Frege's Sense and Reference: Solving Puzzles of Identity and Meaning

Introduction

In our last lesson, we explored Gottlob Frege's revolutionary break from 2,000 years of Aristotelian logic. By replacing the subject-predicate analysis with a more powerful function-argument structure and introducing quantifiers, he created a formal language—predicate logic—capable of expressing the complex statements found in mathematics and science.

However, this new logical clarity immediately brought difficult philosophical puzzles into sharper focus. This lesson addresses the learning outcome: Apply Frege's distinction between sense (Sinn) and reference (Bedeutung) to solve puzzles about identity and meaning.

We will investigate the very puzzle that led Frege to develop one of the most foundational theories in analytic philosophy and the philosophy of language. We'll define his key terms, see how his theory elegantly resolves the initial puzzle, and then apply it to more complex problems involving belief and knowledge.

1. The Puzzle of Informative Identity

At first glance, statements of identity seem straightforward. But Frege noticed a deep problem. Consider two statements:

1. Hesperus is Hesperus.
2. Hesperus is Phosphorus.

"Hesperus" was the ancient name for the celestial body seen in the evening (the Evening Star), and "Phosphorus" was the name for the one seen in the morning (the Morning Star). Statement (1) is trivial. It is something we know a priori (independently of experience) and seems uninformative. It is, as Kant would say, analytic.

Statement (2), however, is different. It expresses a genuine astronomical discovery: the evening star and the morning star are the very same object, the planet Venus. This knowledge is a posteriori (known through empirical observation) and highly informative. It has, in Frege's terms, a different cognitive value than statement (1).

The puzzle is this: If the meaning of a name is simply the object it refers to, then since 'Hesperus' and 'Phosphorus' both refer to Venus, statement (2) should say the same thing as "Venus is Venus"—which is just as trivial as statement (1). But it clearly isn't. How can an identity statement a = b be a significant discovery if 'a' and 'b' just name the same thing?

The following video and reading explore this foundational puzzle in detail. The video uses a modern example, while the reading provides a more formal breakdown.

2. Frege's Solution: The Distinction Between Sense and Reference

To solve this puzzle, Frege proposed that a name (or any designating expression) has two components to its meaning:

• Reference (Bedeutung): The object in the world that the expression picks out.

  • The reference of 'Hesperus' is the planet Venus.
  • The reference of 'Phosphorus' is also the planet Venus.

• Sense (Sinn): The "mode of presentation" of the reference. It is the particular way an object is conceived of or described, and it determines the reference. The sense is an objective, public piece of information, not a private mental image.

  • The sense of 'Hesperus' might be something like, the brightest celestial body visible in the evening sky.
  • The sense of 'Phosphorus' might be something like, the brightest celestial body visible in the morning sky.

This distinction elegantly solves the identity puzzle. The statement "Hesperus is Phosphorus" is informative because, while the names share a reference, they have different senses. The statement tells us that two different modes of presentation pick out one and the same object. The cognitive value of a statement is associated with its sense, not its reference.

The relationship can be visualized as a triangle:

The Fregean Triangle illustrates the relationship between an expression (the sign or name), its sense (the mode of presentation), and its reference (the object). The expression expresses a sense, which in turn determines a reference.

Watch the next part of the Kaplan video, which explains this solution.

3. Application to a Harder Puzzle: Indirect Contexts

Frege's distinction proves its power when applied to more complex puzzles, particularly those involving what are called opaque contexts or indirect discourse. These are parts of a sentence governed by verbs like "believes," "knows," "doubts," or "says that."

Consider this scenario:

1. An ancient astronomer, Babylos, looks at the sky in the evening and forms the belief: "Hesperus is visible." (This is TRUE)
2. We know that Hesperus is Phosphorus. (This is TRUE)

Can we substitute 'Phosphorus' for 'Hesperus' in statement (1) and preserve the truth?
3. Babylos believes that Phosphorus is visible [in the evening]. (This is FALSE. Babylos believes Phosphorus only appears in the morning).

Here, the principle of substituting co-referential terms fails. Why?

Frege's Solution: In ordinary discourse, words refer to their customary reference. But in an indirect context (like "Babylos believes that..."), words shift their reference. Inside such a clause, a name no longer refers to its customary object but instead refers to its customary sense.

• In "Babylos believes that Hesperus is visible," the word 'Hesperus' refers to the sense of Hesperus.
• In "Babylos believes that Phosphorus is visible," the word 'Phosphorus' refers to the sense of Phosphorus.

Since the senses of 'Hesperus' and 'Phosphorus' are different, we are no longer substituting co-referential terms within that belief context. The two belief statements are about different things (different senses), which is why they can have different truth values.

The following video explains this crucial application in detail.

4. The Sense and Reference of a Sentence

Frege extended his theory to cover entire declarative sentences.

• The sense of a sentence is the thought it expresses (what philosophers today would call a proposition). The sentences "Hesperus is a planet" and "Phosphorus is a planet" have the same reference but express different thoughts.
• The reference of a sentence is its truth value—either The True or The False. Frege's argument, rooted in the principle of compositionality, is that if you substitute one co-referential part for another (like 'Phosphorus' for 'Hesperus'), the reference of the whole sentence must not change. The only thing that consistently remains unchanged in such substitutions is the sentence's truth value.

This might seem strange, but it creates a beautifully systematic theory where the truth value of a whole sentence is determined by the references of its parts.

Conclusion

Today, we've unpacked one of the most influential ideas in the history of analytic philosophy. By addressing the puzzle of informative identity statements, Frege developed a powerful theory of meaning that distinguishes what a word points to from how it points to it.

Key Takeaways:

• Frege's Puzzle: Statements of the form a = b can be informative and empirical in a way that a = a cannot. This is difficult to explain if the meaning of a name is just the object it refers to.
• The Solution: Frege proposed two levels of meaning. The reference (Bedeutung) is the object itself. The sense (Sinn) is the "mode of presentation"—an objective description or condition that picks out the reference.
• Cognitive Value: The informativeness of a statement comes from its sense. "Hesperus is Phosphorus" is informative because it teaches us that two different senses determine the same reference.
• Indirect Contexts: In contexts of belief, knowledge, etc. ("X believes that..."), a name's reference shifts from its customary object to its customary sense. This solves the puzzle of why substituting co-referential names in belief reports can change the sentence's truth value.
• Sentences: For Frege, the sense of a sentence is the thought it expresses, and its reference is its truth value (The True or The False).

Preview of the next lesson:

With Frege's powerful logic (from Lesson 1) and his theory of meaning (from this lesson) in hand, we are now ready to tackle his grand philosophical project: logicism. In the next lesson, we will see how Frege attempted to use these tools to achieve the monumental goal of reducing all of arithmetic to pure logic, demonstrating that the truths of mathematics are simply complex logical truths. This will take us to the very heart of your interest in the philosophy of mathematics.