Definite description
Encyclopedia
A definite description is a denoting
phrase
in the form of "the X" where X is a noun-phrase or a singular common noun
. The definite description is proper if X applies to a unique individual or object. For example: "the first person in space
" and "the 42nd President of the United States of America", are proper. The definite descriptions "the person in space" and "the Senator from Ohio" are improper because the noun phrase X applies to more than one thing, and the definite descriptions "the first man on Mars" and "the Senator from Washington D.C." are improper because X applies to nothing. Improper descriptions raise some difficult questions about the law of excluded middle
, denotation
, modality
, and mental content.
is presently a republic
, and has no king. Bertrand Russell
pointed out that this raises a puzzle about the truth value of the sentence "The present King of France is bald."
The sentence does not seem to be true: if we consider all the bald things, the present King of France isn't among them, since there is no present King of France. But if it is false, then one would expect that the negation
of this statement, that is, "It is not the case that the present King of France is bald," or its logical equivalent
, "The present King of France is not bald," is true. But this sentence doesn't seem to be true either: the present King of France is no more among the things that fail to be bald than among the things that are bald. We therefore seem to have a violation of the Law of Excluded Middle
.
Is it meaningless, then? One might suppose so (and some philosophers have; see below) since "the present King of France" certainly does fail to refer. But on the other hand, the sentence "The present King of France is bald" (as well as its negation) seem perfectly intelligible, suggesting that "the Present King of France" can't be meaningless.
Russell proposed to resolve this puzzle via his theory of descriptions
. A definite description like "the present King of France", he suggested, isn't a referring
expression, as we might naively suppose, but rather an "incomplete symbol" that introduces quantificational structure into sentences in which it occurs. The sentence "the present King of France is bald", for example, is analyzed as a conjunction of the following three quantified statements:
More briefly put, the claim is that "The present King of France is bald" says that some x is such that x is presently King of France, and that any y is presently King of France only if y = x, and that x is bald:
This is false, since it is not the case that some x is presently King of France.
The negation of this sentence, i.e. "The present King of France is not bald", is ambiguous. It could mean one of two things, depending on where we place the negation 'not'. On one reading, it could mean that there is no one who is presently King of France and bald:
On this disambiguation, the sentence is true (since there is indeed no x that is presently King of France).
On a second, reading, the negation could be construed as attaching directly to 'bald', so that the sentence means that there is presently a King of France, but that this King fails to be bald:
On this disambiguation, the sentence is false (since there is no x that is presently King of France).
Thus, whether "the present King of France is not bald" is true or false depends on how it is interpreted at the level of logical form
: if the negation
is construed as taking wide scope
(as in ~∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]), it is true, whereas if the negation is construed as taking narrow scope
(with the existential quantifier taking wide scope
, as in ∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & ~B(x)]), it is false. In neither case does it lack a truth value.
So we do not have a failure of the Law of Excluded Middle
: "the present King of France is bald" (i.e. ∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]) is false, because there is no present King of France. The negation of this statement is the one in which 'not' takes wide scope: ~∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]. This statement is true because there does not exist anything which is presently King of France.
, among others, has defended Russell's theory, and incorporated it into the theory of generalized quantifiers
. On this view, 'the' is a quantificational determiner like 'some', 'every', 'most' etc. The definite description 'the' has the following denotation (using lambda
notation):
Given the denotation of the predicates 'present King of France' (again PKoF for short) and 'bald (B for short)'
we then get the Russellian truth conditions: 'The present King of France is bald' is true just in case ∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]. On this view, definite descriptions like 'the present King of France' do have a denotation (specifically, definite descriptions denote a function from properties to truth values -- they are in that sense not syncategorematic, or "incomplete symbols"); but the view retains the essentials of the Russellian analysis, yielding exactly the truth conditions Russell argued for.
context, it can be symbolized by
, so that
means "the
such that 
", and
is equivalent to "There is exactly one
and it has the property 
":
Denotation
This word has distinct meanings in other fields: see denotation . For the opposite of Denotation see Connotation.*In logic, linguistics and semiotics, the denotation of a word or phrase is a part of its meaning; however, the part referred to varies by context:** In grammar and literary theory, the...
phrase
Phrase
In everyday speech, a phrase may refer to any group of words. In linguistics, a phrase is a group of words which form a constituent and so function as a single unit in the syntax of a sentence. A phrase is lower on the grammatical hierarchy than a clause....
in the form of "the X" where X is a noun-phrase or a singular common noun
Noun
In linguistics, a noun is a member of a large, open lexical category whose members can occur as the main word in the subject of a clause, the object of a verb, or the object of a preposition .Lexical categories are defined in terms of how their members combine with other kinds of...
