Russells Work in Analytic Philosophy

In much the same way that Russell used logic in an attempt to clarify issues in the
foundations of mathematics, he also used logic in an attempt to clarify issues in
philosophy. As one of the founders of analytic philosophy, Russell made
significant contributions to a wide variety of areas, including metaphysics,
epistemology, ethics and political theory. His advances in logic and metaphysics
also had significant influence on Rudolf Carnap and the Vienna Circle.
According to Russell, it is the philosophers job to discover a logically ideal
language a language that will exhibit the nature of the world in such a way that
we will not be misled by the accidental, imprecise surface structure of natural
language. As Russell writes, Ordinary language is totally unsuited for expressing
what physics really asserts, since the words of everyday life are not sufficiently
abstract. Only mathematics and mathematical logic can say as little as the physicist
means to say (1931, 82). Just as atomic facts (the association of properties and
relations with individuals) combine to form molecular facts in the world itself,
such a language will allow for the description of such combinations using logical
connectives such as and and or. In addition to the existence of atomic and
molecular facts, Russell also held that general facts (facts about all of something)
are needed to complete our picture of the world. Famously, he vacillated on
whether negative facts are also required.
The reason Russell believes many ordinarily accepted statements are open to doubt
is that they appear to refer to entities that may be known only through inference.
Thus, underlying Russells various projects was not only his use of logical
analysis, but also his long-standing aim of discovering whether, and to what extent,
knowledge is possible. There is one great question, he writes in 1911. Can
human beings know anything, and if so, what and how? This question is really the
most essentially philosophical of all questions (quoted in Slater 1994, 67).
Motivating this question was the traditional problem of the external world. If our
knowledge of the external world comes through inferences to the best explanation,
and if such inferences are always fallible, what guarantee do we have that our
beliefs are reliable? Russells response to this question was partly metaphysical
and partly epistemological. On the metaphysical side, Russell developed his
famous theory of logical atomism, in which the world is said to consist of a
complex of logical atoms (such as little patches of colour) and their properties
and relations. (The theory was crucial for influencing Wittgensteins theory of the
same name.) Together these atoms and their properties form the atomic facts
which, in turn, combine to form logically complex objects. What we normally take
to be inferred entities (for example, enduring physical objects) are then understood
as logical constructions formed from the immediately given entities of sensation,
viz., sensibilia.
On the epistemological side, Russell argues that it is also important to show how
each questionable entity may be reduced to, or defined in terms of, another entity
(or entities) whose existence is more certain. For example, on this view, an
ordinary physical object that normally might be thought to be known only through
inference may be defined instead
as a certain series of appearances, connected with each other by continuity and by
certain causal laws. More generally, a thing will be defined as a certain series
of aspects, namely those which would commonly be said to be of the thing. To say
that a certain aspect is an aspect of a certain thing will merely mean that it is one of
those which, taken serially, are the thing. (1914a, 106107)
The reason we are able to do this, says Russell, is that
our world is not wholly a matter of inference. There are things that we know
without asking the opinion of men of science. If you are too hot or too cold, you
can be perfectly aware of this fact without asking the physicist what heat and cold
consist of. We may give the name data to all the things of which we are aware
without inference. (1959, 23)
We can then use these data (or sensibilia or sense data) with which we are
directly acquainted to construct the relevant objects of knowledge. Similarly,
numbers may be reduced to collections of classes; points and instants may be
reduced to ordered classes of volumes and events; and classes themselves may be
reduced to propositional functions.
It is with these kinds of examples in mind that Russell suggests we adopt what he
calls the supreme maxim in scientific philosophizing, namely the principle that
Whenever possible, logical constructions, or as he also sometimes puts it,
logical fictions, are to be substituted for inferred entities (1914c, 155; cf.
1914a, 107, and 1924, 326). Anything that resists construction in this sense may be
said to be an ontological atom. Such objects are atomic, both in the sense that they
fail to be composed of individual, substantial parts, and in the sense that they exist
independently of one another. Their corresponding propositions are also atomic,
both in the sense that they contain no other propositions as parts, and in the sense
that the members of any pair of true atomic propositions will be logically
independent of one another. Russell believes that formal logic, if carefully
developed, will mirror precisely, not only the various relations between all such
propositions, but their various internal structures as well.
It is in this context that Russell also introduces his famous distinction between two
kinds of knowledge of truths: that which is direct, intuitive, certain and infallible,
and that which is indirect, derivative, uncertain and open to error (1905, 41f; 1911,
1912, and 1914b). To be justified, every indirect knowledge claim must be capable
of being derived from more fundamental, direct or intuitive knowledge claims. The
kinds of truths that are capable of being known directly include both truths about
immediate facts of sensation and truths of logic. Examples are discussed in The
Problems of Philosophy (1912a) where Russell states that propositions with the
highest degree of self-evidence (what he here calls intuitive knowledge) include
those which merely state what is given in sense, and also certain abstract logical
and arithmetical principles, and (though with less certainty) some ethical
propositions (1912a, 109).
Eventually, Russell supplemented this distinction between direct and indirect
knowledge of truths with his equally famous distinction between knowledge by
acquaintance and knowledge by description. As Russell explains, I say that I am
