On Denoting
Bertrand Russell
We’ve got some serious problems
A Puzzle About Identity
• Indiscernibility of Identicals: If a = b then whatever is true of one
is true of the other.
If a is identical with b, whatever is true of the one is true of the other,
and either may be substituted for the other in any proposition without
altering the truth or falsehood of that proposition. Now George IV
wished to know whether Scott was the author of Waverly, and in fact
Scott was the author of Waverly. Hence we may substitute Scott for
the author of Waverly and thereby prove that George IV wished to
know whether Scott was Scott.
• Problem: contexts in which substitutivity salve veritate fails, e.g.
– George IV wondered whether ___ wrote Waverly.
– Necessarily ___ is odd.
A Puzzle About Excluded Middle
• The Law of Excluded Middle: For any proposition, P, either P or
not-P
By the Law of Excluded Middle, either ‘A is B’ or ‘A is not B’ must
be true. Hence either ‘the present King of France is bald’ or ‘the
present King of France is not bald’ must be true. Yet if we
enumerated the things that are bald and then the things that are not
bald, we should not find the present King of France in either list.
• Problem: When the subject term of a sentence fails to refer,
excluded middle fails, e.g.
– The present King of France is bald.
– George W. Bush’s son is a plumber
A Puzzle About Negative Existentials
• A proposition cannot be about a non-entity: to be either true or
false the subject term of a singular statement must refer to
something.
Consider the proposition ‘A differs from B.’ If this is true, there is a
difference between A and B, which fact may be expressed in the
form‘the difference between A and B subsists.’ But if it is false that A
differs from B, then there is no difference between A and B, which
fact may be may be expressed in the form, ‘the difference between
A and B does not subsist.’ But how can a non-entity be the subject
of a proposition?...[I]f A and B do not differ, to suppose either that
there is, or that there is not, such an object as ‘the difference
between A and B’ seems equally impossible.
• Problem: Negative existentials, sentences of the form “x doesn’t
exist” seem to commit us to the existence of x and then say of it that
it doesn’t exist—which is a contradiction.
– Santa Claus doesn’t exist
The Source of Our Problems
• We are mislead by language!
• “Surface grammar” obscures the true logical form of propositions
– Our failure to recognize the true logical form beneath the surface
is responsible for the puzzles
• Subject-predicate form of some sentences is misleading
• Russell proposes an account of the true logical form of propositions
that provides solutions to the puzzles
– He criticizes alternative approaches of Meinong and Frege
– And shows how his account deals with the puzzles
An Ideal Language
• Disagreement about what philosophy is supposed to be doing
– Analyzing ordinary language or
– Translating into an ideal language that reveals true “logical
form”
• Russell proposes translating ordinary language into an ideal
language that will avoid puzzles and paradoxes.
• The language that reveals “logical form” and so allows us to explain
validity and to provide solutions without getting into crazy
metaphysics is . . .
• PREDICATE LOGIC WITH IDENTITY!
Predicate Logic Vocabulary
• Connectives:  ,  , • ,  , and 
• Individual Constants: lower case letters of the alphabet (a, b, c,…, u,
v, w)
• Predicates: upper case letters (A, B, C,…, X, Y, Z)
• Variables: x, y and z
• Quantifiers:
– Existential ( [variable]), e.g. (x), (y)…
– Universal ([variable]), e.g. (x), (y)…
• Identity (a special predicate): =
Basic Predicate Logic Translation
• Singular propositions
– Russell was a philosopher.
Pr
– Russell was Moore’s friend.
Frm
– Russell authored Principia Mathematica with Whitehead.
Arwp
• General propositions
– Everything is material
(x)Mx
– There is a God.
(∃x)Gx
Identity
• Identity is an equivalence relation
– Reflexive: everything is identical to itself
x=x
– Symmetric: if one thing is identical to another the other is identical
to the first
If x = y then y = x
– Transitive: if one thing’s identical to a second and the second’s
identical to a third then the first is identical to the third
If x = y and y = z, then x = z
• Identity is an indiscernibility relation
– Indiscernibility of Identicals: if x = y then whatever is true of x is
true of y and vice versa.
