Names, Descriptions and
Quantifiers
(1) Singular terms
Singular term: a word or phrase that refers
to an individual object; its semantic value is an object.
This notion contrasts with
general terms:
General terms: a word or phrase that refers to/is true of objects that satisfy some
general condition; its semantic value is a set.
Note, these definitions are
in terms of semantic properties rather than grammatical ones, i.e., there is no syntactic necessary and sufficient conditions on what
is to count as a singular/general term.
Prima facie, the following provide examples:
(i)
Singular terms
a. Proper names:
b. Demonstratives: this,
that, now, he, I, here, etc.
c. Definite descriptions:
the present Prime Minister, the even prime, etc.
(ii) General terms
a. Count nouns: house,
people, book, etc.
b. Mass nouns: snow, beer,
rain, etc.
c. Generics: the leopard,
cars, etc.
d. Indefinite descriptions:
a man, a number, etc.
e. Quantified formulae: Some
number, every country, etc.
In a 1905 paper - ‘On
Denoting’ - Bertrand Russell argued that this classification is mistaken: the only genuine singular terms are
demonstratives; definite descriptions and names are in fact general terms.
(2) In two steps
Russell’s argument for the
new classification proceeds in two steps:
(i)
Names are ‘truncated’ definite descriptions.
(ii) Definite descriptions
are general quantified formulae.
Since Russell’s time, a
certain consensus has arisen that (ii) is true, with some qualifications, while
(i) is false. That is, names are singular terms, but
definite descriptions are not.
Let us first consider the
background to Russell’s theory of 1905.
Background
(1) Some Russellean Theses and a
Misunderstanding
Russell held a very simple
view of meaning: the meaning of a name is the object it stands for and the
meaning of a predicate is the universal it stands for. Such a view issues in
what we may call Russellian propositions: a proposition is a
complex consisting of the very objects which are the values of the words which
express the proposition. Let us dub this the radical constituency thesis.
Radical Constituency Thesis: Necessarily, a judgement expresses a proposition
just if the constituents of the proposition are the very ‘objects’ the
judgement is about.
So, if S says that Joe is
tall, then S is entertaining the proposition
(i)
<Joe, Tallness>,
where these constituents are not
representations or mental entities, but the very objects: the actual person Joe
and the abstract universal Tallness. This position is an extensional one, i.e., if ‘a’ and ‘b’ refer to the same object,
then ‘a’ means the same as ‘b’; the proposition is individuated with respect to
extension alone. Patently, if there is
no person Joe, then there is no corresponding proposition:
(ii) <…, Tallness>.
Russellian propositions are object dependent: if no object, then no
thought. In particular, this gives us a simple picture of names:
Object Theory of Reference: The meaning of a name is the object it stands for:
“the name is merely a means of pointing to the thing, and does not occur in
what you are asserting” (Russell, Lectures
on Logical Atomism, 1918, p.245). Thus, if two names have the same
referent, then they mean the same thing. If there are no objects to enter into
the proposition, then we have said nothing!
Object dependency: If a sentence has an empty term (a singular term with no reference),
then no proposition is expressed by
it.
“Whenever the grammatical
subject of a proposition can be supposed not to exist without rendering the
proposition meaningless, it is plain that the grammatical subject is not a
proper name, i.e., not a name directly representing some object” (Russell, PM, p.66)
Here is the key problem for
this picture: we can have thoughts, it seems, when there is no object to be a
constituent of the proposition:
(iii) Pegasus is swift <?, Swiftness>
(iv) The king of
Meinong, faced with just this problem,
appealed to a doctrine of subsistence.
Since ‘Pegasus is swift’ expresses a perfectly coherent meaning, Pegasus must be in some sense so as to feature in the
corresponding proposition. This sense of being is subsistence. Now Russell is
most often read as essaying just this kind of Meinongian
solution in Principles of Mathematics (1903).
We are obliged, if we follow this reading, to view Russell’s articulation of
the mature theory of descriptions in 1905 as motivated by a rejection of the
subsistence doctrine, such is what is new
about the theory of descriptions.
I think that such a view is mistaken, worse, it elides the profundity of Russell’s
theory.
