# Chapter 25 Multiple generality

So far, we have only considered sentences that require one-place predicates and one quantifier. The full power of FOL really comes out when we start to use many-place predicates and multiple quantifiers. For this insight, we largely have Gottlob Frege (1879) to thank, but also C. S. Peirce.

## 25.1 Many-placed predicates

All of the predicates that we have considered so far concern properties that objects might have. Those predicates have one gap in them, and to make a sentence, we simply need to slot in one term. They are one-place predicates.

However, other predicates concern the relation between two things. Here are some examples of relational predicates in English:

• blank loves blank

• blank is to the left of blank

• blank is in debt to blank

These are two-place predicates. They need to be filled in with two terms in order to make a sentence. Conversely, if we start with an English sentence containing many singular terms, we can remove two singular terms, to obtain different two-place predicates. Consider the sentence ‘Vinnie borrowed the family car from Nunzio’. By deleting two singular terms, we can obtain any of three different two-place predicates

• Vinnie borrowed blank from blank

• blank borrowed the family car from blank

• blank borrowed blank from Nunzio

and by removing all three singular terms, we obtain a three-place predicate:

• blank borrowed blank from blank

Indeed, there is no in principle upper limit on the number of places that our predicates may contain.

## 25.2 Mind the gap(s)!

We have used the same symbol, ‘blank’, to indicate a gap formed by deleting a term from a sentence. However, as Frege emphasized, these are different gaps. To obtain a sentence, we can fill them in with the same term, but we can equally fill them in with different terms, and in various different orders. The following are three perfectly good sentences, obtained by filling in the gaps in ‘blank loves blank’ in different ways; but they all have distinctively different meanings:

1. 1.

Karl loves Imre.

2. 2.

Imre loves Karl.

3. 3.

Karl loves Karl.

The point is that we need to keep track of the gaps in predicates, so that we can keep track of how we are filling them in. To keep track of the gaps, we assign them variables. Suppose we want to symbolize the preceding sentences. Then I might start with the following representation key:

domain:

people

$i$:

Imre

$k$:

Karl

$L(x,y)$:

blank${}_{x}$ loves blank${}_{y}$

Sentence 1 will be symbolized by ‘$L(k,i)$’, sentence 2 will be symbolized by ‘$L(i,k)$’, and sentence 3 will be symbolized by ‘$L(k,k)$’. Here are a few more sentences that we can symbolize with the same key:

1. 4.

Imre loves himself.

2. 5.

Karl loves Imre, but not vice versa.

3. 6.

Karl is loved by Imre.

Sentence 4 can be paraphrased as ‘Imre loves Imre’, and so symbolized by ‘$L(i,i)$’. Sentence 5 is a conjunction. We can paraphrase it as ‘Karl loves Imre, and Imre does not love Karl’, and so symbolize it as ‘$L(k,i)\wedge\neg L(i,k)$’. Sentence 6 can be paraphrased by ‘Imre loves Karl’, and so symbolized as ‘$L(i,k)$’. In this last case, of course, we have lost the difference in tone between the active and passive voice; but we have at least preserved the truth conditions.

But the relationship between ‘Imre loves Karl’ and ‘Karl is loved by Imre’ highlights something important. To see what, suppose we add another entry to our symbolization key:

$M(x,y)$:

blank${}_{y}$ loves blank${}_{x}$

The entry for ‘$M$’ uses exactly the same English word—‘loves’—as the entry for ‘$L$’. But the gaps have been swapped around! (Just look closely at the subscripts.) And this matters.

To explain: when we see a sentence like ‘$L(k,i)$’, we are being told to take the first name (i.e., ‘$k$’) and associate its value (i.e., Karl) with the gap labelled ‘$x$’, then take the second name (i.e., ‘$i$’) and associate its value (i.e., Imre) with the gap labelled ‘$y$’, and so come up with: Karl loves Imre. The sentence ‘$M(i,k)$’ also tells us to take the first name (i.e., ‘$i$’) and plug its value into the gap labelled ‘$x$’, and take the second name (i.e., ‘$k$’) and plug its value into the gap labelled ‘$y$’, and so come up with: Imre loves Karl.