acquainted with an object when I have a direct cognitive relation to that object, i.e.
when I am directly aware of the object itself. When I speak of a cognitive relation
here, I do not mean the sort of relation which constitutes judgment, but the sort
which constitutes presentation (1911, 209). Later, he clarifies this point by adding
that acquaintance involves, not knowledge of truths, but knowledge of things
(1912a, 44). Thus, while intuitive knowledge and derivative knowledge both
involve knowledge of propositions (or truths), knowledge by acquaintance and
knowledge by description both involve knowledge of things (or objects). This
distinction is slightly complicated by the fact that, even though knowledge by
description is in part based upon knowledge of truths, it is still knowledge of
things, and not of truths. (I am grateful to Russell Wahl for reminding me of this
point.) Since it is things with which we have direct acquaintance that are the least
questionable members of our ontology, it is these objects upon which Russell
ultimately bases his epistemology.
Also relevant was Russells reliance upon his so-called regressive method (Irvine
1989, Mayo-Wilson 2011) and his eventual abandoning of foundationalism in
favour of a more recognizably coherentist approach to knowledge (Irvine 2004).
As Russell puts it, even in logic and mathematics
We tend to believe the premises because we can see that their consequences are
true, instead of believing the consequences because we know the premises to be
true. But the inferring of premises from consequences is the essence of induction;
thus the method in investigating the principles of mathematics is really an
inductive method, and is substantially the same as the method of discovering
general laws in any other science. (1907, 273274)
Russells contributions to metaphysics and epistemology are also unified by his
views concerning the centrality of both scientific knowledge and the importance of
there being an underlying methodology common to both philosophy and science.
In the case of philosophy, this methodology expresses itself through Russells use
of logical analysis (Hager 1994, Irvine 2004). In fact, Russell often claims that he
has more confidence in his methodology than in any particular philosophical
conclusion.
This broad conception of philosophy arose in part from Russells idealist origins
(Hylton 1990a, Griffin 1991). This is so, even though Russell tells us that his one,
true revolution in philosophy came as a result of his break from idealism. Russell
saw that the idealist doctrine of internal relations led to a series of contradictions
regarding asymmetrical (and other) relations necessary for mathematics. As he
reports,
It was towards the end of 1898 that Moore and I rebelled against both Kant and
Hegel. Moore led the way, but I followed closely in his footsteps. [Our rebellion
centred upon] the doctrine that fact is in general independent of experience.
Although we were in agreement, I think that we differed as to what most interested
us in our new philosophy. I think that Moore was most concerned with the
rejection of idealism, while I was most interested in the rejection of monism.
(1959, 54)
The two ideas were closely connected through the so-called doctrine of internal
relations. In contrast to this doctrine, Russell proposed his own new doctrine
of external relations:
The doctrine of internal relations held that every relation between two terms
expresses, primarily, intrinsic properties of the two terms and, in ultimate analysis,
a property of the whole which the two compose. With some relations this view is
plausible. Take, for example, love or hate. If A loves B, this relation exemplifies
itself and may be said to consist in certain states of mind of A. Even an atheist
must admit that a man can love God. It follows that love of God is a state of the
man who feels it, and not properly a relational fact. But the relations that interested
me were of a more abstract sort. Suppose that A and B are events, and A is earlier
than B. I do not think that this implies anything in A in virtue of which,
independently of B, it must have a character which we inaccurately express by
mentioning B. Leibniz gives an extreme example. He says that, if a man living in
Europe has a wife in India and the wife dies without his knowing it, the man
undergoes an intrinsic change at the moment of her death. (1959, 54)
This is the type of doctrine Russell opposed, especially with respect to the
asymmetrical relations necessary for mathematics. For example, consider two
numbers, one of which is found earlier than the other in a given series:
If A is earlier than B, then B is not earlier than A. If you try to express the relation
of A to B by means of adjectives of A and B, you will have to make the attempt by
means of dates. You may say that the date of A is a property of A and the date of B
is a property of B, but that will not help you because you will have to go on to say
that the date of A is earlier than the date of B, so that you will have found no
escape from the relation. If you adopt the plan of regarding the relation as a
property of the whole composed of A and B, you are in a still worse predicament,
for in that whole A and B have no order and therefore you cannot distinguish
between A is earlier than B and B is earlier than A. As asymmetrical relations
are essential in most parts of mathematics, this doctrine was important. (1959, 54
55)
Thus, by the end of 1898 Russell had abandoned the idealism that he had been
encouraged to adopt as a student at Cambridge, along with his original Kantian
methodology. In its place he adopted a new, pluralistic realism. As a result, he
soon became famous as an advocate of the new realism and of his new
philosophy of logic, emphasizing as he did the importance of modern logic for
philosophical analysis. The underlying themes of this revolution included Russells
belief in pluralism, his emphasis on anti-psychologism and his belief in the
importance of science. Each of these themes remained central to his philosophy for
the remainder of his life (Hager 1994, Weitz 1944).
Russells most important writings relating to these topics include Knowledge by
Acquaintance and Knowledge by Description (1911), The Problems of
Philosophy (1912a), Our Knowledge of the External World (1914a), On the
Nature of Acquaintance (1914b, published more completely in Collected Papers,
Vol. 7), The Philosophy of Logical Atomism (1918, 1919), Logical Atomism
(1924), The Analysis of Mind (1921), The Analysis of Matter (1927a), Human
Knowledge: Its Scope and Limits (1948), and Theory of Knowledge (CP, Vol. 7).