Ordinary language is deceptive!
• According to Russell, what appear to be simple singular statements
are often really more complicated existentially quantified statements.
– I met a man
• Mm (wrong! “a man” is not the name of an individual!)
• (∃x)Mx (correct translation)
• Even ordinary language statements that include ordinary proper
names are deceptive: ordinary proper names are really “disguised
descriptions”
– Apollo lives on Mount Olympus
• (∃x)(x is sun god and x is Leto’s son and Artemis’ twin
brother and…and x lives on Mount Olympus
• Puzzles arise because we think statements that involve denoting
expressions are singular statements when they are really existential
and general—and involve quantifiers and complicated logical
Denoting
•
The subject of denoting is of very great importance, not only in
logic and mathematics, but also in the theory of knowledge.
•
No general characterization of denoting is given, only a list of
“denoting phrases.”
•
–
A man
–
Some man
–
Any man
–
Every man
–
All men
–
The present King of England
“A phrase is denoting solely in virtue of its form.”
Denoting Phrases and Denotation
• Russell considers three cases
1. Denoting phrases that do not denote anything, e.g. “the
present King of France” [in 1905]
2. Denoting phrases that denote one definite object, e.g. “the
present King of England” [in 1905]
3. Denoting phrases that denote ambiguously, e.g. “a man”
• We want an account that will accommodate all three kinds of cases
without paradox
• And explain how we can think and talk about many things with which
we are not “acquainted” but only know by description.
Paraphrasing Away Denoting Phrases
I use “C(x) to mean a proposition in which x is a constituent, where
x, the variable, is essentially and wholly undetermined.
• We paraphrase away everything, nothing, and something as follows:
• C(everything) means “C(x) is always true”: (x)Cx
– Everything is material: for all x, x is material
• C(nothing) means “ ‘C(x) is is false’ is always true”: (x)  Cx
– Nothing is free: for all x, it is not the case that x is free
• C(something) means “It is false that ‘C(x) is false’ is always true”:
 (x)  C(x) i.e. (x)Cx
– Something smells: there exists an x such that x smells.
More Translations
• I met a man
– “I met x and x is human” is not always false: (x)(Mx  Hx)
• All men are mortal
– “If x is human then x is mortal” is always true: (x)(Hx  Mx)
• No men are perfect
– “If x is human then x is perfect is false” is always true: (x)(Hx 
 Px)
• Some men are philosophers
– “x is human and x is a philosopher” is not always false: (x)(Hx
 Px)
Definite Descriptions
• The denoting phrase in the sentence “The father of Charles II was
executed” involves:
– Existence: x was father of Charles II, for some value of x.
– Uniqueness: if y was father of Charles II, then y is identical with
x, for any value of x who was father of Charles II and any value
of y.
• To convey the uniqueness condition we need to introduce an
additional special predicate, viz IDENTITY to produce the following
translation:
– There exists an x such that x is the father of Charles II
– For all y, if y is the father of Charles II then x = y
– x was executed
Denoting Phrases All Gone!
• We have now paraphrased away all denoting phrases
– A man
– Some man
– Any man
– Every man
– All men
– The present King of England
• We’ve gotten rid of them in favor of logical machinery--and this will
enable us to solve all three puzzles
The Identity Puzzle
• Indiscernibility of Identiticals (substitutivity): If a is identical with
b…either may be substituted for the other…without altering the truth
or falsity of the proposition
• Apparent Counterexample: George IV wished to know whether Scott
was the author of Waverly
– Scott = the author of Waverly
– By substitutivity principle, George IV wished to know whether
Scott was Scott
– But George IV did not wish to know whether Scott was Scott (he
knew that!)
• So, contrary to the principle we have a case where substituting a
different name for the same objects makes a true proposition false—
which is unacceptable!
Substitutivity Failures
• There are some contexts in which it looks like we cannot freely
substitute identicals for identicals
• These include intentional contexts which occur in sentences that
ascribe certain mental states, e.g.
