TFL. We just have to use a few other symbols as well.

## 5.4 Conditional

Consider these sentences:

1. 25.

If Jean is in Paris, then Jean is in France.

2. 26.

Jean is in France only if Jean is in Paris.

Let’s use the following symbolization key:

$P$:

Jean is in Paris

$F$:

Jean is in France

Sentence 25 is roughly of this form: ‘if $P$, then $F$’. We will use the symbol ‘$\rightarrow$’ to symbolize this ‘if…, then…’ structure. So we symbolize sentence 25 by ‘$(P\rightarrow F)$’. The connective is called the conditional . Here, ‘$P$’ is called the antecedent of the conditional ‘$(P\rightarrow F)$’, and ‘$F$’ is called the consequent .

Sentence 26 is also a conditional. Since the word ‘if’ appears in the second half of the sentence, it might be tempting to symbolize this in the same way as sentence 25. That would be a mistake. Your knowledge of geography tells you that sentence 25 is unproblematically true: there is no way for Jean to be in Paris that doesn’t involve Jean being in France. But sentence 26 is not so straightforward: were Jean in Dieppe, Lyon, or Toulouse, Jean would be in France without being in Paris, thereby rendering sentence 26 false. Since geography alone dictates the truth of sentence 25, whereas travel plans (say) are needed to know the truth of sentence 26, they must mean different things.

In fact, sentence 26 can be paraphrased as ‘If Jean is in France, then Jean is in Paris’. So we can symbolize it by ‘$(F\rightarrow P)$’.

A sentence can be symbolized as $(\mathscr{A}\rightarrow\mathscr{B})$ if it can be paraphrased in English as ‘If A, then B’ or ‘A only if B’.

In fact, the conditional can represent many English expressions. Consider:

1. 27.

For Jean to be in Paris, it is necessary that Jean be in France.

2. 28.

It is a necessary condition on Jean’s being in Paris that she be in France.

3. 29.

For Jean to be in France, it is sufficient that Jean be in Paris.

4. 30.

It is a sufficient condition on Jean’s being in France that she be in Paris.

If we think about it, all four of these sentences mean the same as ‘If Jean is in Paris, then Jean is in France’. So they can all be symbolized by ‘$(P\rightarrow F)$’.

It is important to bear in mind that the connective ‘$\rightarrow$’ tells us only that, if the antecedent is true, then the consequent is true. It says nothing about a causal connection between two events (for example). In fact, we lose a huge amount when we use ‘$\rightarrow$’ to symbolize English conditionals. We will return to this in section 10.3 and chapter 13.

## 5.5 Biconditional

Consider these sentences:

1. 31.

Laika is a dog only if she is a mammal.

2. 32.

Laika is a dog if she is a mammal.

3. 33.

Laika is a dog if and only if she is a mammal.

We will use the following symbolization key:

$D$:

Laika is a dog

$M$:

Laika is a mammal

For reasons discussed above, sentence 31 can be symbolized by ‘$(D\rightarrow M)$’.

Sentence 33 says something stronger than either sentence 31 or sentence 32. It can be paraphrased as ‘Laika is a dog if Laika is a mammal, and Laika is a dog only if Laika is a mammal’. This is just the conjunction of sentences 31 and 32. So we can symbolize it as ‘$(D\rightarrow M)\wedge(M\rightarrow D)$’. We call this a biconditional , because it amounts to stating both directions of the conditional.

We could treat every biconditional this way. So, just as we do not need a new TFL symbol to deal with exclusive or, we do not really need a new TFL symbol to deal with biconditionals. Because the biconditional occurs so often, however, we will use the symbol ‘$\leftrightarrow$’ for it. We can then symbolize sentence 33 with the TFL sentence ‘$(D\leftrightarrow M)$’.

The expression ‘if and only if’ occurs a lot especially in philosophy, mathematics, and logic. For brevity, we can abbreviate it with the snappier word ‘iff’. We will follow this practice. So ‘if’ with only one ‘f’ is the English conditional. But ‘iff’ with two ‘f’s is the English biconditional. Armed with this we can say:

A sentence can be symbolized as $(\mathscr{A}\leftrightarrow\mathscr{B})$ if it can be paraphrased in English as ‘A if⁠f B’; that is, as ‘A if and only if B’.

A word of caution. Ordinary speakers of English often use ‘if …, then…’ when they really mean to use something more like ‘…if and only if …’. Perhaps your parents told you, when you were a child: ‘if you don’t eat your greens, you won’t get any dessert’. Suppose you ate your greens, but that your parents refused to give you any dessert, on the grounds that they were only committed to the conditional (roughly ‘if you get dessert, then you will have eaten your greens’), rather than the biconditional (roughly, ‘you get dessert if⁠f you eat your greens’). Well, a tantrum would rightly ensue. So, be aware of this when interpreting people; but in your own writing, make sure you use the biconditional if⁠f you mean to.

