Definite Descriptions in One Easy Lesson
A definite description is ‘the’ + a noun phrase: ‘the present king of France,’ ‘the first dog born at sea,’ ‘the bed’, ‘the smallest positive number that is both greater than any multiple of 3 and divisible by 7’
Question: How do definite descriptions function semantically? That is, how do we figure out whether a sentence of the form ‘the G is F’ is true?
Observation: In general, we’ll have to know both (i) what ‘G’ and ‘F’ mean and (ii) how things are in the world.
Frege’s claim: Definite descriptions are referring expressions. In order to figure out whether ‘a is F’ is true, where ‘a’ is a referring expression, we need to do two things: (i) identify the particular object, o, to which ‘a’ refers, (ii) determine whether o is F.
Why this is plausible: Grammatically, definite descriptions behave like proper names (‘Richard Nixon’, ‘China’, etc.): they fit into sentences in exactly the same places proper names do. And proper names are usually thought to be referring expressions. Moreover, it is easy to identify an object that is particularly relevant for assessing the truth of sentences containing a definite description: namely, the object that fits the description.
Russell’s claim : Definite descriptions are not referring expressions, but quantifier phrases. In order to figure out whether ‘aG is F’ is true, where ‘aG’ is a quantifier phrase, we need only to determine how many things are G, how many are F, and how many are both G and F. We need not identify any particular object. This is what distinguishes quantifier phrases from referring expressions, semantically.
Some notation: Let #(F) denote the number of Fs, #(F&G) the number of things that are both F and G, etc.
Examples of quantifier phrases: ‘No successful farmer is a good software engineer.’ ‘No’ is a quantifier; ‘no successful farmer’ is a quantifier phrase. A sentence of the form ‘no G is F’ is true iff #(G&F) = 0. So ‘No successful farmer is a good software engineer’ is true iff the number of things that are both successful farmers and good software engineers is 0. Some other examples of quantifiers: ‘every’, ‘several’, ‘most’, ‘no more than one’, ‘at least one’.
Exercise: using the # notation, give the truth conditions for sentences of the form ‘every G is F’, ‘several Gs are Fs’, ‘most Gs are Fs’, ‘no more than one G is F’.
Observation: Quantifier phrases are not referring expressions, and referring expressions are not quantifier phrases. Are there noun phrases that are neither quantifier phrases nor referring expressions? That’s an open question, but many philosophers and linguists think not.
Before seeing how one might argue for Russell’s claim, let’s try something easier. We’ll argue that indefinite descriptions (‘a’ + a noun phrase, e.g. ‘a man’, ‘a cat with nine tails’) are quantifier phrases, not referring expressions.
Problems with taking indefinite descriptions to be referring expressions:
(I1) What does an indefinite description like ‘a man’ refer to, when there is more than one man?
some particular man, say Joe?
no particular man, but a man in general? (early Russell [1903, sec. 62]: “what is denoted is essentially not each separate man, but a kind of combination of all men. … it is with this very paradoxical object that propositions are concerned in which the corresponding concept is used as denoting.”
(I2) What does an indefinite description like ‘a sixteen-legged cat’ refer to, when there is no sixteen-legged cat?
not any particular sixteen-legged cat (since there aren’t any), but a sixteen-legged cat in general.
Solution to these problems: ‘A man’ is not a referring expression. It functions semantically as a quantifier phrase, = ‘at least one man’. The same goes for ‘a sixteen-legged cat’ and all other indefinite descriptions.
Now let’s consider a parallel argument for Russell’s claim.
with taking definite descriptions to be referring expressions:
(D1) What does a definite description refer to when the description is satisfied by more than one thing (‘the vice-president of Motorola’)?
one of the things that fits the description—but which?
(D2) What does a definite description refer to when the description isn’t satisfied by anything (‘the present king of France’)?
something that does not exist (though it has some kind of ‘being’): cf. Meinong’s golden mountain.
Solution to these problems: ‘The vice-president of Motorola’ is not a referring expression. It functions semantically as a quantifier phrase (see below for details). The same goes for ‘the present king of France’ and all other definite descriptions. Terminological note: although Russell denies that definite descriptions refer, he says that they denote the unique object that satisfies the description, if there is one.
‘The’ as a quantifier: ‘The G is F’ is true iff #(G) ³ 1 [there is at least one G] and #(G) £ 1 [there is at most one G] and #(G&F) = #(G) [every G is F]. Some other, equivalent, specifications of the truth-conditions:
‘The G is F’ is true iff #(G) = 1 and #(G&F) = #(G)
‘The G is F’ is true iff #(G) = 1 and #(G&F) = 1
Exercise: Convince yourself that these are all equivalent.
Another way to think of ‘the G’: Our approach has been to take ‘the G’ seriously as a meaningful unit and say how it operates semantically. Russell’s actual approach was a bit different. He argued that ‘the G’ is an incomplete symbol and showed us how to eliminate it in any context in which it occurs, i.e., how to rewrite sentences containing it as sentences containing only the quantifiers ‘at least one’ ($x) and ‘every’ ("x). So, for example, ‘The king of France is bald’ comes out: ‘There is at least one king of France and there is at most one king of France and every king of France is bald’. [The quantifier ‘at most one’ can be defined using ‘every’, and ‘is identical to’.] This approach yields the same truth conditions as our account, but it amounts to saying that the grammatical form of ‘The G is F’ is misleading as to its logical form: ‘the G’ does not function as a semantic unit. If we allow ourselves a more generous notion of a quantifier, the semantics meshes better with the grammar.
For logic buffs only: The iota-operator is sometimes used to symbolize definite descriptions. (ix)(fx) means ‘the object x such that fx.’ On Russell’s theory, y(ix)(fx) is defined as ($x)(fx & ("y)(fy É y=x) & yx).
Exercise: Verify that the following are equivalent: ($x)(fx & ("y)(fy É y=x) & yx), ($x)(("y)(fy º y=x) & yx),
($x)fx & ("x)("y)((fx & fy) É y=x) & ("x)(fx É yx).
Strawson’s view : On Russell’s view, ‘The present king of France is bald’ is false (because it implies that there is a present king of France). On Strawson’s view, the question of its truth or falsity doesn’t arise: the speaker presupposes that there is a present king of France, and when this presupposition is not satisfied, nothing definite has been said. When the presupposition is satisfied, Strawson holds, the definite description acts as a referring expression. But consider: ‘The present king of France does not exist,’ ‘This morning my father had breakfast with the present king of France’ (Neale ), ‘The present king of France is responsible for California’s power troubles.’ Strawson will have to say these sentences can’t be used to make claims, even false ones.
References: Frege, Gottlob . “On Sense and Reference,” in Geach and Black, eds., Translations from the Philosophical Writings of Gottlob Frege (Oxford: Blackwell, 1952), 56-78.
Russell, Bertrand . Principles of Mathematics. London: George Allen and Unwin.
Russell, Bertrand . “On Denoting.” Mind 14, 479-93. Reprinted in Ostertag .
Strawson, P. F. . “On Referring.” Mind 59, 320-44. Reprinted in Ostertag .
A good anthology: Ostertag, Gary, ed. . Definite Descriptions: A Reader. Cambridge: MIT Press.
A nice modern defense of Russell: Neale, Stephen . Descriptions. Cambridge: MIT Press.