(2) The Theory of Principles of Mathematics (1903)
In Principles,
Russell had a theory of denoting concepts
(DCs). DCs correspond
to quantifier terms (all, some, none, a/an, the,
etc.). The idea was that DCs are exceptions to the
general rule about Russellian propositions: the DC is
a constituent of the proposition, rather than the object the DC ‘refers’ to.
For example, the proposition expressed by
(i)
Every number has a successor
is not
(ii) <{x: x
` N}, Succession> (where {x: x ` N} is the set of numbers)
but is, rather,
(iii) <*Every number*,
Succession> (where ‘*’s mark a denoting concept).
In simple terms, we get to
think about the infinite set of numbers through the DC; we cannot, as it were,
think about the set directly, i.e., it doesn’t enter into the proposition. Now
the significance of this exception to the general rule is that DCs can be empty without the proposition being incomplete:
the DC is there in the proposition, it is an independent matter whether the DC
has a value or not. For example, the proposition expressed by ‘The King of
France is bald’ is
(iv) <*The King of
This is a perfectly good
proposition, even though there is no King of France; for the DC takes the place
of the missing king, if you will.
In short, Russell already
had the means to reject Meinongian subsistence in 1903, he didn’t need a theory of
descriptions to do that. I must say
that Principles and other writings
from the pre-1905 period are ambiguous as to whether Russell exploited the
potential of DCs in the way suggested or, confusedly,
accepted Meinongianism in spite of his new found
resources. Here is what I think.
Russell was uncomfortable
with DCs precisely because they were exceptions to
the general RCT thesis governing propositions. But, so long as one holds to the
general constituency view and the thought that ‘The King of France is bald’ is
expressive of a proposition, i.e., is meaningful, it looks impossible to
negotiate an alternative. Why? Well, ‘The king of
Why should we think that the
‘The king of
(3) Surface Form vs.
Logical Form
The heart of Russell’s
theory of descriptions, then, is the claim that definite descriptions, phrases
of the form ‘The so and so’, are not singular terms, i.e., they do not contribute an object to the
propositions they express. For Russell, this means that surface form is misleading as to logical form. Consider:
(SUBJECT) (PREDICATE)
Tony is balding
The present prime
minister is balding
Assuming for the moment that
‘Tony’ is a paradigmatic singular term, if we were to take the subject +
predicate form as our guide, we would treat ‘Tony’ as making the same semantic
contribution as ‘The present PM’ to the propositions respectively expressed.
That is, the one proposition is expressed: <Tony, Baldingness>.
Russell thinks this is mistaken. Russell’s fundamental move is to claim that
although ‘The present PM’ is a grammatical subject, it is an incomplete symbol with respect to
logical form, i.e., the phrase does not designate a propositional constituent.
Sentences featuring definite descriptions in fact express three distinct propositions, each one a general proposition. Thus:
(i)a.
Something is Prime Minister - ($x)[PM(x)].
b. Only that thing is Prime Minister - ("y)(PM(y) → x = y)
c. The thing is balding - B(x)]
Put all three together, and
one has:
(ii) ($x)[PM(x) & ("y)(PM(y) → x = y) & B(x)]
(There is at least
It follows that sentences
with definite descriptions as subjects do not express particular (object dependent) thoughts, rather, they express general (object independent) thoughts. One does not need to have a particular
object in mind to express the thought that the
so and so is F.
This notion of an incomplete symbol in fact applies to all
quantifier noun phrases as regimented in first-order logic. Consider:
(i)
Every number has a successor,
where ‘every number’ is the
subject and ‘has a successor’ is the predicate. Does the formalisation of (i) contain a constituent that corresponds to the subject?
The standard formalisation
is
(ii) ("x)( Fx
→ Gx),
where ‘F’ corresponds to ‘number’
and ‘G’ corresponds to ‘has a successor’, but there isn’t a constituent here
which corresponds to ‘every number’. Consider:
(iii) Fx
(iv) ("x)( Fx
(v) ("x)( Fx)
(iii)
is an open sentence - ‘x is a
number’.
(iv) is
not a formula at all.
(v)
is a formula, but says that ‘Everything is a number’.
Whereas (i)
is formed by predicating a property to a subject, (ii)
is formed by binding all free variables in an open sentence by a quantifier.
The meaning of the subject is, as it were, smeared across the quantifier and
the antecedent of the conditional.
