Chapter 28 Definite descriptions
Consider sentences like:

1.
Nick is the traitor.

2.
The traitor went to Cambridge.

3.
The traitor is the deputy.
These are definite descriptions: they are meant to pick out a unique object. They should be contrasted with indefinite descriptions, such as ‘Nick is a traitor’. They should equally be contrasted with generics, such as ‘The whale is a mammal’ (when it’s inappropriate to ask which whale). The question we face is: How should we deal with definite descriptions in FOL?
28.1 Treating definite descriptions as terms
One option would be to introduce new names whenever we come across a definite description. This is probably not a great idea. We know that the traitor—whoever it is—is indeed a traitor. We want to preserve that information in our symbolization.
A second option would be to use a new definite description operator, such as ‘$\iota $’. The idea would be to symbolize ‘the $F$’ as ‘$\iota xF(x)$’ (think ‘the $x$ such that $F(x)$’); or to symbolize ‘the $G$’ as ‘$\iota xG(x)$’, etc. Expressions of the form $\iota \U0001d4cd\mathcal{A}(\U0001d4cd)$ would then behave like names. If we were to follow this path, we could use the following symbolization key:
 domain:

people
 $T(x)$:

blank_{x} is a traitor
 $D(x)$:

blank_{x} is a deputy
 $C(x)$:

blank_{x} went to Cambridge
 $n$:

Nick
Then, we could symbolize sentence 1 with ‘$n=\iota xT(x)$’, sentence 2 with ‘$C(\iota xT(x))$’, and sentence 3 with ‘$\iota xT(x)=\iota xD(x)$’.
However, it would be nice if we didn’t have to add a new symbol to FOL. And we might be able to make do without one.
28.2 Russell’s analysis
Bertrand Russell offered an analysis of definite descriptions. Very briefly put, he observed that, when we say ‘the $F$’ in the context of a definite description, our aim is to pick out the one and only thing that is $F$ (in the appropriate context). Thus Russell analyzed the notion of a definite description as follows:^{1}^{1} 1 Bertrand Russell, “On denoting”, Mind 14 (1905), pp. 479–93; also Russell, Introduction to Mathematical Philosophy, 1919, London: Allen and Unwin, ch. 16.
The $\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$F$}$ is $\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$G$}$ iff

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there is at least one $\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$F$}$, and

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there is at most one $\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$F$}$, and

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every $\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$F$}$ is $\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$G$}$.
Note a very important feature of this analysis: ‘the’ does not appear on the rightside of the equivalence. Russell is aiming to provide an understanding of definite descriptions in terms that do not presuppose them.
Now, one might worry that we can say ‘the table is brown’ without implying that there is one and only one table in the universe. But this is not (yet) a fantastic counterexample to Russell’s analysis. The domain of discourse is likely to be restricted by context (e.g., to salient objects in my vicinity).
If we accept Russell’s analysis of definite descriptions, then we can symbolize sentences of the form ‘the $F$ is $G$’ using our strategy for numerical quantification in FOL. After all, we can deal with the three conjuncts on the righthand side of Russell’s analysis as follows:
In fact, we could express the same point rather more crisply, by recognizing that the first two conjuncts just amount to the claim that there is exactly one $F$, and that the last conjunct tells us that that object is $G$. So, equivalently, we could offer:
Using these sorts of techniques, we can now symbolize sentences 1 to 3 without using any newfangled fancy operator, such as ‘$\iota $’.
Sentence 1 is exactly like the examples we have just considered. So we would symbolize it by
$\exists x[T(x)\wedge \forall y(T(y)\to x=y)$  $\wedge x=n].$  
Sentence 2 poses no problems either:  
$\exists x[T(x)\wedge \forall y(T(y)\to x=y)$  $\wedge C(x)].$ 
Sentence 3 is a little trickier, because it links two definite descriptions. But, deploying Russell’s analysis, it can be paraphrased by ‘there is exactly one traitor, $x$, and there is exactly one deputy, $y$, and $x=y$’. So we can symbolize it by:
Note that the formula ‘$x=y$’ must fall within the scope of both quantifiers!
28.3 Empty definite descriptions
One of the nice features of Russell’s analysis is that it allows us to handle empty definite descriptions neatly.
France has no king at present. Now, if we were to introduce a name, ‘$k$’, to name the present King of France, then everything would go wrong: remember from chapter 23 that a name must always pick out some object in the domain, and whatever we choose as our domain, it will contain no present kings of France.
Russell’s analysis neatly avoids this problem. Russell tells us to treat definite descriptions using predicates and quantifiers, instead of names. Since predicates can be empty (see chapter 24), this means that no difficulty now arises when the definite description is empty.
Indeed, Russell’s analysis helpfully highlights two ways to go wrong in a claim involving a definite description. To adapt an example from Stephen Neale^{2}^{2} 2 Stephen Neale, Descriptions, MIT Press, 1990. suppose Alex claims:

4.
I am dating the present king of France.
Using the following symbolization key:
 $a$:

Alex
 $K(x)$:

blank_{x} is a present king of France
 $D(x,y)$:

blank_{x} is dating blank_{y}
(Note that the symbolization key speaks of a present King of France, not the present King of France; i.e., it employs an indefinite, rather than a definite, description.) Sentence 4 would be symbolized by ‘$\exists x\left[(K(x)\wedge \forall y(K(y)\to x=y))\wedge D(a,x)\right]$’. Now, this can be false in (at least) two ways, corresponding to these two different sentences:

5.
There is no one who is both the present King of France and such that he and Alex are dating.

6.
There is a unique present King of France, but Alex is not dating him.
Sentence 5 might be paraphrased by ‘It is not the case that: the present King of France and Alex are dating’. It will then be symbolized by ‘$\neg \exists x\left[(K(x)\wedge \forall y(K(y)\to x=y))\wedge D(a,x)\right]$’. We might call this outer negation, since the negation governs the entire sentence. Note that the sentence is true if there is no present King of France.
Sentence 6 can be symbolized by ‘$\exists x\left[(K(x)\wedge \forall y(K(y)\to x=y))\wedge \neg D(a,x)\right]$’. We might call this inner negation, since the negation occurs within the scope of the definite description. Note that its truth requires that there is a present King of France, albeit one who is not dating Alex.
28.4 Possessives, ‘both’, ‘neither’
We can use Russell’s analysis of definite descriptions also to deal with singular possessive constructions in English. For instance, ‘Smith’s murderer’ means something like ‘the person who murdered Smith’, i.e., it is a disguised definite description. On Russell’s analysis, the sentence

7.
Smith’s murderer is insane.
can be false in one of three ways. It can be false because the one person who murdered Smith is not, in fact, insane. But it can also be false if the definite description is empty, namely if either noone murdered Smith (e.g., if Smith met with an unfortunate accident) or if more than one person murdered Smith.
To symbolize sentences containing singular possessives such as ‘Smith’s murderer’ you should paraphrase them using an explicit definite description, e.g., ‘The person who murdered Smith is insane’ and then symbolize it according to Russell’s analysis. In our case, we would use the symbolization key:
 Domain:

people
 $I(x)$:

blank_{x} is insane
 $M(x,y)$:

blank_{x} murdered blank_{y}
 $m$:

Smith
Our symbolization then reads, ‘$\exists x\left[M(x,m)\wedge \forall y(M(y,m)\to x=y)\wedge I(x)\right]$’.
Two other determiners that we can extend Russell’s analysis to are ‘both’ and ‘neither’. To say ‘both $F$s are $G$’ is to say that there are exactly two $F$s, and each of them is $G$. To say that ‘neither $F$ is $G$’, is to also say that there are exactly two $F$s, and neither of them is $G$. In FOL, the symbolizations would read, respectively,
$\exists x\exists y\left[F(x)\wedge F(y)\wedge \neg x=y\wedge \forall z(F(z)\to (x=z\vee y=z))\wedge G(x)\wedge G(y)\right]$  
$\exists x\exists y\left[F(x)\wedge F(y)\wedge \neg x=y\wedge \forall z(F(z)\to (x=z\vee y=z))\wedge \neg G(x)\wedge \neg G(y)\right]$ 
Compare these symbolizations with the symbolizations of ‘exactly two $F$s are $G$s’ from section 26.5, i.e., of ‘there are exactly two things that are both $F$ and $G$’:
The difference between the symbolization of this and that of ‘both $F$s are $G$s’ lies in the antecedent of the conditional. For ‘exactly two $F$s are $G$s’, we only require that there are no $F$s that are also $G$s other than $x$ and $y$, whereas for ‘both $F$s are $G$s’, there cannot be any $F$s, whether they are $G$s or not, other than $x$ and $y$. In other words, ‘both $F$s are $G$s’ implies that exactly two $F$s are $G$s. However, ‘exactly two $F$s are $G$s’ does not imply that both $F$s are $G$s (there might be a third $F$ which isn’t a $G$).
28.5 The adequacy of Russell’s analysis
How good is Russell’s analysis of definite descriptions? This question has generated a substantial philosophical literature, but we will restrict ourselves to two observations.
One worry focusses on Russell’s treatment of empty definite descriptions. If there are no $F$s, then on Russell’s analysis, both ‘the $F$ is $G$’ and ‘the $F$ is non$G$’ are false. P. F. Strawson suggested that such sentences should not be regarded as false, exactly, but involve presupposition failure, and so need to be treated as neither true nor false.^{3}^{3} 3 P. F. Strawson, “On referring”, Mind 59 (1950), pp. 320–34.
If we agree with Strawson here, we will need to revise our logic. For, in our logic, there are only two truth values (True and False), and every sentence is assigned exactly one of these truth values.
But there is room to disagree with Strawson. Strawson is appealing to some linguistic intuitions, but it is not clear that they are very robust. For example: isn’t it just false, not ‘gappy’, that Tim is dating the present King of France?
Keith Donnellan raised a second sort of worry, which (very roughly) can be brought out by thinking about a case of mistaken identity.^{4}^{4} 4 Keith Donnellan, “Reference and definite descriptions”, Philosophical Review 77 (1966), pp. 281–304. Two men stand in the corner: a very tall man drinking what looks like a gin martini; and a very short man drinking what looks like a pint of water. Seeing them, Malika says:

8.
The gindrinker is very tall!
Russell’s analysis will have us render Malika’s sentence as:

8^{′}.
There is exactly one gindrinker [in the corner], and whoever is a gindrinker [in the corner] is very tall.
Now suppose that the very tall man is actually drinking water from a martini glass; whereas the very short man is drinking a pint of (neat) gin. By Russell’s analysis, Malika has said something false, but don’t we want to say that Malika has said something true?
Again, one might wonder how clear our intuitions are on this case. We can all agree that Malika intended to pick out a particular man, and say something true of him (that he was tall). On Russell’s analysis, she actually picked out a different man (the short one), and consequently said something false of him. But maybe advocates of Russell’s analysis only need to explain why Malika’s intentions were frustrated, and so why she said something false. This is easy enough to do: Malika said something false because she had false beliefs about the men’s drinks; if Malika’s beliefs about the drinks had been true, then she would have said something true.^{5}^{5} 5 Interested parties should read Saul Kripke, “Speaker reference and semantic reference”, in: French et al. (eds.), Contemporary Perspectives in the Philosophy of Language, University of Minnesota Press, 1977, pp. 6–27.
To say much more here would lead us into deep philosophical waters. That would be no bad thing, but for now it would distract us from the immediate purpose of learning formal logic. So, for now, we will stick with Russell’s analysis of definite descriptions, when it comes to putting things into FOL. It is certainly the best that we can offer, without significantly revising our logic, and it is quite defensible as an analysis.
Practice exercises
A. Using the following symbolization key:
 domain:

people
 $K(x)$:

blank_{x} knows the combination to the safe
 $S(x)$:

blank_{x} is a spy
 $V(x)$:

blank_{x} is a vegetarian
 $T(x,y)$:

blank_{x} trusts blank_{y}
 $h$:

Hofthor
 $i$:

Ingmar
symbolize the following sentences in FOL:

1.
Hofthor trusts a vegetarian.

2.
Everyone who trusts Ingmar trusts a vegetarian.

3.
Everyone who trusts Ingmar trusts someone who trusts a vegetarian.

4.
Only Ingmar knows the combination to the safe.

5.
Ingmar trusts Hofthor, but no one else.

6.
The person who knows the combination to the safe is a vegetarian.

7.
The person who knows the combination to the safe is not a spy.
B. Using the following symbolization key:
 domain:

cards in a standard deck
 $B(x)$:

blank_{x} is black
 $C(x)$:

blank_{x} is a club
 $D(x)$:

blank_{x} is a deuce
 $J(x)$:

blank_{x} is a jack
 $M(x)$:

blank_{x} is a man with an axe
 $O(x)$:

blank_{x} is oneeyed
 $W(x)$:

blank_{x} is wild
symbolize each sentence in FOL:

1.
All clubs are black cards.

2.
There are no wild cards.

3.
There are at least two clubs.

4.
There is more than one oneeyed jack.

5.
There are at most two oneeyed jacks.

6.
There are two black jacks.

7.
There are four deuces.

8.
The deuce of clubs is a black card.

9.
Oneeyed jacks and the man with the axe are wild.

10.
If the deuce of clubs is wild, then there is exactly one wild card.

11.
The man with the axe is not a jack.

12.
The deuce of clubs is not the man with the axe.
C. Using the following symbolization key:
 domain:

animals in the world
 $B(x)$:

blank_{x} is in Farmer Brown’s field
 $H(x)$:

blank_{x} is a horse
 $P(x)$:

blank_{x} is a Pegasus
 $W(x)$:

blank_{x} has wings
symbolize the following sentences in FOL:

1.
There are at least three horses in the world.

2.
There are at least three animals in the world.

3.
There is more than one horse in Farmer Brown’s field.

4.
There are three horses in Farmer Brown’s field.

5.
There is a single winged creature in Farmer Brown’s field; any other creatures in the field must be wingless.

6.
The Pegasus is a winged horse.

7.
The animal in Farmer Brown’s field is not a horse.

8.
The horse in Farmer Brown’s field does not have wings.
D. In this chapter, we symbolized ‘Nick is the traitor’ by ‘$\exists x(T(x)\wedge \forall y(T(y)\to x=y)\wedge x=n)$’. Explain why these would be equally good symbolisations:

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$T(n)\wedge \forall y(T(y)\to n=y)$

‣
$\forall y(T(y)\leftrightarrow y=n)$