Kripke Wittgenstein Rule Following Paradox

Overview

The Kripke–Wittgenstein rule-following paradox is a central puzzle in late 20th-century philosophy of language and mind. Developed by Saul Kripke in his 1982 book Wittgenstein on Rules and Private Language, it is presented as an interpretation of Ludwig Wittgenstein’s later work, especially the discussions of rules, meaning, and private language in the Philosophical Investigations. Kripke’s reconstruction yields a striking form of skepticism about meaning: it appears that there is no fact that constitutes my meaning one thing rather than another by any word or symbol.

The paradox targets the idea that to understand a word is to grasp a rule for its correct use. Kripke argues that no matter what facts we cite about a speaker—her past uses, mental imagery, dispositions, or internal representations—these facts seem equally compatible with many different rules. If so, there is no determinate fact about what rule she was following, and thus no determinate fact about what she meant.

The Quus Argument and the Skeptical Problem

Kripke dramatizes the paradox using a fictional “bizarre” function, which he calls quus and symbolizes with “⊕”. Define quus as:

• For any natural numbers x and y:
  • If x, y < 57, then x ⊕ y = x + y (ordinary addition).
  • Otherwise, x ⊕ y = 5.

Now consider your own history of doing addition. You have only ever computed sums of numbers below 57. Any behavior you have exhibited in the past—your spoken answers, written calculations, and even your internal experiences—are all consistent with the hypothesis that, in fact, you were following the quus rule rather than plus. After all, in all of the cases you have encountered (x, y < 57) the two rules give exactly the same outputs.

The skeptical reasoning runs roughly as follows:

1. Rule-determination requirement: To mean addition by “+” is to be following a specific rule—plus, not quus or anything else.
2. Underdetermination of rules: All the physical, behavioral, and psychological facts about your finite past are compatible with multiple rules (plus, quus, and infinitely many others) that coincide on your past inputs but diverge on untried ones.
3. Against hidden inner rules: Simply appealing to an inner mental “grasp” of the rule seems to reintroduce the same problem: whatever inner state you had can be redescribed as grasping a quus-like rule that agrees with plus on all the applications you have actually made.
4. Skeptical conclusion: Therefore, there is no fact of the matter that you meant plus rather than quus; more generally, there is no fact that constitutes anyone’s meaning anything definite by any word.

Kripke interprets this as a radical, global skepticism about meaning and rule-following, analogous to Hume’s skepticism about induction or personal identity. The paradox does not just question whether we have justification for our meaning claims but seems to deny that there are any truths about meaning to justify in the first place.

Kripke’s Skeptical Solution and Community Standards

Kripke attributes to Wittgenstein not a direct refutation of the skeptical problem, but what he calls a “skeptical solution”. In Humean fashion, such a solution accepts the skeptical conclusion at the level of metaphysical facts, while explaining why our ordinary linguistic and normative practices nonetheless remain intact.

According to this reading, Wittgenstein abandons the idea that there are private, inner facts—mental items or platonic rules—that determine meaning. Instead, we look to public, communal practices:

• What counts as following a rule or using a word correctly is grounded in the regularities of use within a linguistic community.
• To say that “plus” means standard addition is to say, roughly, that speakers of the language are disposed, under normal conditions, to make certain moves and to treat some outputs as correct and others as incorrect.
• Normativity (the distinction between correct and incorrect applications) is not underwritten by private mental episodes but by shared standards embedded in our practices of training, correction, justification, and agreement.

On this view, the skeptical challenge—that no inner fact determines meaning—is accepted, but its disruptive force is defused by relocating meaning in outward, social criteria of correctness, rather than in private facts.

Critics disagree about whether this genuinely solves a paradox or merely redescribes our practices. Proponents hold that once we recognize that meaning is use in a rule-governed social practice, the demand for a further inner fact becomes misguided.

Critical Responses and Significance

Kripke’s reconstruction has provoked extensive debate, both about its philosophical implications and its fidelity to Wittgenstein.

Major lines of response include:

• Deflationary or “straight” interpretations of Wittgenstein: Some commentators argue that Kripke exaggerates the skeptical dimension; Wittgenstein, on this view, is clarifying how talk of rules and meanings actually works, not endorsing a deep metaphysical skepticism.
• Realist responses about meaning-facts: Others insist that there are determinate facts about meaning—perhaps neurophysiological states, internal representations, or dispositional profiles—that distinguish plus from quus, even if we cannot specify them simply. They deny the underdetermination premise.
• Dispositional theories: Some philosophers attempt to ground rule-following in idealized dispositions to respond, arguing that when properly idealized (e.g., corrected for fatigue, error, and finite limits), these dispositions do determine a unique function like standard addition, not quus.
• Normative and pragmatic accounts: Another tradition emphasizes the normative character of rule-following: what fixes meaning are not descriptive facts alone but patterns of justification, entitlement, and practice-dependent norms. Kripke’s skeptical problem is taken to show that purely descriptive accounts of meaning are inadequate.

The paradox has substantial implications:

• In philosophy of language, it challenges reductive theories of meaning that seek to ground semantic facts purely in psychological or physical states.
• In epistemology, it parallels skeptical problems about justification, raising questions about our knowledge of our own meanings and of logical and mathematical rules.
• In philosophy of mind and cognitive science, it informs debates about representation: what it is for a mental or computational state to represent one function rather than another.

Whether or not Kripke’s reading is ultimately accepted as a faithful interpretation of Wittgenstein, the Kripke–Wittgenstein rule-following paradox remains a touchstone for contemporary discussions of meaning, normativity, and the social character of language. It forces a reconsideration of how rules can guide action and how words can have determinate meanings in the absence of privileged inner facts.