Rule-Following and Skepticism in Understanding

Hello! In our last lesson, we explored Wittgenstein's concepts of 'language-games' and 'family resemblance'. We saw that the meaning of a word is its use within a practical activity, and that general terms are often held together by a network of overlapping similarities, not by a single, strict definition. This leads to the conclusion that many of our concepts have "blurred edges."

This lesson will address a deeply unsettling question that arises from this conclusion: If rules and concepts don't have sharp, pre-defined boundaries, what makes any particular application of a rule correct? How can we be sure that we are understanding each other, or even ourselves? This brings us to today's learning outcome: Analyze the rule-following considerations and their skeptical implications for understanding.

We will delve into what is often called the "rule-following paradox," a set of arguments found in Wittgenstein's Philosophical Investigations but given its most famous and forceful expression by the philosopher Saul Kripke. Given your background in mathematics, you will find the logical structure of this problem, which uses a mathematical example at its core, particularly engaging. This is one of the most debated topics in modern analytic philosophy and has profound implications for the nature of meaning, language, and thought.

1. The Skeptical Challenge: Kripkenstein's Paradox

Saul Kripke's book Wittgenstein on Rules and Private Language presented a radical interpretation of Wittgenstein's arguments. Kripke frames the issue as a skeptical challenge posed to our ordinary assumption that we mean determinate things by our words. His interpretation is so influential that the views he develops are often attributed to a hybrid philosopher, "Kripkenstein."

The central question Kripkenstein's skeptic asks is: What fact about you makes it the case that you are following one rule rather than another, given that your past behavior is consistent with an infinite number of different rules?

To get a clear overview of this problem, let's watch a short video.

The video introduces the now-famous "plus" versus "quus" example. Let's formalize it:

• Addition (plus): The familiar function $x+y$.
• Quaddition (quus): A bizarre function Kripke defines, for example, as: $$x\text{ quus }y=\{\begin{array}{ll}x+y& \text{if }x,y<57\\ 5& \text{otherwise}\end{array}$$

The skeptic's challenge is this: Suppose you've never before computed with numbers 57 or greater. Now, someone asks you for the answer to "68 + 57". You confidently answer "125". The skeptic then argues:

"All your past behavior is perfectly consistent with the rule you were following for '+' being quus, not plus. If you were following the quus rule, the correct answer to '68 + 57' would be 5. What fact about your past mental states or behavior makes it true that you meant plus rather than quus?"

This is not just an epistemological problem ("How do you know which rule you meant?"). It's a metaphysical one: What fact constitutes your meaning one thing rather than another?

2. A Deeper Dive into the Argument

To fully appreciate the force of this skeptical argument, we'll examine its structure more closely. Wittgenstein first approaches the problem with a simpler example of continuing a series, which Kripke later develops into the more radical plus/quus form.

As the video shows, the problem is not about weird mathematical functions. It applies to any word. For example, what fact makes it true that by "green" I mean the color green, and not the property grue (green if observed before the year 2050, and blue otherwise)? All my past uses of "green" are consistent with me meaning grue.

3. The Failed Search for a Meaning-Constituting Fact

Kripke's skeptic proceeds by systematically examining every possible candidate for the "fact" that constitutes meaning and showing why each one fails. This methodical demolition of alternatives is a classic piece of analytic philosophy.

Let's recap the main reasons these candidates fail:

• Past Behavior/Explicit Instructions: Any finite set of examples or verbal instructions is itself open to multiple interpretations (e.g., plus vs. quus, count vs. quount). This leads to an infinite regress.
• Dispositions: This is a strong contender. The idea is that meaning plus is just being disposed to give the sum in response to a '+' query. Kripke argues this fails for three key reasons, which the Stanford Encyclopedia of Philosophy article on Rule-Following explains well as conditions a meaning-fact must meet.
  • The Extensionality Problem (Finitude & Error): Our dispositions are finite (we can't compute infinitely large numbers) and fallible (we make mistakes). But the addition rule is infinite and determines the correct answer, regardless of our mistakes. So, our actual dispositions don't match the rule of addition.
  • The Normativity Problem: Dispositions are descriptive—they tell us what we will do. Meaning is normative—it tells us what we should do to be correct. A disposition to err doesn't make the error correct.
• Mental States (Images, Feelings): An image of adding two piles of apples is just another signpost needing interpretation. A feeling of "understanding" doesn't guarantee future correctness.
• Primitivism/Platonism: Saying meaning is a primitive, unanalyzable state or that we grasp an abstract Platonic Form of Addition doesn't solve the problem. It just names the mystery. It doesn't explain how our finite minds connect with this infinite rule to guide our actions in new cases.

4. The Skeptical Conclusion: The Vanishing of Meaning

After systematically refuting every candidate for a meaning-constituting fact, the skeptic reaches a breathtaking conclusion. As Kripke puts it:

"There can be no such thing as meaning anything by any word. Each new application of a word is a leap in the dark; any present intention could be interpreted so as to accord with anything we may choose to do. So there can be neither accord nor conflict."

This is the ultimate skeptical implication. The problem isn't just that we can't be certain about what we mean. The problem is that there seems to be no fact at all that we mean one thing rather than another. Meaning itself seems to dissolve.

This conclusion seems insane and self-refuting. After all, the skeptical argument itself uses words that we presume to have meaning. Kripke acknowledges this, calling it an "incredible and self-defeating" conclusion that must be addressed.

Conclusion

This lesson has taken us to the edge of a philosophical abyss. By relentlessly pursuing the question "What makes our use of a word correct?", the rule-following considerations seem to lead to a paradox that threatens the very possibility of meaning.

Key Takeaways:

• The rule-following paradox, articulated by Kripke as a skeptical challenge, questions what fact about a person constitutes their meaning one thing (e.g., plus) rather than another (quus), given that all past behavior is consistent with infinitely many rules.
• The skeptic systematically refutes all candidate "meaning facts," including past usage, explicit instructions, mental images, and dispositions, by showing they cannot satisfy the extensionality (determining the correct set of applications) and normativity (accounting for correctness/incorrectness) of meaning.
• The devastating skeptical conclusion is that there appears to be no fact of the matter about what we mean by our words. This isn't just a problem of knowledge; it's a metaphysical problem suggesting that determinate meaning is an illusion.

Preview of the Next Lesson:

How do we escape this paradox? Kripke argues that Wittgenstein doesn't refute the skeptic but offers a "skeptical solution." He accepts that there are no meaning-facts about isolated individuals, but argues this shows that meaning isn't a private mental affair at all. It's an essentially public, social practice. This leads directly to Wittgenstein's famous Private Language Argument, which we will reconstruct in our next lesson. It claims that a language only one person could understand is logically impossible, precisely for the reasons uncovered by the rule-following paradox.