– The Minoans didn’t believe that the Morning Star was identical
to the Evening Star.
– George IV wondered whether Scott was the author of Waverly.
• The Problem: everyone knows that everything is identical with itself
BUT
– The Minoans believed that the Morning Star was identical to the
Morning Star, but not that the Morning Star was identical to the
Evening Star.
– George IV didn’t wonder whether Scott was identical to Scott
but did wonder whether Scott was identical to the author of
Waverly.
Frege’s Puzzle
• How can true identity statements be informative?
• The Morning Star = The Evening Star
• What makes this true is the fact that “The Morning Star” and “The
Evening Star” refer to the same object, viz. the planet Venus so it
looks like the identity statement just says that Venus, like everything
else, is identical to itself!
• But everyone knows that everything is identical to itself so what do
we know that the Minoans didn’t know? And what did George IV
wonder about? Surely not whether Scott was identical to himself!
• Russell discusses Frege’s way of dealing with this and related
issues (pp 4 – 5) by distinguishing the meaning and denotation of
denoting expressions: we’ll deal with this when we get to Frege—
not now.
Scope Ambiguity
Guest: “I thought your yacht was larger than it is.”
Touchy Yacht Owner: “No, my yacht is not larger than it is.”
1 The size I thought your yacht was is greater than the size your yacht
is.
– There’s a certain size, x, and I thought that x was the size of
your yacht but x is greater than the size of your yacht.
2 I thought the size of your yacht was greater than the size of your
yacht.
– I thought that there’s a certain size, x which is the size of your
yacht but x is greater than the size of your yacht.
In 1 the Guest does not believe that the size of the Owner’s yacht is
greater than it is. Likewise, in the correct reading “George IV
Russell’s Solution
George IV wished to know whether Scott was the author of Waverly
1
George IV wished to know whether one and only one man wrote
Waverly and Scott was that man
–
2
Secondary occurrence of “the author of Waverly”
George IV wished to know whether (x)[Axw  (y)(Ayw  y=x) 
x=s]
One and only one man wrote Waverly and George IV wished to know
whether Scott was that man
–
Primary occurrence of “the author of Waverly”
(x)[Axw  (y)(Ayw  y=x)  George IV wished to know whether
x=s]
1, not 2, is the correct interpretation of the original sentence. What
George IV didn’t know but wanted to know, was whether the
proposition (x)[Axw  (y)(Ayw  y=x)  x=s] was true, and that is not
the same proposition as s=s which is the result of substituting “s” for
The Excluded Middle Puzzle
Excluded Middle: P or not P
• Consider the sentence P: “The
present king of France is bald.”
• P can’t be true since there is no
present king of France.
• Since it’s not true it must be false
• Therefore we conclude that the
present King of France is not bald,
i.e. not P
• But that’s also false since there is no
present king of France
• But this seems to violate Excluded
Middle since we deny both P and not
P
Russell’s Solution
“It’s false that the present King of France is bald” is ambiguous.
1
There is an entity which is now king of France and is not bald.
(∃x)(Kx • ∼Bx)
2
It is false that there is an entity which is now king of France and is
bald.
∼ (∃x)(Kx • Bx)
•
There’s a scope ambiguity concerning negation!
•
1is false: there is no x who is King of France and is either bald or
non-bald.
•
2 is true.
The Negative Existentials Puzzle
• Negative Existential: A claim to the effect that something doesn’t
exist, e.g.
– Santa Claus doesn’t exist
– “There are no unicorns.”
• Problem: the following argument
1 If an individual denies the existence of something, then he refers
to what he says does not exist.
2 Things which do not exist cannot be referred to or mentioned; no
statement can be about them.
3 Therefore, if an individual denies the existence of something,
then what he says does not exist exists.
The Paradox of Non-Being
What can be spoken of and
thought must be;
for it is possible for it to be,
but it is not possible for
“nothing” to be.
Parmenides posed the Paradox of Non-Being
What’s the problem???
• (F) If Ferdinand is not drowned, then Ferdinand is my only son (p. 3)
– Intuitively names don’t have sense (dictionary-meaning): one
cannot give a definition of, e.g. “Ferdinand.”