## 5.6 Unless

We have now introduced all of the connectives of TFL. We can use them together to symbolize many kinds of sentences. An especially difficult case is when we use the English-language connective ‘unless’:

1. 34.

Unless you wear a jacket, you will catch a cold.

2. 35.

You will catch a cold unless you wear a jacket.

These two sentences are clearly equivalent. To symbolize them, we will use the symbolization key:

$J$:

You will wear a jacket

$D$:

You will catch a cold

Both sentences mean that if you do not wear a jacket, then you will catch a cold. With this in mind, we might symbolize them as ‘$(\neg J\rightarrow D)$’.

Equally, both sentences mean that if you do not catch a cold, then you must have worn a jacket. With this in mind, we might symbolize them as ‘$(\neg D\rightarrow J)$’.

Equally, both sentences mean that either you will wear a jacket or you will catch a cold. With this in mind, we might symbolize them as ‘$(J\vee D)$’.

All three are correct symbolizations. Indeed, in chapter 12 we will see that all three symbolizations are equivalent in TFL.

If a sentence can be paraphrased as ‘Unless $A$, $B$,’ then it can be symbolized as ‘$(\mathscr{A}\vee\mathscr{B})$’.

Again, though, there is a little complication. ‘Unless’ can be symbolized as a conditional; but as we said above, people often use the conditional (on its own) when they mean to use the biconditional. Equally, ‘unless’ can be symbolized as a disjunction; but there are two kinds of disjunction (exclusive and inclusive). So it will not surprise you to discover that ordinary speakers of English often use ‘unless’ to mean something more like the biconditional, or like exclusive disjunction. Suppose someone says: ‘I will go running unless it rains’. They probably mean something like ‘I will go running if⁠f it does not rain’ (i.e., the biconditional), or ‘either I will go running or it will rain, but not both’ (i.e., exclusive disjunction). Again: be aware of this when interpreting what other people have said, but be precise in your writing.

## Practice exercises

A. Using the symbolization key given, symbolize each English sentence in TFL.

$M$:

Those creatures are men in suits

$C$:

Those creatures are chimpanzees

$G$:

Those creatures are gorillas

1. 1.

Those creatures are not men in suits.

2. 2.

Those creatures are men in suits, or they are not.

3. 3.

Those creatures are either gorillas or chimpanzees.

4. 4.

Those creatures are neither gorillas nor chimpanzees.

5. 5.

If those creatures are chimpanzees, then they are neither gorillas nor men in suits.

6. 6.

Unless those creatures are men in suits, they are either chimpanzees or they are gorillas.

B. Using the symbolization key given, symbolize each English sentence in TFL.

$A$:

Mister Ace was murdered

$B$:

The butler did it

$C$:

The cook did it

$D$:

The Duchess is lying

$E$:

Mister Edge was murdered

$F$:

The murder weapon was a frying pan

1. 1.

Either Mister Ace or Mister Edge was murdered.

2. 2.

If Mister Ace was murdered, then the cook did it.

3. 3.

If Mister Edge was murdered, then the cook did not do it.

4. 4.

Either the butler did it, or the Duchess is lying.

5. 5.

The cook did it only if the Duchess is lying.

6. 6.

If the murder weapon was a frying pan, then the culprit must have been the cook.

7. 7.

If the murder weapon was not a frying pan, then the culprit was either the cook or the butler.

8. 8.

Mister Ace was murdered if and only if Mister Edge was not murdered.

9. 9.

The Duchess is lying, unless it was Mister Edge who was murdered.

10. 10.

If Mister Ace was murdered, he was done in with a frying pan.

11. 11.

Since the cook did it, the butler did not.

12. 12.

Of course the Duchess is lying!

C. Using the symbolization key given, symbolize each English sentence in TFL.

$E_{1}$:

Ava is an electrician

$E_{2}$:

Harrison is an electrician

$F_{1}$:

Ava is a firefighter

$F_{2}$:

Harrison is a firefighter

$S_{1}$:

Ava is satisfied with her career

$S_{2}$:

Harrison is satisfied with his career

1. 1.

Ava and Harrison are both electricians.

2. 2.

If Ava is a firefighter, then she is satisfied with her career.

3. 3.

Ava is a firefighter, unless she is an electrician.

4. 4.

