In
formal semantics and
philosophy of language
In analytic philosophy, philosophy of language investigates the nature of language and the relations between language, language users, and the world. Investigations may include inquiry into the nature of meaning, intentionality, reference, ...
, a definite description is a
denoting phrase
In syntax and grammar, a phrase is a group of words or singular word acting as a grammatical unit. For instance, the English expression "the very happy squirrel" is a noun phrase which contains the adjective phrase "very happy". Phrases can consi ...
in the form of "the X" where X is a noun-phrase or a singular common
noun
A noun () is a word that generally functions as the name of a specific object or set of objects, such as living creatures, places, actions, qualities, states of existence, or ideas.Example nouns for:
* Living creatures (including people, alive, ...
. 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 some Country" are ''improper'' because X applies to nothing. Improper descriptions raise some difficult questions about the
law of excluded middle
In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradi ...
,
denotation,
modality
Modality may refer to:
Humanities
* Modality (theology), the organization and structure of the church, as distinct from sodality or parachurch organizations
* Modality (music), in music, the subject concerning certain diatonic scales
* Modaliti ...
, and
mental content
The mind is the set of faculties responsible for all mental phenomena. Often the term is also identified with the phenomena themselves. These faculties include thought, imagination, memory, will, and sensation. They are responsible for vario ...
.
Russell's analysis
As
France
France (), officially the French Republic ( ), is a country primarily located in Western Europe. It also comprises of overseas regions and territories in the Americas and the Atlantic, Pacific and Indian Oceans. Its metropolitan area ...
is
currently a republic, it has no king.
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
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 is not 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 does not 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
In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradi ...
.
Is it meaningless, then? One might suppose so (and some philosophers have) 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" cannot be meaningless.
Russell proposed to resolve this puzzle via his
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 (commonly abbreviated as RTD). In short, Russell argued that the ...
. A definite description like "the present King of France", he suggested, is not a
referring
Reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another object. The first object in this relation is said to ''refer to'' the second object. It is called a '' name'' ...
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 currently King of France:
(using 'Kx' for 'x is currently King of France')
# for any x and y, if x is currently King of France and y is currently King of France, then x=y (i.e. there is at most one thing which is currently King of France):
# for every x that is currently King of France, x is bald:
(using 'B' for 'bald')
More briefly put, the claim is that "The present King of France is bald" says that some x is such that x is currently King of France, and that any y is currently King of France only if y = x, and that x is bald:
This is ''false'', since it is ''not'' the case that some is currently 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 currently King of France and bald:
On this disambiguation, the sentence is ''true'' (since there is indeed no x that is currently 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 currently 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 currently 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
In logic, logical form of a statement is a precisely-specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambiguou ...
: if the
negation is construed as taking wide scope (as in the first of the above), it is true, whereas if the negation is construed as taking narrow scope (as in the second of the above), 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
In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradi ...
: "the present King of France is bald" (i.e.
) is false, because there is no present King of France.
The negation of this statement is the one in which 'not' takes wide scope:
. This statement is ''true'' because there does not exist anything which is currently King of France.
Generalized quantifier analysis
Stephen Neale,
among others, has defended Russell's theory, and incorporated it into the theory of
generalized quantifiers In formal semantics, a generalized quantifier (GQ) is an expression that denotes a set of sets. This is the standard semantics assigned to quantified noun phrases. For example, the generalized quantifier ''every boy'' denotes the set of sets of ...
. On this view, 'the' is a quantificational determiner like 'some', 'every', 'most' etc. The determiner 'the' has the following denotation (using
lambda
Lambda (}, ''lám(b)da'') is the 11th letter of the Greek alphabet, representing the voiced alveolar lateral approximant . In the system of Greek numerals, lambda has a value of 30. Lambda is derived from the Phoenician Lamed . Lambda gave ri ...
notation):
(That is, the definite article 'the' denotes a function which takes a pair of
properties
Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property.
Property may also refer to:
Mathematics
* Property (mathematics)
Philosophy and science
* Property (philosophy), in philosophy an ...
and to truth
if, and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicon ...
, there exists something that has the property , only one thing has the property , and that thing also has the property .) Given the denotation of the
predicates
Predicate or predication may refer to:
* Predicate (grammar), in linguistics
* Predication (philosophy)
* several closely related uses in mathematics and formal logic:
**Predicate (mathematical logic)
**Propositional function
**Finitary relation, ...
'present King of France' (again for short) and 'bald' ( for short)
we then get the Russellian truth conditions via two steps of
function application: 'The present King of France is bald' is true if, and only if,
. 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
In logic and linguistics, an expression is syncategorematic if it lacks a denotation but can nonetheless affect the denotation of a larger expression which contains it. Syncategorematic expressions are contrasted with categorematic expressions, whi ...