So, ‘$L(i,k)$’ and ‘$M(k,i)$’ both symbolize ‘Imre loves Karl’, whereas ‘$L(k,i)$’ and ‘$M(i,k)$’ both symbolize ‘Karl loves Imre’. Since love can be unrequited, these are different claims.

One last example might be helpful. Suppose we add this to our symbolisation key:

$P(x,y)$:

blank${}_{x}$ prefers blank${}_{x}$ to blank${}_{y}$

Now the sentence ‘$P(i,k)$’ symbolizes ‘Imre prefers Imre to Karl’, and ‘$P(k,i)$’ symbolizes ‘Karl prefers Karl to Imre’. And note that we could have achieved the same effect, if we had instead specified:

$P(x,y)$:

blank${}_{x}$ prefers themselves to blank${}_{y}$

In any case, the overall moral of this is simple. When dealing with predicates with more than one place, pay careful attention to the order of the gaps!

## 25.3 The order of quantifiers

Consider the sentence ‘everyone loves someone’. This is potentially ambiguous. It might mean either of the following:

1. 7.

For every person x, there is some person that x loves.

2. 8.

There is some particular person whom every person loves.

Sentence 7 can be symbolized by ‘$\forall x\exists y\,L(x,y)$’. Suppose that our domain of discourse is restricted to Imre, Juan and Karl. Suppose also that Karl loves Imre but not Juan, that Imre loves Juan but not Karl, and that Juan loves Karl but not Imre. Then sentence 7 is true.

Sentence 8 is symbolized by ‘$\exists y\forall x\,L(x,y)$’. Sentence 8 is not true in the situation just described, since no single person is loved by everyone. All three of Juan, Imre and Karl would have to converge on (at least) one object of love.

The point of the example is to illustrate that the order of the quantifiers matters a great deal. Indeed, to switch them around is called a quantifier shift fallacy. Here is an example, which comes up in various forms throughout the philosophical literature:

• For every person, there is some truth they cannot know. ($\forall\exists$)

• There is some particular truth that no person can know. ($\exists\forall$)

This argument form is obviously invalid. It’s just as bad as:11 1 Thanks to Rob Trueman for the example.

• Every dog has its day. ($\forall\exists$)

• There is a day for all the dogs. ($\exists\forall$)

The order of quantifiers is also important in definitions in mathematics. For instance, there is a big difference between pointwise and uniform continuity of functions:

• A function $f$ is pointwise continuous if

$\forall\epsilon\forall x\forall y\exists\delta(\left|x-y\right|<\delta\to\left% |f(x)-f(y)\right|<\epsilon)$
• A function $f$ is uniformly continuous if

$\forall\epsilon\exists\delta\forall x\forall y(\left|x-y\right|<\delta\to\left% |f(x)-f(y)\right|<\epsilon)$

The moral is simple: take great care with the order of your quantifiers!

## 25.4 Stepping-stones to symbolization

As we are starting to see, symbolization in FOL can become a bit tricky. So, when symbolizing a complex sentence, you should lay down several stepping-stones. As usual, the idea is best illustrated by example. Consider this symbolisation key:

domain:

people and dogs

$D(x)$:

blank${}_{x}$ is a dog

$F(x,y)$:

blank${}_{x}$ is a friend of blank${}_{y}$

$O(x,y)$:

blank${}_{x}$ owns blank${}_{y}$

$g$:

Geraldo

Now let’s try to symbolize these sentences:

1. 9.

Geraldo is a dog owner.

2. 10.

Someone is a dog owner.

3. 11.

All of Geraldo’s friends are dog owners.

4. 12.

Every dog owner is a friend of a dog owner.

5. 13.

