Definite Descriptions, Naming, and Problems for Identity
1. Russel’s Definite Descriptions: Here are three things we’ve been assuming all along:
(1) Any grammatically correct statement formed from meaningful terms has a meaning.
(2) All meaningful statements have truth values.
(3) The Law of Excluded Middle is a theorem. That is, ‘P  ¬P’ is a theorem. What this
means is that, for any statement, P, either P is true, or its negation (¬P) is true. Or,
in other words, every statement is either true or false.
Now consider this statement: “The present king of France is bald.”
This seems to be a meaningful statement. After all, unlike “The poople is a moople”, the
statement is grammatically correct, and we understand all of the terms. So, according to
(1) - (3), it must have a truth value. So, which one is it. Is this statement true or false?
(Keep in mind that there is no present king of France.) Either answer seems weird:
Can’t Be True (?) It couldn’t be true, because then that would mean that France DOES
presently have a bald king (but it doesn’t).
Can’t Be False (?) But, if the statement were false, this means that its NEGATION is true.
OR, in other words, it seems that this would entail that “The present king of France is
NOT bald” is true. (Right?) But, that statement can’t be true either…
This is a big deal, though it may not seem so at first. But, recognize that, to get around
this problem, we would have to reject one of the three claims above—all of which are a
part of the bedrock of logic. It would be absurd to deny any of them. Consider:
(1) Surely, it would be difficult to deny that “The present king of France” has a
meaning. Doesn’t this statement MEAN something?
(2) And surely, if a statement asserts something in a meaningful way, then what it
asserts is either true or false, right?
(3) Finally, the law of excluded middle must be true. For, if a statement is NOT TRUE,
then it must follow that the DENIAL of that statement IS true, right?
Now, some philosophers have tried to reject (1). They would say that, while such
statements APPEAR to be meaningful, they are not meaningful if the subject does not
have a ‘referent’. That is, if ‘the present king of France’ does not REFER to anything
(because there is no such individual), then a statement is meaningless even if it is
formed correctly from meaningful terms. If you’re sympathetic to this solution, you
might want to modify (1) to say:
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(1) Modified: Any grammatically correct statement formed from meaningful terms
has a meaning, but only if all subjects/names have referents.
But, Bertrand Russell didn’t like this solution. “The present king of France is bald” really
seems to be a meaningful assertion! After all, no one, upon hearing the claim, would
respond with, “I’m sorry. I don’t understand what you’ve just said. That’s meaningless.”
So, he proposed another solution. Thus far, it may have seemed like “The present king of
France is bald” should be expressed as something like ‘Bk’ and that its negation (“The
present king of France is NOT bald”) should be expressed as ‘¬Bk’. Russell disagreed.
Russell agreed that the expression ‘the present king of France’ does not name any
particular individual, even though it appears to. Russell called these sorts of expressions,
which seem to pick out exactly one individual ‘definite descriptions’, and pointed out
that such expressions NEVER name a particular individual, appearances to the contrary.
Of course, GRAMATICALLY they appear to name individuals—e.g., the statement at hand
seems to be ABOUT that individual who is the present king of France—but LOGICALLY,
definite descriptions do not name anything. Rather, they are claims about existence. To
see what he means, let’s symbolize the statement using the tools of Predicate Logic:
Take ‘Bx=x is bald’ and ‘Kx=x is a present king of France’. Here’s the wff:
(ꓱx){[Kx  (ꓯy)(Ky  y=x)]  Bx}
Literally: “There exists an x such that x is the present king of France, and anything that is
the present king of France is identical to x (i.e., there is no more than ONE present king
of France), and x is bald.”
If I said that ‘Albert is fat’, we’d symbolize this as ‘Fa’. And ‘a’ here DOES name an
individual (namely, Albert). But, if I say that “The present king of France is bald” I do NOT
symbolize this as ‘Bk’. Rather, because it is a definite DESCRIPTION, I must instead write
it as an existential claim, which says that there exists an x which has certain properties. It
doesn’t NAME anything at all!
Now, if Russell is correct, then it is easy to see that “The present king of France is
bald” is FALSE! After all, there does NOT exist an x such that x is king of France and x is
bald. Problem solved. Pretty cool.
Interestingly, “The present king of France is NOT bald” is ALSO false on Russell’s view:
(ꓱx){[Kx  (ꓯy)(Ky  y=x)]  ¬Bx}
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Wait a minute. If “The king of France is bald” is false AND “The king of France is not
bald” is false, isn’t the Law of Excluded Middle violated? Out of every pair, ‘P’ and ‘¬P’,
one MUST be true and the other MUST be false. But here, it seems like ‘P’ and ‘¬P’ are
BOTH false.
Don’t freak out. The Law of Excluded Middle is not violated. The statement above
negates only a tiny portion of ‘P’ rather than the whole thing. But, if ‘P’ is the ENTIRE
existentially quantified statement, then its negation is really the following:
¬(ꓱx){[Kx  (ꓯy)(Ky  y=x)]  Bx}
Notice that the dash is out FRONT, and not before the ‘Bx’. Now, THIS statement IS true.
Application: Maybe this knowledge will help you someday. Imagine that you are NOT
cheating on your loved one, but that your loved one tries to trap you by accusing:
“Your secret lover must enjoy sleeping with you behind my back.”
It may seem as if you cannot agree OR disagree. For, if you say:
(1) That’s not true! (then it seems as if your secret lover does NOT enjoy sleeping with you)
(2) That’s true! (then your secret lover DOES enjoy it)
Either way, you’ve confessed to a crime you did not commit. So what you should say is:
“You’re suggesting that there exists an x such that x is my secret lover, and x enjoys
sleeping with me behind your back. But, the following assertion:
(ꓱx)(Lx  Ex)
is false, because there exists no ‘x’ who is my secret lover.”