, or "incomplete symbols"); but the view retains the essentials of the Russellian analysis, yielding exactly the truth conditions Russell argued for.
Fregean analysis
The Fregean analysis of definite descriptions, implicit in the work of
Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic ph ...
and later defended by
Strawson among others, represents the primary alternative to the Russellian theory. On the Fregean analysis, definite descriptions are construed as
referring expressions rather than
quantificational expressions. Existence and uniqueness are understood as a
presupposition of a sentence containing a definite description, rather than part of the content asserted by such a sentence. The sentence 'The present King of France is bald', for example, is not used to claim that there exists a unique present King of France who is bald; instead, that there is a unique present King of France is part of what this sentence ''presupposes'', and what it ''says'' is that this individual is bald. If the presupposition fails, the definite description
fails to refer, and the sentence as a whole fails to express a
proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
.
The Fregean view is thus committed to the kind of
truth value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false'').
Computing
In some pro ...
gaps (and failures of the
law of excluded middle
In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradi ...
) that the Russellian analysis is designed to avoid. Since there is currently no King of France, the sentence 'The present King of France is bald' fails to express a proposition, and therefore fails to have a truth value, as does its
negation, 'The present King of France is not bald'. The Fregean will account for the fact that these sentences are nevertheless ''meaningful'' by relying on speakers' knowledge of the conditions under which either of these sentences ''could'' be used to express a true proposition. The Fregean can also hold on to a restricted version of the law of excluded middle: for any sentence whose presuppositions are met (and thus expresses a proposition), either that sentence or its negation is true.
On the Fregean view, the definite article 'the' has the following denotation (using
lambda
Lambda (}, ''lám(b)da'') is the 11th letter of the Greek alphabet, representing the voiced alveolar lateral approximant . In the system of Greek numerals, lambda has a value of 30. Lambda is derived from the Phoenician Lamed . Lambda gave ri ...
notation):
(That is, 'the' denotes a function which takes a property and yields the unique object that has property , if there is such a , and is undefined otherwise.) The presuppositional character of the existence and uniqueness conditions is here reflected in the fact that the definite article denotes a
partial function
In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is de ...
on the set of properties: it is only defined for those properties which are true of exactly one object. It is thus undefined on the denotation of the predicate 'currently King of France', since the property of currently being King of France is true of no object; it is similarly undefined on the denotation of the predicate 'Senator of the US', since the property of being a US Senator is true of more than one object.
Mathematical logic
Following the example of ''
Principia Mathematica
The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
'', it is customary to use a definite description operator symbolized using the "turned" (rotated) Greek lower case iota character "℩". The notation ℩
means "the unique
such that
", and
is equivalent to "There is exactly one
and it has the property
":
See also
*
Lambert's law (logic)
*
Philosophy of language
In analytic philosophy, philosophy of language investigates the nature of language and the relations between language, language users, and the world. Investigations may include inquiry into the nature of meaning, intentionality, reference, ...
*
John Searle
*
Vacuous truth
In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. For example, the statement "she d ...
References
Bibliography
*
Donnellan, Keith, "Reference and Definite Descriptions," in ''
Philosophical Review
''The Philosophical Review'' is a quarterly journal of philosophy edited by the faculty of the Sage School of Philosophy at Cornell University and published by Duke University Press (since September 2006).
Overview
The journal publishes original ...
'' 75 (1966): 281–304.
* Neale, Stephen, ''Descriptions'', MIT Press, 1990.
* Ostertag, Gary (ed.). (1998) ''Definite Descriptions: A Reader'' Bradford, MIT Press. (Includes Donnellan (1966), Chapter 3 of Neale (1990), Russell (1905), and Strawson (1950).)
* Reimer, Marga and Bezuidenhout, Anne (eds.) (2004), ''Descriptions and Beyond'', Clarendon Press, Oxford
* Russell, Bertrand, "
On Denoting," in ''
Mind'' 14 (1905): 479–493
Online text
* Strawson, P. F., "On Referring," in ''Mind'' 59 (1950): 320–344.
External links
*
{{Formal semantics
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Bertrand Russell
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