Every dog owner’s friend owns a dog of a friend.

Sentence 9 can be paraphrased as, ‘There is a dog that Geraldo owns’. This can be symbolized by ‘$\exists x(D(x)\wedge O(g,x))$’.

Sentence 10 can be paraphrased as, ‘There is some $y$ such that $y$ is a dog owner’. Dealing with part of this, we might write ‘$\exists y(y\text{ is a dog owner})$’. Now the fragment we have left as ‘$y$ is a dog owner’ is much like sentence 9, except that it is not specifically about Geraldo. So we can symbolize sentence 10 by:

$\exists y\exists x(D(x)\wedge O(y,x))$

We should pause to clarify something here. In working out how to symbolize the last sentence, we wrote down ‘$\exists y(y\text{ is a dog owner})$’. To be very clear: this is neither an FOL sentence nor an English sentence: it uses bits of FOL (‘$\exists$’, ‘$y$’) and bits of English (‘dog owner’). It is really just a stepping-stone on the way to symbolizing the entire English sentence with a sentence of FOL. You should regard it as a bit of rough-working-out, on a par with the doodles that you might absent-mindedly draw in the margin of this book, whilst you are concentrating fiercely on some problem.

Sentence 11 can be paraphrased as, ‘Every $x$ who is a friend of Geraldo is a dog owner’. Using our stepping-stone tactic, we might write

$\forall x\bigl{[}F(x,g)\rightarrow x\text{ is a dog owner}\bigr{]}$

Now the fragment that we have left to deal with, ‘$x$ is a dog owner’, is structurally just like sentence 9. However, it would be a mistake for us simply to write

$\forall x\bigl{[}F(x,g)\rightarrow\exists x(D(x)\wedge O(x,x))\bigr{]}$

for we would here have a clash of variables. The scope of the universal quantifier, ‘$\forall x$’, is the entire conditional, so the ‘$x$’ in ‘$D(x)$’ should be governed by that, but ‘$D(x)$’ also falls under the scope of the existential quantifier ‘$\exists x$’, so the ‘$x$’ in ‘$D(x)$’ should be governed by that. Now confusion reigns: which ‘$x$’ are we talking about? Suddenly the sentence becomes ambiguous (if it is even meaningful at all), and logicians hate ambiguity. The broad moral is that a single variable cannot serve two quantifier-masters simultaneously.

To continue our symbolization, then, we must choose some different variable for our existential quantifier. What we want is something like:

$\forall x\bigl{[}F(x,g)\rightarrow\exists z(D(z)\wedge O(x,z))\bigr{]}$

Sentence 12 can be paraphrased as ‘For any $x$ that is a dog owner, there is a dog owner who $x$ is a friend of’. Using our stepping-stone tactic, this becomes

$\forall x\bigl{[}\mbox{x is a dog owner}\rightarrow\exists y(\mbox{y is a % dog owner}\wedge F(x,y))\bigr{]}$

Completing the symbolization, we end up with

$\forall x\bigl{[}\exists z(D(z)\wedge O(x,z))\rightarrow\exists y\bigl{(}% \exists z(D(z)\wedge O(y,z))\wedge F(x,y)\bigr{)}\bigr{]}$

Note that we have used the same letter, ‘$z$’, in both the antecedent and the consequent of the conditional, but that these are governed by two different quantifiers. This is ok: there is no clash here, because it is clear which quantifier that variable falls under. We might graphically represent the scope of the quantifiers thus:

$\overbrace{\forall x\bigl{[}\overbrace{\exists z(D(z)\wedge O(x,z))}^{\text{% scope of 1st }\exists z\text{'}}\rightarrow\overbrace{\exists y(\overbrace{% \exists z(D(z)\wedge O(y,z))}^{\text{scope of 2nd }\exists z\text{'}}\wedge F% (x,y))\bigr{]}}^{\text{scope of }\exists y\text{'}}}^{\text{scope of }% \forall x\text{'}}$

This shows that no variable is being forced to serve two masters simultaneously.

