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: Bill, France, 4, etc.

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.








(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 France is bald             <?, baldness>


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 France*, Baldness>.


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 France’ must contribute some object to the proposition. The object cannot be the king of France, for there is no such object. Here we have an exception, DCs are a way of making sense of it. Thus, although DCs solve the Meinong problem of empty names, they only do so by creating an exception to the general RCT. Russell’s mature theory of descriptions was essentially proposed as a way out of this dilemma.


Why should we think that the ‘The king of France’ must contribute some object to the proposition? Russell’s diagnosis is: there is a dogma that propositions must reflect grammatical form, i.e., for each subject and predicate there is a corresponding propositional constituent. Russell’s greatness lies in his seeing that this is mistaken: surface form need not be a guide to logical form (the form of the proposition, that which determines truth conditions). If we can see our way to such a view, then there need not be any exceptions to the general RCT.


(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 one PM and at most one PM and anything which is PM is B)


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)( FxGx),


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 France is bald. But since there is no such person it might seem that he is neither bald nor not bald (excluding wigs). This comes down to an issue of scope of negation.


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 France’ has primary occurrence is false, because there is no King of France. Thus:


(iii) ($x)[Fx & ("y)(Fyy = 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)(Fyy = 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.




(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


(where½A½’ 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, Bismarck, Julius Caesar, etc?


“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) Bismarck is an astute diplomat = The first chancellor of Germany is an astute diplomat.


Not everyone will have the same description, but they will have descriptions of the one proposition that includes Bismarck (the one Bismarck can use) and this enables them to communicate and be talking of the same thing.


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. ‘Bismarck was the first Chancellor of Germany’ is clearly an empirical truth.


(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 Germany’s first Chancellor, then WWI would have started sooner’. But if ‘Bismarckjust means ‘the first Chancellor of Germany’, then this statement would be meaningless.


Similarly, (a) is true, but (b) is false:


(a) Necessarily, Bismarck was Bismarck

(b) Necessarily, Bismarck was the first Chancellor of Germany


(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 France.) You should understand the difference:


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 France, then ‘The king of France is bald’ is false; for Strawson, ‘The king of France is bald’ would be neither true nor false in such circumstances. Strawson takes this reading to be intuitively correct. Is it?


(ii) It is easy to find instances where Strawson’s intuition is simply wrong:

(a) The king of France exists (clearly false)

(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 France isn’t bald, because there is no king of France


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.




(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.