2. Problems for =E: We have learned that the following sequent is valid:
S218: Bls , s=c Ⱶ Blc
1
2
1, 2
(1) Bls
(2) s=c
(3) Blc
(your textbook uses different letters)
A
A
1, 2, =E
But, think about the following interpretation:
Bxy=x believes y can fly
l=Lois Lane
s=Superman
c=Clark Kent
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On this interpretation, the sequent above reads as: “Lois Lane believes that Superman
can fly. Superman is Clark Kent. Therefore, Lois Lane believes that Clark Kent can fly.”
Something has gone horribly wrong! Now consider this one:
Hasm , s=m Ⱶ Hass
1
2
1, 2
(1) Hasm
(2) s=m
(3) Hass
A
A
1, 2, =E
This is a permissible use of =E. But, think about the following interpretation:
Hxyz=x hopes that y is not z
a=Alice
s=Samuel Clemens
m=Mark Twain
Imagine that Alice reads in the paper that Samuel Clemens has been arrested for a
crime. Furthermore, the police suspect that Mr. Clemens is actually the famous author
who goes by the pseudonym ‘Mark Twain’. In that case, the sequent above reads as:
“Alice hopes that Samuel Clemens is not Mark Twain. Samuel Clemens is Mark
Twain. Therefore, Alice hopes that Samuel Clemens is not Samuel Clemens.”
That doesn’t seem right!
The problems pointed out above for =E occur in epistemic contexts. Epistemology is
the study of knowledge, but epistemic issues also include belief, desire, hope, and so on.
There are also problems for =E in modal contexts. Modality is the study of possibility
and necessity. Consider this inference:
1. Necessarily, Ben Franklin is Ben Franklin.
2. Ben Franklin is the inventor of the bifocals.
3. Therefore, necessarily, Ben Franklin is the inventor of the bifocals.
In modal logic, we symbolize ‘necessarily’ as ‘□’, so we could express this as:
□b=b , b=i Ⱶ □b=i
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While the premises seem true, it should not follow that “Ben Franklin is the inventor of
the bifocals” is a NECESSARY truth. What this would mean is that there is NO WAY that
Ben Franklin could have failed to invent the bifocals. But, surely he could have failed to
be their inventor! (for instance, it seems possible that someone else could have invented
them first)
Solution? We might try to use Russell’s tools to solve this issue. For instance, whether or
not Superman is identical to Clark Kent depends on what we MEAN by ‘Superman’ or
‘Clark Kent’. If ‘Superman’ is defined in terms of a definite description, then it seems that
Superman is NOT identical to Clark Kent. Consider these definitions:
Superman=The costumed superhero who flies around and saves the planet.
Clark Kent=The glasses-wearing reporter who works for the Daily Planet.
If THIS is how we designate the names ‘Superman’ and ‘Clark Kent’ then it is clear that
the argument is invalid. In this way, we might use Russell’s definite descriptions to solve
the puzzle. In conclusion, we might claim that =E can only be used upon an identity
claim such as ‘x=y’ when it is a claim where ‘x’ and ‘y’ are pure NAMES; in logic, we say
that =E is permissible only when ‘x’ and ‘y’ are extensional rather than intensional. To
see the difference between these two uses of a term, consider:
Extension of ‘dog’: Fido, Sparky, Lassie, Benji, etc.
Intension of ‘dog’: furry, four-legged, domesticated, playful, mammal, etc.
It turns out that epistemic and modal contexts often use names intensionally rather than
extensionally. So, in such contexts, the use of =E is impermissible. For instance, here is
the actual argument being made about Superman:
1. Lois Lane believes that the costumed superhero who flies around and saves the
planet and goes by the name ‘Superman’ can fly.
2. Superman=Clark Kent.
3. Therefore, Lois Lane believes that the glasses-wearing reporter who works for the
Daily Planet and goes by the name ‘Clark Kent’ can fly.
The conclusion does not follow in this case. The referent of (1) is a DESCRIPTION of
Superman, and not an individual (Superman). Meanwhile, the referents of (2) are the
INDIVIDUALS ‘Superman’ and ‘Clark Kent’. So, (3) does not follow from (1) and (2).
Similarly, the argument about Mark Twain is invalid if in (1) and (3), ‘Samuel Clemens’
refers to the description, ‘the man whom Alice has just read about, suspected of a crime’
and ‘Mark Twain’ refers to ‘the famous author of books such as Tom Sawyer and
Huckleberry Finn’, while those names refer to the individual man in (2).
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Also, the claim, “Necessarily, Ben Franklin = Ben Franklin” refers to the INDIVIDUAL, Ben
Franklin, while “Ben Franklin = the inventor of the bifocals” refers to the INDIVIDUAL on
the left, but to the DESCRIPTION on the right. But, we cannot put an ‘=’ sign between
a name and a definite description.
Saul Kripke pointed out that identity claims are only necessarily true when both sides of
the ‘=’ sign are either names or else descriptions that identify what it is to be that
individual (such as ‘water is H2O‘). For instance,
Necessarily, the first postmaster general is the inventor of the bifocals.
Necessarily, Ben Franklin is the inventor of the bifocals.
Necessarily, Cicero is Tully.
Necessarily, Hesperus is Phosphorus.
Necessarily, water is H2O.
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FALSE
FALSE
TRUE
TRUE
TRUE