Sentence 13 is the trickiest yet. First we paraphrase it as ‘For any $x$ that is a friend of a dog owner, $x$ owns a dog which is also owned by a friend of $x$’. Using our stepping-stone tactic, this becomes:

$\forall x\bigl{[}x\text{ is a friend of a dog owner}\rightarrow x\text{ owns a% dog which is owned by a friend of }x\bigr{]}$

Breaking this down a bit more:

$\forall x\bigl{[}\exists y(F(x,y)\wedge y\text{ is a dog owner})\rightarrow% \exists y((D(y)\wedge O(x,y))\wedge y\text{ is owned by a friend of }x)\bigr{]}$

And a bit more:

$\forall x\bigl{[}\exists y(F(x,y)\wedge\exists z(D(z)\wedge O(y,z)))% \rightarrow\exists y((D(y)\wedge O(x,y))\wedge\exists z(F(z,x)\wedge O(z,y)))% \bigr{]}$

And we are done!

There is one subtle issue we should briefly address. We paraphrased sentence 10 as ‘There is some $y$ such that $y$ is a dog owner’. Now our domain includes people and dogs, and ‘someone’ includes people, but (at least arguably) does not include dogs. To be more correct, we should have paraphrased sentence 10 as ‘There is some $y$ such that $y$ is a person and a dog owner’. A more accurate symbolization of ‘Someone is a dog owner’ would require that we add a predicate for ‘blank is a person’ to our symbolization key:

$P(x)$:

blank${}_{x}$ is a person

Then we can give a better symbolization of sentence 10:

$\exists y(P(y)\wedge\exists x(D(x)\wedge O(y,x)))$

‘Everyone’ and ‘no one’ have to be treated similarly: ‘Everyone is a friend of Geraldo’ and ‘No one is a friend of Geraldo’ would be symbolized, respectively, as

 $\displaystyle\forall x(P(x)\rightarrow F(x,g))$ $\displaystyle\forall x(P(x)\rightarrow\neg F(x,g)).$

Only ‘someone’, ‘everyone’, and ‘no one’ require this treatment. In particular, we do not need to explicitly state in the symbolization of sentence 9 that Geraldo is a person. Neither do we have to ensure in sentence 11 that Geraldo’s friend $x$ is a person. Although it may be true that only people can own dogs or be friends with Geraldo, it is not part of what the sentences say, and so does not need to be taken into account when we symbolize them.22 2 You might object: but the sentences also don’t say that dogs aren’t people. E.g., if we’re talking about the fictional world of Mickey Mouse, Goofy should be included in ‘everyone’, but Pluto should not be. That’s why we picked the predicate ‘blank is a person’ and not ‘blank is a human’: in that domain, Goofy would fall under both ‘blank is a person’ and ‘blank is a dog’, but Pluto would only fall under ‘blank is a dog’.

## 25.5 Supressed quantifiers

Logic can often help to get clear on the meanings of English claims, especially where the quantifiers are left implicit or their order is ambiguous or unclear. The clarity of expression and thinking afforded by FOL can give you a significant advantage in argument, as can be seen in the following takedown by British political philosopher Mary Astell (1666–1731) of her contemporary, the theologian William Nicholls. In Discourse IV: The Duty of Wives to their Husbands of his The Duty of Inferiors towards their Superiors, in Five Practical Discourses (London 1701), Nicholls argued that women are naturally inferior to men. In the preface to the 3rd edition of her treatise Some Reflections upon Marriage, Occasion’d by the Duke and Duchess of Mazarine’s Case; which is also considered, Astell responded as follows:

’Tis true, thro’ Want of Learning, and of that Superior Genius which Men as Men lay claim to, she [Astell] was ignorant of the Natural Inferiority of our Sex, which our Masters lay down as a Self-Evident and Fundamental Truth. She saw nothing in the Reason of Things, to make this either a Principle or a Conclusion, but much to the contrary; it being Sedition at least, if not Treason to assert it in this Reign.
For if by the Natural Superiority of their Sex, they mean that every Man is by Nature superior to every Woman, which is the obvious meaning, and that which must be stuck to if they would speak Sense, it wou’d be a Sin in any Woman to have Dominion over any Man, and the greatest Queen ought not to command but to obey her Footman, because no Municipal Laws can supersede or change the Law of Nature; so that if the Dominion of the Men be such, the Salique Law,33 3 The Salique law was the common law of France which prohibited the crown be passed on to female heirs. as unjust as English Men have ever thought it, ought to take place over all the Earth, and the most glorious Reigns in the English, Danish, Castilian, and other Annals, were wicked Violations of the Law of Nature!
If they mean that some Men are superior to some Women this is no great Discovery; had they turn’d the Tables they might have seen that some Women are Superior to some Men. Or had they been pleased to remember their Oaths of Allegiance and Supremacy, they might have known that One Woman is superior to All the Men in these Nations, or else they have sworn to very little purpose.44 4 In 1706, England was ruled by Queen Anne. And it must not be suppos’d, that their Reason and Religion wou’d suffer them to take Oaths, contrary to the Laws of Nature and Reason of things.55 5 Mary Astell, Reflections upon Marriage, 1706 Preface, iii–iv, and Mary Astell, Political Writings, Patricia Springborg (ed.), Cambridge University Press, 1996, pp. 9–10.

We can symbolize the different interpretations Astell offers of Nicholls’ claim that men are superior to women: He either meant that every man is superior to every woman, i.e.,

$\forall x(M(x)\rightarrow\forall y(W(y)\rightarrow S(x,y)))$

or that some men are superior to some women,

$\exists x(M(x)\wedge\exists y(W(y)\wedge S(x,y))).$

The latter is true, but so is

$\exists y(W(y)\wedge\exists x(M(x)\wedge S(y,x)))$

(some women are superior to some men), so that would be “no great discovery.” In fact, since the Queen is superior to all her subjects, it’s even true that some woman is superior to every man, i.e.,

$\exists y(W(y)\land\forall x(M(x)\rightarrow S(y,x))).$

But this is incompatible with the “obvious meaning” of Nicholls’ claim, i.e., the first reading. So what Nicholls claims amounts to treason against the Queen!

## Practice exercises

A. Using this symbolization key:

domain:

all animals

$A(x)$:

blank${}_{x}$ is an alligator

$M(x)$:

blank${}_{x}$ is a monkey

$R(x)$:

blank${}_{x}$ is a reptile

$Z(x)$:

blank${}_{x}$ lives at the zoo

$L(x,y)$:

blank${}_{x}$ loves blank${}_{y}$

$a$:

Amos

$b$:

Bouncer

$c$:

Cleo

symbolize each of the following sentences in FOL:

1. 1.

If Cleo loves Bouncer, then Bouncer is a monkey.

2. 2.

If both Bouncer and Cleo are alligators, then Amos loves them both.

3. 3.

Cleo loves a reptile.

4. 4.

Bouncer loves all the monkeys that live at the zoo.

5. 5.

All the monkeys that Amos loves love him back.

6. 6.

Every monkey that Cleo loves is also loved by Amos.

7. 7.

There is a monkey that loves Bouncer, but sadly Bouncer does not reciprocate this love.

B. Using the following symbolization key:

domain:

all animals

$D(x)$:

blank${}_{x}$ is a dog

$S(x)$:

blank${}_{x}$ likes samurai movies

$L(x,y)$:

blank${}_{x}$ is larger than blank${}_{y}$

$r$:

Rave

$h$:

Shane

$d$:

Daisy

symbolize the following sentences in FOL:

1. 1.

Rave is a dog who likes samurai movies.

2. 2.

Rave, Shane, and Daisy are all dogs.

3. 3.

