Chapter 26 Identity
Consider this sentence:

1.
Pavel owes money to everyone.
Let the domain be people; this will allow us to symbolize ‘everyone’ with a universal quantifier. Offering the symbolization key:
 $O(x,y)$:

blank_{x} owes money to blank_{y}
 $p$:

Pavel
we can symbolize sentence 1 by ‘$\forall xO(p,x)$’. But this has a (perhaps) odd consequence. It requires that Pavel owes money to every member of the domain (whatever the domain may be). The domain certainly includes Pavel. So this entails that Pavel owes money to himself. And maybe we did not want to say that. Maybe we meant to leave it open if Pavel owes money to himself, something we could have expressed more precisely by using either one of the following:

2.
Pavel owes money to everyone else.

3.
Pavel owes money to everyone other than Pavel.
But we do not have any way for dealing with the italicized words yet. The solution is to add another symbol to FOL.
26.1 Adding identity
The symbol ‘$=$’ will be a twoplace predicate. Since it will have a special meaning, we shall write it a bit differently: we put it between two terms, rather than out front. (This should also be familiar; consider a mathematical equation like $\frac{1}{2}=0.5$.) And the special meaning for ‘$=$’ is given by the fact that we always adopt the following symbolization key:
 $x=y$:

blank_{x} is identical to blank_{y}
This does not mean merely that the objects in question are indistinguishable, or that all of the same things are true of them. Rather, it means that the objects in question are the very same object.
To put this to use, suppose we want to symbolize this sentence:

4.
Pavel is Mister Chekov.
Let us add to our symbolization key:
 $c$:

Mister Chekov
Now sentence 4 can be symbolized as ‘$p=c$’. This tells us that the names ‘$p$’ and ‘$c$’ both name the same thing.
We can also now deal with sentences 2 and 3. Both of these sentences can be paraphrased as ‘Everyone who is not Pavel is owed money by Pavel’. Paraphrasing some more, we get: ‘For all $x$, if $x$ is not Pavel, then $x$ is owed money by Pavel’. Now that we are armed with our new identity symbol, we can symbolize this as ‘$\forall x(\neg x=p\to O(p,x))$’.
This last sentence contains the formula ‘$\neg x=p$’. That might look a bit strange, because the symbol that comes immediately after the ‘$\neg $’ is a variable, rather than a predicate, but this is not a problem. We are simply negating the entire formula ‘$x=p$’.
26.2 ‘Only’ and ‘except’
In addition to sentences that use the word ‘else’, and ‘other than’, identity is helpful when symbolizing some sentences that contain the words ‘only’, and ‘except’. Consider:

5.
Only Pavel owes money to Hikaru.
Let ‘$h$’ name Hikaru. Plausibly, sentence 5 is true if, and only if, both of the following conditions hold:

6.
Pavel owes money to Hikaru.

7.
No one who is not Pavel owes money to Hikaru.
Sentence 7 can be symbolized by any one of:
$\neg \exists x(\neg x=p\wedge O(x,h)),$  
$\forall x(\neg x=p\to \neg O(x,h)),$  
$\forall x(O(x,h)\to x=p).$ 
Thus, we can symbolize sentence 5 as the conjunction of one of the above with the symbolization of sentence 6, ‘$O(p,h)$’, or more compactly using ‘$\leftrightarrow $’ as ‘$\forall x(O(x,h)\leftrightarrow x=p)$’.

8.
Everyone except Pavel owes money to Hikaru.
Sentence 8 can be treated similarly, although now of course Pavel does not owe Hikaru money. We can paraphrase it as ‘Everyone who is not Pavel owes Hikaru money, and Pavel does not’. Consequently, it can be symbolized as ‘$\forall x(\neg x=p\to O(x,h))\wedge \neg O(p,h)$’, or more concisely, ‘$\forall x(\neg x=p\leftrightarrow O(x,h))$’. Other locutions akin to ‘except’ such as ‘but’ or ‘besides’ (as used in ‘no one but Pavel’ or ‘someone besides Hikaru’) can be treated in similar ways.
The above treatment of socalled “exceptives” is not uncontentious. Some linguists think that sentence 8 does not entail that Pavel doesn’t owe Hikaru money, and so the symbolization should just be ‘$\forall x(\neg x=p\to O(x,h))$’. There are also uses of ‘except’ that clearly do not have that entailment, especially in mathematical writing. For instance, you may read in a calculus textbook that “the function $f$ is defined everywhere except possibly at $a$”. That means only that for every point $x$ other than $a$, $f$ is defined at $x$. It is not required that $f$ is undefined at $a$; it’s left open whether $f$ is or is not defined at $a$.
26.3 There are at least…
We can also use identity to say how many things there are of a particular kind. For example, consider these sentences:

9.
There is at least one apple.

10.
There are at least two apples.

11.
There are at least three apples.
We will use the symbolization key:
 $A(x)$:

blank_{x} is an apple
Sentence 9 does not require identity. It can be adequately symbolized by ‘$\exists xA(x)$’: There is an apple; perhaps many, but at least one.
It might be tempting to also symbolize sentence 10 without identity, namely as ‘$\exists x\exists y(A(x)\wedge A(y))$’. Roughly, this says that there is some apple $x$ in the domain and some apple $y$ in the domain. Since nothing precludes these from being one and the same apple, this would be true even if there were only one apple. In order to make sure that we are dealing with different apples, we need the identity predicate. Sentence 10 needs to say that the two apples that exist are not identical, so it can be symbolized by ‘$\exists x\exists y((A(x)\wedge A(y))\wedge \neg x=y)$’.
Sentence 11 requires talking about three different apples. Now we need three existential quantifiers, and we need to make sure that each will pick out something different:
Note that it is not enough to use ‘$\neg x=y\wedge \neg y=z$’ to symbolize ‘$x$, $y$, and $z$ are all different.’ For that would be true if $x$ and $y$ were different, but $x=z$. In general, to say that ${x}_{1}$, …, ${x}_{n}$ are all different, we must have a conjunction of $\neg {x}_{i}={x}_{j}$ for every different pair $i$ and $j$.
26.4 There are at most…
Now consider these sentences:

12.
There is at most one apple.

13.
There are at most two apples.
Sentence 12 can be paraphrased as, ‘It is not the case that there are at least two apples’. This is just the negation of sentence 10:
But sentence 12 can also be approached in another way. It means that if you pick out an object and it’s an apple, and then you pick out an object and it’s also an apple, you must have picked out the same object both times. With this in mind, sentence 12 can also be symbolized by:
The two sentences will turn out to be logically equivalent.
Similarly, sentence 13 can be approached in two equivalent ways. It can be paraphrased as, ‘It is not the case that there are three or more distinct apples’, so we can offer:
Alternatively we can read it as saying that if you pick out an apple, and an apple, and an apple, then you will have picked out (at least) one of these objects more than once. Thus:
26.5 There are exactly…
We can now consider statements of exact numerical quantity, like:

14.
There is exactly one apple.

15.
There are exactly two apples.

16.
There are exactly three apples.
Sentence 14 can be paraphrased as, ‘There is at least one apple and there is at most one apple’. This is just the conjunction of sentence 9 and sentence 12. So we can offer:
But it is perhaps more straightforward to paraphrase sentence 14 as, ‘There is a thing $x$ which is an apple, and everything which is an apple is just $x$ itself’. Thought of in this way, we offer:
Similarly, sentence 15 may be paraphrased as, ‘There are at least two apples, and there are at most two apples’. Thus we could offer
More efficiently, though, we can paraphrase it as ‘There are at least two different apples, and every apple is one of those two apples’. Then we offer:
Finally, consider these sentences:

17.
There are exactly two things.

18.
There are exactly two objects.
It might be tempting to add a predicate to our symbolization key, to symbolize the English predicate ‘blank is a thing’ or ‘blank is an object’, but this is unnecessary. Words like ‘thing’ and ‘object’ do not sort wheat from chaff: they apply trivially to everything, which is to say, they apply trivially to every thing. So we can symbolize either sentence with either of the following:
$\exists x\exists y\neg x=y\wedge \neg \exists x\exists y\exists z((\neg x=y\wedge \neg y=z)\wedge \neg x=z)$  
$\exists x\exists y\left[\neg x=y\wedge \forall z(x=z\vee y=z)\right]$ 
Practice exercises
A. Consider the sentence,

19.
Every officer except Pavel owes money to Hikaru.
Symbolize this sentence, using ‘$F(x)$’ for ‘blank_{x} is an officer’. Are you confident that your symbolization is true if, and only if, sentence 19 is true? What happens if every officer owes money to Hikaru, Pavel does not, but Pavel isn’t an officer?
B. Explain why:

1.
‘$\exists x\forall y(A(y)\leftrightarrow x=y)$’ is a good symbolization of ‘there is exactly one apple’.

2.
‘$\exists x\exists y[\neg x=y\wedge \forall z(A(z)\leftrightarrow (x=z\vee y=z))]$’ is a good symbolization of ‘there are exactly two apples’.