Chapter 25 Multiple generality

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:

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  blank loves blank

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  blank is to the left of blank

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  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

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  Vinnie borrowed blank from blank

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  blank borrowed the family car from blank

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  blank borrowed blank from Nunzio

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

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  blank borrowed blank from blank

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

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.
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   Karl loves Imre.

2. 2.
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   Imre loves Karl.

3. 3.
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   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.
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   Imre loves himself.

2. 5.
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   Karl loves Imre, but not vice versa.

3. 6.
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   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 also come up with: Karl loves Imre.

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!

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

1. 7.
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   For every person x, there is some person that x loves.

2. 8.
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   There is some particular person whom every person loves.

Sentence 7 can be symbolized by ‘$\forall x\exists yL(x,y)$’. 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 ‘$\exists y\forall xL(x,y)$’. 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:

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

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  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.

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

• ∴
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  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:

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  A function $f$ is pointwise continuous if
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  A function $f$ is uniformly continuous if

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

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.
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   Geraldo is a dog owner.

2. 10.
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   Someone is a dog owner.

3. 11.
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   All of Geraldo’s friends are dog owners.

4. 12.
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   Every dog owner is a friend of a dog owner.

5. 13.
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   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:

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

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

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:

This adequately symbolizes sentence 11.

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

Completing the symbolization, we end up with

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:

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:

Breaking this down a bit more:

And a bit more:

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:

‘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

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’.

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.,

or that some men are superior to some women,

The latter is true, but so is

(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.,

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!