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 "Über 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 relative 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.

1 Frege's Puzzles

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 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?

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 Substitution. If a name, say n, appears in a true sentence S, and the identity sentence n=m is true, then the Principle of 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 Substitution seems to take the following form, where S is a sentence, n and m are names, and S(n) differs from S(m) onlyby 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 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 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?

2 Frege's Theory of Sense and Denotation

To explain these puzzles, Frege suggested that in addition to having a denotation, names and descriptions also express a sense. The sense of an expression accounts for its cognitive significance- it 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.

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 "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, guaranteeing that the sense of the whole expression will be 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 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 a function which serves as the denotation of the predicate "loves Mary". Let us refer to that function as d[Lm]. 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.

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]. 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 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 Substitution---the 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.