4. Russells Theory of Definite Descriptions

Russells philosophical method has at its core the making and testing of hypotheses
through the weighing of evidence. Hence Russells comment that he wished to
emphasize the scientific method in philosophy. His method also requires the
rigorous analysis of problematic propositions using the machinery of first-order
logic. It was Russells belief that by using the new logic of his day, philosophers
would be able to exhibit the underlying logical form of natural-language
statements. A statements logical form, in turn, would help resolve various
problems of reference associated with the ambiguity and vagueness of natural
language.
Since the introduction of the modern predicate calculus, it has been common to use
three separate logical notations (Px, x = y, and x) to represent three
separate senses of the natural-language word is: the is of predication, e.g.
Cicero is wise; the is of identity, e.g. Cicero is Tully; and the is of existence,
e.g. Cicero is. It was Russells suggestion that, just as we use logic to make clear
these distinctions, we can also use logic to discover other ontologically significant
distinctions, distinctions that should be reflected in the analysis we give of each
sentences correct logical form.
On Russells view, the subject matter of philosophy is then distinguished from that
of the sciences only by the generality and a prioricity of philosophical statements,
not by the underlying methodology of the discipline. In philosophy, just as in
mathematics, Russell believed that it was by applying logical machinery and
insights that advances in analysis would be made.
Russells most famous example of his new analytic method concerns so-called
denoting phrases, phrases that include both definite descriptions and proper names.
Like Alexius Meinong, Russell had initially adopted the view that every denoting
phrase (for example, Scott, the author ofWaverley, the number two, the
golden mountain) denoted, or referred to, an existing entity. On this view, even
fictional and imaginary entities had to be real in order to serve as truth-makers for
true sentences such as Unicorns have exactly one horn. By the time his landmark
article, On Denoting, appeared in 1905, Russell had modified his extreme
realism, substituting in its place the view that denoting phrases need not possess a
theoretical unity. As Russell puts it, the assumption that every denoting phrase
must refer to an existing entity was the type of assumption that exhibited a failure
of that feeling for reality which ought to be preserved even in the most abstract
studies (1919a, 165).
While logically proper names (words such as this or that which refer to
sensations of which an agent is immediately aware) do have referents associated
with them, descriptive phrases (such as the smallest number less than pi) should
be viewed merely as collections of quantifiers (such as all and some)
and propositional functions (such as x is a number). As such, they are not to be
viewed as referring terms but, rather, as incomplete symbols. In other words,
they are to be viewed as symbols that take on meaning within appropriate contexts,
but that remain meaningless in isolation.
Put another way, it was Russells insight that some phrases may contribute to the
meaning (or reference) of a sentence without themselves being meaningful. As he
explains,
If the author of Waverley meant anything other than Scott, Scott is the author
of Waverley would be false, which it is not. If the author of Waverley meant
Scott, Scott is the author of Waverley would be a tautology, which it is not.
Therefore, the author of Waverley means neither Scott nor anything else i.e.
the author of Waverley means nothing, Q.E.D. (1959, 85)
If Russell is correct, it follows that in a sentence such as
(1) The present King of France is bald,
the definite description The present King of France plays a role quite different
from the role a proper name such as Scott plays in the sentence
(2) Scott is bald.
Letting K abbreviate the predicate is a present King of France and B abbreviate
the predicate is bald, Russell assigns sentence (1) the logical form
(1) There is an x such that

