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While pursuing his investigations into mathematics and logic, in order to ground those investigations, Frege was led to develop a philosophy of language. 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 denotation (reference) 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 (reference).

First Puzzle

The statement ‘a=a’ has a cognitive significance (or meaning) that is different from the cognitive significance of ‘a=b’. We can learn that ‘Mark Twain=Mark Twain’ is true simply by inspecting it (knowable a priori); but we can't learn the truth of ‘Mark Twain=Samuel Clemens’ simply by inspecting it, we have to examine the world to see whether the two persons are the same (discovered a posteriori). 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? And why ‘Mark Twain=Samuel Clemens’ seems informative?

Second Puzzle

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 John believes that Mark Twain wrote Huckleberry Finn. And Mark Twain=Samuel Clemens, therefore, John believes that Samuel Clemens wrote Huckleberry Finn. But this argument is not valid. There are circumstances in which the premises are true and the conclusion false. 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. This law was stated by Leibniz as, "those things are the same of which one can be substituted for another without loss of truth," a sentiment with which Frege was in full agreement. As Frege understands this, it means that if two expressions have the same reference, they should be able to replace each other within any proposition without changing the truth-value of that proposition. Normally, this poses no problem. However, it is not always true that they can replace one another without changing the truth of a sentence. 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?

Frege's Solution

Frege suggested that in addition to having a denotation (reference), 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 (reference) of the term. The expressions ‘4’ and ‘8/2’ have the same denotation (reference) 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 (reference). However, because the senses of these expressions are different--in the first sentence, the object is presented the same way twice, and in the second, it is presented in two different ways, it is informative to learn of the second statement.

Using the distinction between sense and denotation (reference), 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, 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 (reference). 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.

Sense of a Sentence

The reference of the whole proposition depends on the references of the parts and the sense of the proposition depends of the senses of the parts. Frege even suggests that the sense of a whole proposition is composed of the senses of the component expressions. Frege calls the sense of a sentence a thought, he supposes that there are an infinite number of thoughts and the denotation (reference) of a sentence is one of the two truth values.

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 don’t denote their ordinary denotation (reference)s but rather the senses they ordinarily express. For example:

John believes that Mark Twain wrote Huckleberry Finn.

John believes that Samuel Clemens wrote Huckleberry Finn.

Not only do the words ‘Mark Twain’, ‘wrote’ and ‘Huckleberry Finn’ denote their ordinary senses, but also the entire sub-sentence ‘Mark Twain wrote Huckleberry Finn’ also denotes its ordinary sense (namely, a thought). And since the thought denoted by ‘Samuel Clemens wrote Huckleberry Finn’ in this context differs from the thought denoted by ‘Mark Twain wrote Huckleberry Finn’.

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 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.