. The definite description is proper if X applies to a unique individual or object. For example: "the first person in space
Yuri Gagarin
Yuri Alekseyevich Gagarin was a Soviet pilot and cosmonaut. He was the first human to journey into outer space, when his Vostok spacecraft completed an orbit of the Earth on April 12, 1961....
" and "the 42nd President of the United States of America", are proper. The definite descriptions "the person in space" and "the Senator from Ohio" are improper because the noun phrase X applies to more than one thing, and the definite descriptions "the first man on Mars" and "the Senator from Washington D.C." are improper because X applies to nothing. Improper descriptions raise some difficult questions about the law of excluded middle
Law of excluded middle
In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....
, denotation
Denotation
This word has distinct meanings in other fields: see denotation . For the opposite of Denotation see Connotation.*In logic, linguistics and semiotics, the denotation of a word or phrase is a part of its meaning; however, the part referred to varies by context:** In grammar and literary theory, the...
, modality
Linguistic modality
In linguistics, modality is what allows speakers to evaluate a proposition relative to a set of other propositions.In standard formal approaches to modality, an utterance expressing modality can always roughly be paraphrased to fit the following template:...
, and mental content.
Russell's analysis
FranceFrance
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
is presently a republic
Republic
A republic is a form of government in which the people, or some significant portion of them, have supreme control over the government and where offices of state are elected or chosen by elected people. In modern times, a common simplified definition of a republic is a government where the head of...
, and has no king. Bertrand Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
pointed out that this raises a puzzle about the truth value of the sentence "The present King of France is bald."
The sentence does not seem to be true: if we consider all the bald things, the present King of France isn't among them, since there is no present King of France. But if it is false, then one would expect that the negation
Negation
In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...
of this statement, that is, "It is not the case that the present King of France is bald," or its logical equivalent
Logical equivalence
In logic, statements p and q are logically equivalent if they have the same logical content.Syntactically, p and q are equivalent if each can be proved from the other...
, "The present King of France is not bald," is true. But this sentence doesn't seem to be true either: the present King of France is no more among the things that fail to be bald than among the things that are bald. We therefore seem to have a violation of the Law of Excluded Middle
Law of excluded middle
In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....
.
Is it meaningless, then? One might suppose so (and some philosophers have; see below) since "the present King of France" certainly does fail to refer. But on the other hand, the sentence "The present King of France is bald" (as well as its negation) seem perfectly intelligible, suggesting that "the Present King of France" can't be meaningless.
Russell proposed to resolve this puzzle via his theory of descriptions
Theory of descriptions
The theory of descriptions is the philosopher Bertrand Russell's most significant contribution to the philosophy of language. It is also known as Russell's Theory of Descriptions...
. A definite description like "the present King of France", he suggested, isn't a referring
Reference
Reference is derived from Middle English referren, from Middle French rèférer, from Latin referre, "to carry back", formed from the prefix re- and ferre, "to bear"...
expression, as we might naively suppose, but rather an "incomplete symbol" that introduces quantificational structure into sentences in which it occurs. The sentence "the present King of France is bald", for example, is analyzed as a conjunction of the following three quantified statements:
- there is an x such that x is presently King of France: ∃x[PKoF(x)] (using 'PKoF' for 'presently King of France')
- for any x and y, if x is presently King of France and y is presently King of France, then x=y (i.e. there is at most one thing which is presently King of France): ∀x∀yKoF(x) & PKoF(y)] → y=x]
- for every x that is presently King of France, x is bald: ∀x[PKoF(x) → B(x)]
More briefly put, the claim is that "The present King of France is bald" says that some x is such that x is presently King of France, and that any y is presently King of France only if y = x, and that x is bald:
-
- ∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]
This is false, since it is not the case that some x is presently King of France.