i. Kx,
ii. for any y, if Ky then y=x, and
iii. Bx.
Alternatively, in the notation of the predicate calculus, we write
(1) x[(Kx & y(Ky y=x)) & Bx].
In contrast, by allowing s to abbreviate the name Scott, Russell assigns sentence
(2) the very different logical form
(2) Bs.
This distinction between logical forms allows Russell to explain three important
puzzles.
The first concerns the operation of the Law of Excluded Middle and how this law
relates to denoting terms. According to one reading of the Law of Excluded
Middle, it must be the case that either The present King of France is bald is true
or The present King of France is not bald is true. But if so, both sentences appear
to entail the existence of a present King of France, clearly an undesirable result,
given that France is a republic and so has no king. Russells analysis shows how
this conclusion can be avoided. By appealing to analysis (1), it follows that there
is a way to deny (1) without being committed to the existence of a present King of
France, namely by changing the scope of the negation operator and thereby
accepting that It is not the case that there exists a present King of France who is
bald is true.
The second puzzle concerns the Law of Identity as it operates in (so-called) opaque
contexts. Even though Scott is the author of Waverley is true, it does not follow
that the two referring terms Scott and the author of Waverley need be
interchangeable in every situation. Thus, although George IV wanted to know
whether Scott was the author of Waverley is true, George IV wanted to know
whether Scott was Scott is, presumably, false.
Russells distinction between the logical forms associated with the use of proper
names and definite descriptions again shows why this is so. To see this, we once
again let s abbreviate the name Scott. We also let w abbreviate Waverley
and A abbreviate the two-place predicate is the author of. It then follows that the
sentence
(3) s=s
is not at all equivalent to the sentence
(4) x[(Axw & y(Ayw y=x)) & x=s].
Sentence (3), for example, is a necessary truth, while sentence (4) is not.
The third puzzle relates to true negative existential claims, such as the claim The
golden mountain does not exist. Here, once again, by treating definite descriptions
as having a logical form distinct from that of proper names, Russell is able to give
an account of how a speaker may be committed to the truth of a negative
existential without also being committed to the belief that the subject term has
reference. That is, the claim that Scott does not exist is false since
(5) ~x(x=s)
is self-contradictory. (After all, there must exist at least one thing that is identical
to s since it is a logical truth that s is identical to itself!) In contrast, the claim that a
golden mountain does not exist may be true since, assuming that G abbreviates the
predicate is golden and M abbreviates the predicate is a mountain, there is
nothing contradictory about
(6) ~x(Gx & Mx).
Russells most important writings relating to his theory of descriptions include not
only On Denoting (1905), but also The Principles of
Mathematics (1903), Principia Mathematica (1910) and Introduction to
Mathematical Philosophy (1919). (See too Kaplan 1970, Kroon 2009 and Stevens
2011.)