Harrison is an unsatisfied electrician.

5. 5.

Neither Ava nor Harrison is an electrician.

6. 6.

Both Ava and Harrison are electricians, but neither of them find it satisfying.

7. 7.

Harrison is satisfied only if he is a firefighter.

8. 8.

If Ava is not an electrician, then neither is Harrison, but if she is, then he is too.

9. 9.

Ava is satisfied with her career if and only if Harrison is not satisfied with his.

10. 10.

If Harrison is both an electrician and a firefighter, then he must be satisfied with his work.

11. 11.

It cannot be that Harrison is both an electrician and a firefighter.

12. 12.

Harrison and Ava are both firefighters if and only if neither of them is an electrician.

D. Using the symbolization key given, symbolize each English-language sentence in TFL.

$J_{1}$:

John Coltrane played tenor sax

$J_{2}$:

John Coltrane played soprano sax

$J_{3}$:

John Coltrane played tuba

$M_{1}$:

Miles Davis played trumpet

$M_{2}$:

Miles Davis played tuba

1. 1.

John Coltrane played tenor and soprano sax.

2. 2.

Neither Miles Davis nor John Coltrane played tuba.

3. 3.

John Coltrane did not play both tenor sax and tuba.

4. 4.

John Coltrane did not play tenor sax unless he also played soprano sax.

5. 5.

John Coltrane did not play tuba, but Miles Davis did.

6. 6.

Miles Davis played trumpet only if he also played tuba.

7. 7.

If Miles Davis played trumpet, then John Coltrane played at least one of these three instruments: tenor sax, soprano sax, or tuba.

8. 8.

If John Coltrane played tuba then Miles Davis played neither trumpet nor tuba.

9. 9.

Miles Davis and John Coltrane both played tuba if and only if Coltrane did not play tenor sax and Miles Davis did not play trumpet.

E. Give a symbolization key and symbolize the following English sentences in TFL.

1. 1.

Alice and Bob are both spies.

2. 2.

If either Alice or Bob is a spy, then the code has been broken.

3. 3.

If neither Alice nor Bob is a spy, then the code remains unbroken.

4. 4.

The German embassy will be in an uproar, unless someone has broken the code.

5. 5.

Either the code has been broken or it has not, but the German embassy will be in an uproar regardless.

6. 6.

Either Alice or Bob is a spy, but not both.

F. Give a symbolization key and symbolize the following English sentences in TFL.

1. 1.

If there is food to be found in the pridelands, then Rafiki will talk about squashed bananas.

2. 2.

Rafiki will talk about squashed bananas unless Simba is alive.

3. 3.

Rafiki will either talk about squashed bananas or he won’t, but there is food to be found in the pridelands regardless.

4. 4.

Scar will remain as king if and only if there is food to be found in the pridelands.

5. 5.

If Simba is alive, then Scar will not remain as king.

G. For each argument, write a symbolization key and symbolize all of the sentences of the argument in TFL.

1. 1.

If Dorothy plays the piano in the morning, then Roger wakes up cranky. Dorothy plays piano in the morning unless she is distracted. So if Roger does not wake up cranky, then Dorothy must be distracted.

2. 2.

It will either rain or snow on Tuesday. If it rains, Neville will be sad. If it snows, Neville will be cold. Therefore, Neville will either be sad or cold on Tuesday.

3. 3.

If Zoog remembered to do his chores, then things are clean but not neat. If he forgot, then things are neat but not clean. Therefore, things are either neat or clean; but not both.

H. For each argument, write a symbolization key and symbolize the argument as well as possible in TFL. The part of the passage in italics is there to provide context for the argument, and doesn’t need to be symbolized.

1. 1.

It is going to rain soon. I know because my leg is hurting, and my leg hurts if it’s going to rain.

2. 2.

Spider-man tries to figure out the bad guy’s plan. If Doctor Octopus gets the uranium, he will blackmail the city. I am certain of this because if Doctor Octopus gets the uranium, he can make a dirty bomb, and if he can make a dirty bomb, he will blackmail the city.

3. 3.

A westerner tries to predict the policies of the Chinese government. If the Chinese government cannot solve the water shortages in Beijing, they will have to move the capital. They don’t want to move the capital. Therefore they must solve the water shortage. But the only way to solve the water shortage is to divert almost all the water from the Yangzi river northward. Therefore the Chinese government will go with the project to divert water from the south to the north.

I. We symbolized an exclusive or using ‘$\vee$’, ‘$\wedge$’, and ‘$\neg$’. How could you symbolize an exclusive or using only two connectives? Is there any way to symbolize an exclusive or using only one connective?