– So we may say that the meaning of a name is the object to it
denotes, e.g. “Ferdinand” means that guy
– Therefore, if there is no object a name denotes,
the name is meaningless…
– And so is any sentence in which it occurs.
• If Ferdinand is drowned, i.e. there is no object
“Ferdinand” denotes…
• The sentence (F) is meaningless!
What’s the solution???
• (F) If Ferdinand is not drowned, then Ferdinand is my only son (p. 3)
• But (F) is not meaningless, so we have a choice:
– Meinong’s Solution: in addition to objects that exist, there are
also possible objects that “subsist” and even impossible objects
that don’t either subsist or exist.
– Russell’s Solution: “Ferdinand” is not really a name.
• “Ferdinand is my only son” is not really a singular, subjectpredicate proposition
• Neither is “The king of France is bald” or
“Unicorns have one horn.”
• So let’s compare these two solutions…
Meinong’s Bloated Ontology
Real things
(things that
actually exist)
Possibilia (don’t
actually exist but
could: they
“subsist”)
Impossibilia
(don’t exist, can’t
exist and don’t
even “subsist”)
Round Squares
Married Bachelors
Meinong’s Solution
• It avoids the flat-out contradiction of having to say that some things
that exist don’t exist
– Zeus subsists so in saying that Zeus doesn’t exist, we’re not
saying that something that exists doesn’t exist.
• But talk about impossible objects, e.g. round squares, lands us in
contradictions.
• And we can’t/shouldn’t arbitrarily analyze talk about impossible
objects differently from talk about things that exist or subsist.
• Furthermore, introducing impossible and merely possible
things multiplies objects unnecessarily.
Russell’s Solution
•
[W]e must abandon the view that denotation is what is concerned
in propositions which contain denoting phrases.
•
We have to deal with three kinds of cases, examplified by:
•
1.
Unicorns don’t exist.
2.
The Fountain of Youth doesn’t exist.
3.
Apollo doesn’t exist
Note: in 1 the surface-grammatical subject is a general term, in 2
it’s a definite description, and in 3 it’s a name
Solution to Case 1
Unicorns don’t exist.
It is not the case that there exists an x such that x is a Unicorn
 (x)(Ux)
•
“__is a unicorn” is is a predicate
•
We say: “there’s nothing of which that predicate is true,”
•
i.e. it is not the case that something has the property of being a
unicorn
Solution to Case 2
The Fountain of Youth doesn’t exist.
It is not the case that there is one and only one x that’s
the Fountain of Youth
 (x)[Fx  (y)(Fy  y = x)]
• We note that “the Fountain of Youth” is a definite description and
treat it accordingly.
• We say in effect: “It’s false that there’s a unique thing that’s a
fountain of youth.”
Solution to Case 3
Apollo doesn’t exist
It is not the case that there is one and only one x which is C
 (x)[Cx  (y)(Cy  y=x)]
• C is “what the classical dictionary tells us is meant by ‘Apollo’--a
description.
• Ordinary names are “disguised descriptions”!
• So “Asw” isn’t strictly speaking (according to Russell’s
metaphysical account) the correct translation of “Scott was the
author of Waverly” since neither “Scott” nor “Waverly” are strictly
speaking names.
• But we’ll pretend it is to avoid getting into some heavy metaphysics.
Genuine proper names
• Russell distinguishes knowledge by acquaintance and knowledge by
description.
• Genuine proper names are names of things with which we are
directly acquainted--our current sense data.
• Genuine proper names are simply tags.
• Ordinary names and definite descriptions attach to objects insofar as
the objects have the characteristics that satisfy the descriptions.
They’re “disguised descriptions”
Russell’s Theory of Descriptions
• “On Denoting” was (and by many is) viewed as the “paradigm of
analytic philosophy” and one of the greatest pieces of philosophical
writing of the 20th century!
• It provides the standard analysis of definite descriptions.
• It stood unchallenged for over half a century.
• And then there was Strawson…