Frege's Philosophy of Language

While pursuing his investigations into mathematics and logic (and quite possibly,
in order to ground those investigations), Frege was led to develop a philosophy of
language. His philosophy of language has had just as much, if not more, impact
than his contributions to logic and mathematics. Frege's seminal paper in this field
ber Sinn und Bedeutung (On Sense and Reference, 1892a) is now a classic. In
this paper, Frege considered two puzzles about language and noticed, in each case,
that one cannot account for the meaningfulness or logical behavior of certain
sentences simply on the basis of the denotations of the terms (names and
descriptions) in the sentence. One puzzle concerned identity statements and the
other concerned sentences with subordinate clauses such as propositional attitude
reports. To solve these puzzles, Frege suggested that the terms of a language have
both a sense and a denotation, i.e., that at least two semantic relations are required
to explain the significance or meaning of the terms of a language. This idea has
inspired research in the field for over a century and we discuss it in what follows.
(See Heck and May 2006 for further discussion of Frege's contribution to the
philosophy of language.)

3.1 Frege's Puzzles

3.1.1 Frege's Puzzle About Identity Statements
Here are some examples of identity statements:
117+136 = 253.
The morning star is identical to the evening star.
Mark Twain is Samuel Clemens.
Bill is Debbie's father.
Frege believed that these statements all have the form a=b, where a and b are
either names or descriptions that denote individuals. He naturally assumed that a
sentence of the form a=b is true if and only if the object a just is (identical to) the
object b. For example, the sentence 117+136 = 253 is true if and only if the
number 117+136 just is the number 253. And the statement Mark Twain is
Samuel Clemens is true if and only if the person Mark Twain just is the person
Samuel Clemens.
But Frege noticed (1892) that this account of truth can't be all there is to the
meaning of identity statements. The statement a=a has a cognitive significance
(or meaning) that must be different from the cognitive significance of a=b. We
can learn that Mark Twain=Mark Twain is true simply by inspecting it; but we
can't learn the truth of Mark Twain=Samuel Clemens simply by inspecting it
you have to examine the world to see whether the two persons are the same.
Similarly, whereas you can learn that 117+136 = 117+136 and the morning star
is identical to the morning star are true simply by inspection, you can't learn the
truth of 117+136 = 253 and the morning star is identical to the evening star
simply by inspection. In the latter cases, you have to do some arithmetical work or
astronomical investigation to learn the truth of these identity claims. Now the
problem becomes clear: the meaning of a=a clearly differs from the meaning of
a=b, but given the account of the truth described in the previous paragraph, these
two identity statements appear to have the same meaning whenever they are true!
For example, Mark Twain=Mark Twain is true just in case: the person Mark
Twain is identical with the person Mark Twain. And Mark Twain=Samuel
Clemens is true just in case: the person Mark Twain is identical with the person
Samuel Clemens. But given that Mark Twain just is Samuel Clemens, these two
cases are the same case, and that doesn't explain the difference in meaning between
the two identity sentences. And something similar applies to all the other examples
of identity statements having the forms a=a and a=b.
So the puzzle Frege discovered is: how do we account for the difference in
cognitive significance between a=b and a=a when they are true?
3.1.2 Frege's Puzzle About Propositional Attitude Reports
Frege is generally credited with identifying the following puzzle about
propositional attitude reports, even though he didn't quite describe the puzzle in the
terms used below. A propositional attitude is a psychological relation between a
person and a proposition. Belief, desire, intention, discovery, knowledge, etc., are
all psychological relationships between persons, on the one hand, and propositions,
on the other. When we report the propositional attitudes of others, these reports all
have a similar logical form:
x believes that p
x desires that p
x intends that p
x discovered that p
x knows that p

If we replace the variable x by the name of a person and replace the variable p
with a sentence that describes the propositional object of their attitude, we get
specific attitude reports. So by replacing x by John and p by Mark Twain
wrote Huckleberry Finn in the first example, the result would be the following
specific belief report:
John believes that Mark Twain wrote Huckleberry Finn.

To see the problem posed by the analysis of propositional attitude reports, consider
what appears to be a simple principle of reasoning, namely, the Principle of
Identity Substitution (this is not to be confused with the Rule of Substitution
discussed earlier). If a name, say n, appears in a true sentence S, and the identity
sentence n=m is true, then the Principle of Identity Substitution tells us that the
substitution of the name m for the name n in S does not affect the truth of S. For
example, let S be the true sentence Mark Twain was an author, let n be the name
Mark Twain, and let m be the name Samuel Clemens. Then since the identity
sentence Mark Twain=Samuel Clemens is true, we can substitute Samuel
Clemens for Mark Twain without affecting the truth of the sentence. And
indeed, the resulting sentence Samuel Clemens was an author is true. In other
words, the following argument is valid:
Mark Twain was an author.
Mark Twain=Samuel Clemens.
Therefore, Samuel Clemens was an author.