In general, quantified
subjects at surface form disappear in the formalisation: (ii) contains no
subject at all. In other words, there is no propositional constituent
corresponding to ‘every number’. Russell’s proposal, therefore, is simply that
definite descriptions fall together with quantifier binding rather than
singular terms.
(4) Contextual definition
In effect, Russell offers a
contextual definition of ‘the F’: any context in which a phrase of the form
occurs can be translated into one in which only quantifiers occur. The
definition of all logical constants in terms of ‘¬’ and ‘&’ is another
example of contextual definition.
(5) Principles in place
Russell’s theory allows him
to keep uniformly to his principles:
(i)
The object theory of reference is vacuously satisfied because the ‘The F’ is
not a singular term; it gives way to a conjunction of general propositions.
(ii) Russell’s principle is satisfied because ‘The
F’ does not entail the existence of a particular object one needs to know.
(iii) Just so, the principle
of acquaintance is satisfied.
The Good of the Theory
(1) ‘Excluded Middle’ and Scope
We
want to say that every proposition is either true or false (‘Excluded Middle’). We thus want to
make a decision about The present King of
Names
are insensitive to scope of negation:
‘¬a
is F’ and ‘a is ¬F’ mean the same thing. Another reason why definite descriptions are not
singular terms.
Following
Russell, we may distinguish between two kinds of scope:
(i) Primary occurrence
(wide scope): the definite description is not a constituent of a more
complex clause.
(ii) Secondary occurrence (narrow scope): the
definite description is a constituent of a complex phrase.
Any
sentence in which ‘the king of
(iii)
($x)[Fx
& ("y)(Fy → y = x)
& ¬Gx]
is false.
But
where the definite description has secondary
occurrence, the sentence can be true precisely because the negation has
scope over the first clause:
(iv) ¬[($x)
[Fx & ("y)(Fy → y = x) & Gx]]
Thus:
Russell’s theory preserves excluded middle, even though there is no present
King of France.
(2) The Standard Argument for definite descriptions not
being names
Quantified phrases are
incomplete relative to surface form, and so if definite descriptions are
quantified phrases, then they will be incomplete also. But there is an
independent reason for thinking that definite descriptions do not contribute an
object to the propositions in which they occur.
Consider:
(i)
The author of Waverly is Scott
The is here is
identity. What object does ‘the author of Waverly’ contribute to the
proposition? Either
(a) it
contributes Scott, in which case the sentence becomes a tautology, which it
clearly is not. George IV wanted to know if the author of Waverly was Scott,
not if Scott was Scott. Or,
(b) it
contributes something other than Scott, in which case the identity is false.
Thus:
definite descriptions are not singular terms.
A Logical Excursus
One might think that the
distortion to surface form Russell’s analysis entails suggests that the
analysis is mistaken. There are two responses.
(i)
Why not butcher surface form? If one accepts the distinction between surface and
logical form, then the former does not impose a tight constraint on the latter.
(ii) The distance from
surface form Russell’s analysis exhibits is but an artefact of the reduction of ‘the’ to ‘some’ and
‘every’, the only terms of generality Russell worked with (others are
available, of course, with the employment of negation.) We do not, though, need
to effect such a reduction, we can treat ‘the’ as a
quantifier in its own right:
[the
x] (PM(x), B(x))
This form is due to Mostowski (a Polish logician). Think of the ‘(the x)’ as
expressing a function defined over two sets, the set of PMs
and the set of balding things. The value of the function is true just if there is no member of the
PM set that is not a member of the balding set and there is only one member of the PM set, i.e.,
‘[the x] (PM(x), B(x))’ is true iff
½PM - B½= 0 & ½PM½=1
(where
‘½A½’ means ‘the cardinality of
A’. This gives us the same truth conditions as the Russellian
analysis without beating surface form to a pulp.
Names, Descriptions, and
Acquaintance
(1) Russell’s account
It is crucial to an
understanding of Russell not to conflate his account of descriptions with his
account of names. Russell’s theory of descriptions only pertains to ‘the so and so’-type phrases. What Russell does
propose is that apparent names (singular terms), such as ‘Tony’, are in fact truncated or telescoped descriptions.
This claim is a separate thesis from the one which says that definite
descriptions are not singular terms. Prima
facie, one can accept the theory of descriptions without thinking that
‘Tony’ is, logically speaking, really a description.
Why should we not think that
names are also definite descriptions?