The negation of this sentence, i.e. "The present King of France is not bald", is ambiguous. It could mean one of two things, depending on where we place the negation 'not'. On one reading, it could mean that there is no one who is presently King of France and bald:
-
- ~∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]
On this disambiguation, the sentence is true (since there is indeed no x that is presently King of France).
On a second, reading, the negation could be construed as attaching directly to 'bald', so that the sentence means that there is presently a King of France, but that this King fails to be bald:
-
- ∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & ~B(x)]
On this disambiguation, the sentence is false (since there is no x that is presently King of France).
Thus, whether "the present King of France is not bald" is true or false depends on how it is interpreted at the level of logical form
Logical form
In logic, the logical form of a sentence or set of sentences is the form obtained by abstracting from the subject matter of its content terms or by regarding the content terms as mere placeholders or blanks on a form...
: if the negation
Negation
In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...
is construed as taking wide scope
Scope
The word scope may refer to many different devices or viewing instruments, constructed for many different purposes. It may refer to a telescopic sight, an optical device commonly used on firearms. Other uses of scope or Scopes may refer to:...
(as in ~∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]), it is true, whereas if the negation is construed as taking narrow scope
Scope
The word scope may refer to many different devices or viewing instruments, constructed for many different purposes. It may refer to a telescopic sight, an optical device commonly used on firearms. Other uses of scope or Scopes may refer to:...
(with the existential quantifier taking wide scope
Scope
The word scope may refer to many different devices or viewing instruments, constructed for many different purposes. It may refer to a telescopic sight, an optical device commonly used on firearms. Other uses of scope or Scopes may refer to:...
, as in ∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & ~B(x)]), it is false. In neither case does it lack a truth value.
So we do not have a failure of the Law of Excluded Middle
Law of excluded middle
In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....
: "the present King of France is bald" (i.e. ∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]) is false, because there is no present King of France. The negation of this statement is the one in which 'not' takes wide scope: ~∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]. This statement is true because there does not exist anything which is presently King of France.
Generalized Quantifier Analysis
Stephen NealeStephen Neale
Stephen Roy Albert Neale is Distinguished Professor of Philosophy and Linguistics and holder of the John H. Kornblith Family Chair in the Philosophy of Science and Values at the Graduate Center, City University of New York...
, among others, has defended Russell's theory, and incorporated it into the theory of generalized quantifiers
Generalized quantifier
In linguistic semantics, a generalized quantifier is an expression that denotes a property of a property, also called a higher-order property. This is the standard semantics assigned to quantified noun phrases, also called determiner phrases, in short: DP...
. On this view, 'the' is a quantificational determiner like 'some', 'every', 'most' etc. The definite description 'the' has the following denotation (using lambda
Lambda calculus
In mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. The portion of lambda calculus relevant to computation is now called the untyped lambda calculus...
notation):
-
- λf.λg.[∃x(f(x)=1 & ∀y(f(y)=1 → y=x)) & g(x)=1].
Given the denotation of the predicates 'present King of France' (again PKoF for short) and 'bald (B for short)'
-
- λx.[PKoF(x)]
- λx.[B(x)]
we then get the Russellian truth conditions: 'The present King of France is bald' is true just in case ∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]. On this view, definite descriptions like 'the present King of France' do have a denotation (specifically, definite descriptions denote a function from properties to truth values -- they are in that sense not syncategorematic, or "incomplete symbols"); but the view retains the essentials of the Russellian analysis, yielding exactly the truth conditions Russell argued for.
Symbolic form
When using the definite descriptor in a formal logicFormal logic
Classical or traditional system of determining the validity or invalidity of a conclusion deduced from two or more statements...
context, it can be symbolized by
, so thatmeans "the
such that ", andis equivalent to "There is exactly one
and it has the property ":See also
- Philosophy of languagePhilosophy of languagePhilosophy of language is the reasoned inquiry into the nature, origins, and usage of language. As a topic, the philosophy of language for analytic philosophers is concerned with four central problems: the nature of meaning, language use, language cognition, and the relationship between language...
- Analytic philosophyAnalytic philosophyAnalytic philosophy is a generic term for a style of philosophy that came to dominate English-speaking countries in the 20th century...
- Bertrand RussellBertrand RussellBertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
- John SearleJohn SearleJohn Rogers Searle is an American philosopher and currently the Slusser Professor of Philosophy at the University of California, Berkeley.-Biography:...