Similarly, the following argument is valid.

4>3
4=8/2
Therefore, 8/2 > 3

In general, then, the Principle of Identity Substitution seems to take the following
form, where S is a sentence, n and m are names, and S(n) differs from S(m) only by
the fact that at least one occurrence of m replaces n:
From S(n) and n=m, infer S(m)

This principle seems to capture the idea that if we say something true about an
object, then even if we change the name by which we refer to that object, we
should still be saying something true about that object.
But Frege, in effect, noticed the following counterexample to the Principle of
Identity Substitution. Consider the following argument:
John believes that Mark Twain wrote Huckleberry Finn.
Mark Twain=Samuel Clemens.
Therefore, John believes that Samuel Clemens wrote Huckleberry Finn.

This argument is not valid. There are circumstances in which the premises are true
and the conclusion false. We have already described such circumstances, namely,
one in which John learns the name Mark Twain by reading Huckleberry Finn but
learns the name Samuel Clemens in the context of learning about 19th century
American authors (without learning that the name Mark Twain was a pseudonym
for Samuel Clemens). John may not believe that Samuel Clemens
wrote Huckleberry Finn. The premises of the above argument, therefore, do not
logically entail the conclusion. So the Principle of Identity Substitution appears to
break down in the context of propositional attitude reports. The puzzle, then, is to
say what causes the principle to fail in these contexts. Why aren't we still saying
something true about the man in question if all we have done is changed the name
by which we refer to him?

3.2 Frege's Theory of Sense and Denotation

To explain these puzzles, Frege suggested (1892a) that in addition to having a
denotation, names and descriptions also express a sense.[5] The sense of an
expression accounts for its cognitive significanceit is the way by which one
conceives of the denotation of the term. The expressions 4 and 8/2 have the
same denotation but express different senses, different ways of conceiving the
same number. The descriptions the morning star and the evening star denote the
same planet, namely Venus, but express different ways of conceiving of Venus and
so have different senses. The name Pegasus and the description the most
powerful Greek god both have a sense (and their senses are distinct), but neither
has a denotation. However, even though the names Mark Twain and Samuel
Clemens denote the same individual, they express different senses. (See May
2006b for a nice discussion of the question of whether Frege believed that the
sense of a name varies from person to person.) Using the distinction between sense
and denotation, Frege can account for the difference in cognitive significance
between identity statements of the form a=a and those of the form a=b. Since
the sense of a differs from the sense of b, the components of the sense of a=a
and the sense of a=b are different. Frege can claim that the sense of the whole
expression is different in the two cases. Since the sense of an expression accounts
for its cognitive significance, Frege has an explanation of the difference in
cognitive significance between a=a and a=b, and thus a solution to the first
puzzle.
Moreover, Frege proposed that when a term (name or description) follows a
propositional attitude verb, it no longer denotes what it ordinarily denotes. Instead,
Frege claims that in such contexts, a term denotes its ordinary sense. This explains
why the Principle of Identity Substitution fails for terms following the
propositional attitude verbs in propositional attitude reports. The Principle asserts
that truth is preserved when we substitute one name for another having the same
denotation. But, according to Frege's theory, the names Mark Twain and Samuel
Clemens denote different senses when they occur in the following sentences:
John believes that Mark Twain wrote Huckleberry Finn.
John believes that Samuel Clemens wrote Huckleberry Finn.

If they don't denote the same object, then there is no reason to think that
substitution of one name for another would preserve truth.
Frege developed the theory of sense and denotation into a thoroughgoing
philosophy of language. This philosophy can be explained, at least in outline, by
considering a simple sentence such as John loves Mary. In Frege's view, the
words John and Mary in this sentence are names, the expression loves
signifies a function, and, moreover, the sentence as a whole is a complex name.
Each of these expressions has both a sense and a denotation. The sense and
denotation of the names are basic; but sense and denotation of the sentence as a
whole can be described in terms of the sense and denotation of the names and the
way in which those words are arranged in the sentence alongside the expression
loves. Let us refer to the denotation and sense of the words as follows:
d[j] refers to the denotation of the name John.
d[m] refers to the denotation of the name Mary.
d[L] refers to the denotation of the expression loves.
s[j] refers to the sense of the name John.
s[m] refers to the sense of the name Mary.
s[L] refers to the sense of the expression loves.