Russell’s motivation for
reducing names to descriptions was epistemologically
driven; it simply doesn’t follow from any semantic thesis Russell accepted.
Russell held to a principle:
Russell’s Principle: ‘a’ is a genuinely singular term only if ‘a is
F’ is meaningless, where ‘a’ is empty (lacks a referent).
“It is not possible for a
subject to think about something unless he knows which particular individual he
is thinking about” (Russell, Knowledge by
Acquaintance and Knowledge by
Description, p.159)
Now Russell’s principle does
not license the elimination of names
in favour of descriptions. For the moment, consider that there is a set of
options available. One may, for instance, say that, contra intuition, ‘a is F’ does not
express a thought when ‘a’ is empty, but ‘a’ is still a singular term! Russell
did not consider this option because of an epistemological principle:
Principle of acquaintance: To understand a proposition, one must be acquainted with its constituents.
“Every proposition which we
can understand must be composed wholly of constituents with which we are
acquainted” (ibid., p.209).
To be acquainted to an
object is to have “a direct cognitive relation to that object, i.e., when I am
directly aware of the object itself” (Ibid, p.200
Therefore, if ‘a is F’ is
assumed to be meaningful, where ‘a’ is empty, then ‘a’ cannot be a singular
term, for we cannot be acquainted with its referent - it doesn’t have one. It
thus appears that we are led to view a as complex, as made up out of constituents with which we
may be acquainted.
Acquaintance has no semantic
motivation, it is based upon an empiricist picture of
the mind. If we reject such a picture, then we have removed the motivation for
thinking of names as complexes, descriptions.
So,
by these principles, genuine names (singular terms) are just those things whose
referents are objects with which I am acquainted. But am I acquainted with
Russell,
“When
we say anything about Bismarck, we should like, if we could, to make the judgement
which Bismarck alone can make, namely the judgement in which he himself is a
constituent” (Russell, KAD, p. 208).
It
thus seems to Russell that common proper names are not genuine singular terms, they do not express object-dependent thoughts. For
Russell, names are truncated or telescoped descriptions; e.g.,
(i)
Not
everyone will have the same description, but they will have descriptions of the
one proposition that includes
Russell’s account leaves us
with very few singular terms. When we use definite descriptions and names we
are having general thoughts, not about objects.
The only genuinely referring expressions (“logically proper names”), by
Russell’s principles, are ‘this’, ‘that’ and ‘I’.
(2) Rigidity
Independent of any worries
about Russell’s epistemological assumptions, common proper names appear to be behave quite differently from definite descriptions. Here
are some problems for Russell’s view, based on Kripke’s
notion that names are rigid:
(i)
If names were definite descriptions, then attributing the description to the
name should result in an analytical truth, but this is just false. ‘
(ii) If names were definite
descriptions, then we could make no sense of counterfactual statements
employing the name, but we clearly can. For example, one can say, ‘If Bismarck
weren’t
Similarly, (a) is true, but
(b) is false:
(a) Necessarily,
(b) Necessarily,
(iii) We can refer to an
object with a name even if we know nothing about the object. If I pick up the
name ‘Bill’ in a conversation, then it seems that I can use ‘Bill’ in the
conversation to refer to Bill. In what sense would I fail to refer to him?
Classic Criticisms of
Russell: Strawson and Donnellan
As earlier remarked,
Russell’s theory of definite descriptions is widely accepted; his account of
names as definite descriptions is equally widely rejected. There have been,
however, a number of criticisms of the theory of definite descriptions. here we’ll just look at two standard complaints.
(1) Strawson
Strawson’s objection is essentially
twofold. Firstly, there is the claim that truth attaches to statements, not
sentences. Call this the truth bearer objection. This is of little consequence,
for Russell was clearly concerned with propositions, not sentences. Strawson’s second objection is more interesting.
Strawson claims that ‘The F is G’ presupposes the existence of something
which is F; Russell’s analysis has it that ‘The F is G’ entails the existence of something which is F (Why? Because the
proposition expressed is conjunctive, and a conjunction is true just if each of
its conjuncts are true, and the first conjunct in
Russell’s analysis is that there is a
king of
A presupposes B iff if B is false, then A
is neither true nor false.