Shane is larger than Rave, and Daisy is larger than Shane.

4. 4.

All dogs like samurai movies.

5. 5.

Only dogs like samurai movies.

6. 6.

There is a dog that is larger than Shane.

7. 7.

If there is a dog larger than Daisy, then there is a dog larger than Shane.

8. 8.

No animal that likes samurai movies is larger than Shane.

9. 9.

No dog is larger than Daisy.

10. 10.

Any animal that dislikes samurai movies is larger than Rave.

11. 11.

There is an animal that is between Rave and Shane in size.

12. 12.

There is no dog that is between Rave and Shane in size.

13. 13.

No dog is larger than itself.

14. 14.

Every dog is larger than some dog.

15. 15.

There is an animal that is smaller than every dog.

16. 16.

If there is an animal that is larger than any dog, then that animal does not like samurai movies.

C. Using the symbolization key given, symbolize each English-language sentence into FOL.

domain:

candies

$C(x)$:

blank${}_{x}$ has chocolate in it

$M(x)$:

blank${}_{x}$ has marzipan in it

$S(x)$:

blank${}_{x}$ has sugar in it

$T(x)$:

Boris has tried blank${}_{x}$

$B(x,y)$:

blank${}_{x}$ is better than blank${}_{y}$

1. 1.

Boris has never tried any candy.

2. 2.

Marzipan is always made with sugar.

3. 3.

Some candy is sugar-free.

4. 4.

The very best candy is chocolate.

5. 5.

No candy is better than itself.

6. 6.

Boris has never tried sugar-free chocolate.

7. 7.

Boris has tried marzipan and chocolate, but never together.

8. 8.

Any candy with chocolate is better than any candy without it.

9. 9.

Any candy with chocolate and marzipan is better than any candy that lacks both.

D. Using the following symbolization key:

domain:

people and dishes at a potluck

$R(x)$:

blank${}_{x}$ has run out

$T(x)$:

blank${}_{x}$ is on the table

$F(x)$:

blank${}_{x}$ is food

$P(x)$:

blank${}_{x}$ is a person

$L(x,y)$:

blank${}_{x}$ likes blank${}_{y}$

$e$:

Eli

$f$:

Francesca

$g$:

the guacamole

symbolize the following English sentences in FOL:

1. 1.

All the food is on the table.

2. 2.

If the guacamole has not run out, then it is on the table.

3. 3.

Everyone likes the guacamole.

4. 4.

If anyone likes the guacamole, then Eli does.

5. 5.

Francesca only likes the dishes that have run out.

6. 6.

Francesca likes no one, and no one likes Francesca.

7. 7.

Eli likes anyone who likes the guacamole.

8. 8.

Eli likes anyone who likes the people that he likes.

9. 9.

If there is a person on the table already, then all of the food must have run out.

E. Using the following symbolization key:

domain:

people

$D(x)$:

blank${}_{x}$ dances ballet

$F(x)$:

blank${}_{x}$ is female

$M(x)$:

blank${}_{x}$ is male

$C(x,y)$:

blank${}_{x}$ is a child of blank${}_{y}$

$S(x,y)$:

blank${}_{x}$ is a sibling of blank${}_{y}$

$e$:

Elmer

$j$:

Jane

$p$:

Patrick

symbolize the following sentences in FOL:

1. 1.

All of Patrick’s children are ballet dancers.

2. 2.

Jane is Patrick’s daughter.

3. 3.

Patrick has a daughter.

4. 4.

Jane is an only child.

5. 5.

All of Patrick’s sons dance ballet.

6. 6.

Patrick has no sons.

7. 7.

Jane is Elmer’s niece.

8. 8.

Patrick is Elmer’s brother.

9. 9.

Patrick’s brothers have no children.

10. 10.

Jane is an aunt.

11. 11.

Everyone who dances ballet has a brother who also dances ballet.

12. 12.

Every woman who dances ballet is the child of someone who dances ballet.