Chapter 5 Connectives

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 ‘$\mathrm{\neg }$’. Now we can symbolize sentence 2 with ‘$\mathrm{\neg }B$’.

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

A sentence can be symbolized as $\mathrm{\neg }𝒜$ 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: ‘$\mathrm{\neg }R$’.

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 ‘$\mathrm{\neg }\mathrm{\neg }R$’.

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 ‘$\mathrm{\neg }H$’. 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.

Consider these sentences:

1. 9.
   ​

   Adam is athletic.

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

$A$:

Adam is athletic.

$B$:

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 ‘$(E\wedge \mathrm{\neg }B)$’. 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 $(𝒜\wedge ℬ)$ 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.

$S_2$:

You will get salad.

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: ‘$\mathrm{\neg }(S_1\wedge S_2)$’.

Sentence 19 is a conjunction: you will not get soup, and you will get salad. ‘You will not get soup’ is symbolized by ‘$\mathrm{\neg }{S}_1$’. So to symbolize sentence 19 itself, we offer ‘$(\mathrm{\neg }{S}_1\wedge S_2)$’.

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.

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 $(𝒜\vee ℬ)$ 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 ‘$\mathrm{\neg }{S}_1$’. ‘It is not the case that you get salad’ is symbolized by ‘$\mathrm{\neg }{S}_2$’. So sentence 22 itself is symbolized by ‘$(\mathrm{\neg }{S}_1\vee \mathrm{\neg }{S}_2)$’.

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 ‘$\mathrm{\neg }(S_1\vee S_2)$’.

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 ‘$\mathrm{\neg }(S_1\wedge S_2)$’. 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 ‘$((S_1\vee S_2)\wedge \mathrm{\neg }(S_1\wedge S_2))$’.

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.

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 ‘$\to$’ to symbolize this ‘if…, then…’ structure. So we symbolize sentence 25 by ‘$(P\to F)$’. The connective is called the conditional . Here, ‘$P$’ is called the antecedent of the conditional ‘$(P\to F)$’, 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.

It is important to bear in mind that the connective ‘$\to$’ 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 ‘$\to$’ to symbolize English conditionals. We will return to this in section 10.3 and chapter 13.

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 ‘$(D\to M)$’.

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 ‘$(D\to M)\wedge (M\to D)$’. 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 $(𝒜\leftrightarrow ℬ)$ 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.

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 ‘$(\mathrm{\neg }J\to D)$’.

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 ‘$(\mathrm{\neg }D\to J)$’.

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 ‘$(𝒜\vee ℬ)$’.

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.

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.