We now work toward a theoretical description of the denotation of the sentence as

a whole. On Frege's view, d[j] and d[m] are the real individuals John and Mary,
respectively. d[L] is a function that maps d[m] (i.e., Mary) to the function ( ) loves
Mary. This latter function serves as the denotation of the predicate loves Mary
and we can use the notation d[Lm] to refer to it semantically. Now the
function d[Lm] maps d[j] (i.e., John) to the denotation of the sentence John loves
Mary. Let us refer to the denotation of the sentence as d[jLm]. Frege identifies the
denotation of a sentence as one of the two truth values. Because d[Lm] maps
objects to truth values, it is a concept. Thus, d[jLm] is the truth value The True if
John falls under the concept d[Lm]; otherwise it is the truth value The False. So, on
Frege's view, the sentence John loves Mary names a truth value.[6]
The sentence John loves Mary also expresses a sense. Its sense may be described
as follows. Although Frege doesn't appear to have explicitly said so, his work
suggests that s[L] (the sense of the expression loves) is a function. This function
would map s[m] (the sense of the name Mary) to the sense of the predicate loves
Mary. Let us refer to the sense of loves Mary as s[Lm]. Now again, Frege's work
seems to imply that we should regard s[Lm] as a function which maps s[j] (the
sense of the name John) to the sense of the whole sentence. Let us call the sense
of the entire sentence s[jLm].[7] Frege calls the sense of a sentence a thought, and
whereas there are only two truth values, he supposes that there are an infinite
number of thoughts.
With this description of language, Frege can give a general account of the
difference in the cognitive significance between identity statements of the form
a=a and a=b. The cognitive significance is not accounted for at the level of
denotation. On Frege's view, the sentences 4=8/2 and 4=4 both denote the same
truth value. The function ( )=( ) maps 4 and 8/2 to The True, i.e., maps 4 and 4 to
The True. So d[4=8/2] is identical to d[4=4]; they are both The True. However, the
two sentences in question express different thoughts. That is because s[4] is
different from s[8/2]. So the thought s[4=8/2] is distinct from the thought s[4=4].
Similarly, Mark Twain=Mark Twain and Mark Twain=Samuel Clemens denote
the same truth value. However, given that s[Mark Twain] is distinct from s[Samuel
Clemens], Frege would claim that the thought s[Mark Twain=Mark Twain] is
distinct from the thought s[Mark Twain=Samuel Clemens].
Furthermore, recall that Frege proposed that terms following propositional attitude
verbs denote not their ordinary denotations but rather the senses they ordinarily
express. In fact, in the following propositional attitude report, not only do the
words Mark Twain, wrote and Huckleberry Finn denote their ordinary
senses, but the entire sentence Mark Twain wrote Huckleberry Finn also denotes
its ordinary sense (namely, a thought):
John believes that Mark Twain wrote Huckleberry Finn.

Frege, therefore, would analyze this attitude report as follows: believes that
denotes a function that maps the denotation of the sentence Mark Twain
wrote Huckleberry Finn to a concept. In this case, however, the denotation of the
sentence Mark Twain wrote Huckleberry Finn is not a truth value but rather a
thought. The thought it denotes is different from the thought denoted by Samuel
Clemens wrote Huckleberry Finn in the following propositional attitude report:
John believes that Samuel Clemens wrote Huckleberry Finn.

Since the thought denoted by Samuel Clemens wrote Huckleberry Finn in this
context differs from the thought denoted by Mark Twain wrote Huckleberry Finn
in the same context, the concept denoted by believes that Mark Twain
wrote Huckleberry Finn is a different concept from the one denoted by believes
that Samuel Clemens wrote Huckleberry Finn. One may consistently suppose that
the concept denoted by the former predicate maps John to The True whereas the
concept denoted by the latter predicate does not. Frege's analysis therefore
preserves our intuition that John can believe that Mark Twain wrote Huckleberry
Finn without believing that Samuel Clemens did. It also preserves the Principle of
Identity Substitutionthe fact that one cannot substitute Samuel Clemens for
Mark Twain when these names occur after propositional attitude verbs does not
constitute evidence against the Principle. For if Frege is right, names do not have
their usual denotation when they occur in these contexts.