A entails B iff if B is false, then A is
false. (It is not possible for B to be false and
A to be true)
The difference is that
entailment contraposes:
(i)
Contraposition: P → Q iff ¬Q → ¬P.
Presupposition does not does
not admit contraposition. Thus, for Russell, if there is no king of
(ii) It is easy to find
instances where Strawson’s intuition is simply wrong:
(a)
The king of
(b)
Man U signed the King of France this morning (ditto)
(c)
The King of France does not exist (clearly true)
In general, presupposition
can always be cancelled; that is, if A putatively presupposes B, then we
can always jointly assert the falsity of A and B. This shows that
presupposition simply does not hold, for if it did, the falsity of B would be
enough to show that A lacked a truth value (true or false). Here’s how: A is false because B is false, e.g.,
(iii) The king of
Many have said that knowing p presupposes p. Again, this seems to be just false:
(iv) Mary doesn’t know she is
pregnant, because she isn’t pregnant, it was a phantom pregnancy.
(2) Donnellan
Donnellan argues, in essence, that
‘the’ is ambiguous between attributive and referential interpretations.
Attributive: object independent, as on Russell’s analysis: the definite
description is true of whoever satisfies the description.
Referential: object dependent, e.g., demostratives.
On a referential
interpretation, ‘The F’ functions as a singular term, with ‘F’ simply being
used to ‘point’ to a particular object that may or may not in fact be F. Donnellan’s claim, then, is that, at best, Russell’s
account is partial: it completely ignores referential uses.
An example
(i)
The murderer of Smith is insane
Donnellan suggests that (i),
as said by a detective looking at Smith’s corpse torn to pieces,
is an attributive use - it is true just if whoever murdered Smith is insane. On
the other hand, Donnellan suggests that (i), as said by a trial spectator as the accused of Smith’s
murder is frothing at the mouth, is true just if the accused is insane,
regardless of whether he murdered Smith or not.
(3) One way of answering Donnellan’s
objection is to say that Russell got the semantics correct, with the difference
between attributive and referential interpretations being an issue of
pragmatics (Grice, Kripke). Referential uses are
where we communicate an object dependent proposition by using a
sentence that expresses an object independent
proposition.
The method adopted here
follows the maxim that if an independently motivated pragmatics accounts for a
putative semantic feature, then it is, ceteris
paribus, better to explain the feature pragmatically than to give-up
semantic uniformity.
Grice makes a distinction
between:
Sentence meaning: The truth conditions of a sentence (semantic reference (Kripke)).
Speaker meaning: The proposition the utter intends the audience to entertain (speaker
reference (Kripke)).
Here is an example: a
referee Smith writes a reference to Jones consisting of nothing other than ‘The
candidate has excellent handwriting’. Smith’s sentence simply means that the
candidate has excellent handwriting, what else on Earth can it mean? But Smith
knows that Jones will understand something more, namely, the candidate is no
good. Smith has thus, by exploiting the context, communicated a proposition to
Jones that does not have the truth conditions of the sentence he wrote on the
reference. Just so with Donnellan’s example:
‘The murderer of Smith is insane’ expresses an
attributive proposition true of whoever is the unique satisfier of ‘x murdered Smith’. But it can be used referentially to have a speaker meaning (reference): its
speaker meaning is an object-dependent proposition about, say
Brown, frothing at the mouth in the dock. The object-dependent proposition is
communicated because the audience can work out such a proposition because
he/she realises that the spectator thinks that the man in the dock is the
murderer of Smith, even though he did not say that he was.
(4) Simpler than
ambiguity
There is another good reason
to adopt this method: If we admit the ambiguity of ‘the F’ on the basis of
referential use, then we must say that all quantifiers are ambiguous:
(i)
Most people like football.
I could say this in a room
of three people knowing precisely that only A and B like football and knowing
that A, B and C know
who like football. My audience would
thus get an object-dependent
proposition about A and B. But I have not said anything ambiguous. They can
infer the object dependent proposition from my general proposition and their
knowledge about the people in the room and what I know. We do not have to think
of ‘most’ as ambiguous.
Consider:
(ii) Some boy spilt the milk
I could say this to A in a
situation where everyone present knows that I am accusing A of spilling the
milk; I am trying to get A to admit his crime. Again, an object dependent
proposition is inferred by my audience, but the sentence doesn’t express one.
‘Some’ is not ambiguous.