The Rule That Cannot Explain Itself: Wittgenstein's Paradox and Kripke's Sceptic

The Hook: The Calculation Nobody Has Ever Made

Imagine someone sets you a problem you have never solved in your life. Not because it is hard – it is trivial – but because those particular numbers simply never came up. Say: 68 + 57.

You answer: 125.

Now a sceptic appears. He does not dispute your arithmetical competence. He does not even dispute that 125 is the correct result of addition. He disputes something far stranger: that you ever meant addition by "plus".

Perhaps, he says, you always meant a different function. Call it quaddition, written \(\oplus\). It is defined as:

x \(\oplus\) y = x + y, if x and y are both smaller than 57; otherwise x \(\oplus\) y = 5.

This function is perfectly well defined. It is not meaningless, not contradictory, not even particularly exotic – computer scientists know such functions as special cases with a threshold. And crucially: every single calculation you have ever performed in your life is compatible with it. Because in this thought experiment, you have never computed with numbers of 57 or more.

The sceptic now asks: Which fact about you – about your behaviour, your memory, your mental state, your brain – makes it true that you meant "plus" rather than "quus"?

The obvious answer is: "But I had the addition rule in my head." The sceptic has already anticipated that. Which rule did you have in your head? If you formulate it, it consists of words or symbols – and those can be reinterpreted just as easily. Did you think "add together the elements of two sets"? Then the sceptic asks what you meant by "count": counting or qounting?

This is the core of what philosophy has discussed for some four decades as the rule-following paradox. It is not merely an academic subtlety. In its strictest form it is the question of why a prescription – a law, a specification, a training set, a coding standard – should ever fix exactly one way of going on rather than arbitrarily many. And it is a question anyone must answer who wants to teach a system to behave "correctly".

Part 1: Wittgenstein's Paradox

The passage at issue

Ludwig Wittgenstein formulated the problem in the Philosophical Investigations, published posthumously in 1953. The relevant passage runs roughly from §138 to §242; its hinge is §201. There he states that the paradox consists in the fact that a rule could determine no course of action, because every course of action can be brought into accord with the rule (PI §201).

The argument behind it is simple and devastating. Suppose someone is to continue the series "+2": 2, 4, 6, 8 … At 1000 he suddenly writes 1004, 1008, 1012. You correct him, and he answers in bewilderment: he did exactly what he was told – he went on in just the same way as before. His interpretation of the rule was: "Add 2 up to 1000, then 4." Nothing in the instruction "+2" rules out this interpretation, as long as one looks only at the finitely many steps taken so far.

You could follow up: "No, I meant: add the same number at each step." But "the same" can be reinterpreted too. Every clarification is itself a sign and needs an interpretation in turn. Wittgenstein draws from this not a sceptical but a therapeutic conclusion: there is a grasp of a rule that is not an interpretation, but shows itself in the following – in what one actually does, once one has been trained. And §202 adds: following a rule is a practice; and therefore one cannot follow a rule "privately", because otherwise believing one is following the rule and following it would be the same thing.

Why this is not only philosophy of language

The passage is so consequential because it attacks an assumption we rarely state: that understanding a rule consists in having a content in one's head that fixes all future cases in advance. Crispin Wright took up Wittgenstein's own image of "rails to infinity" for this – as if the rule, like an already laid track, contained every future step.

Wittgenstein's point is that this picture is empty. Not false in the sense of a refuted hypothesis, but empty: it explains nothing, because the question of interpretation is merely postponed.

Part 2: Kripke's Sceptic

A book that caused an earthquake

In 1982, Saul Kripke published a slim volume of about 150 pages with Harvard University Press: Wittgenstein on Rules and Private Language. An Elementary Exposition. It turned Wittgenstein's scattered remarks into a sharply formulated argument – and triggered a debate that continues today.

Kripke was careful enough to distance himself from his own reading. He presented the argument as it struck him, not as faithful Wittgenstein exegesis. The field made of this the half-mocking, half-practical label "Kripkenstein": a figure who holds the theses of the book without being identical with the historical Wittgenstein.

The structure of the argument

What makes Kripke's version so hard to shake off is its generosity towards the opponent. The sceptic concedes:

1. Your memory is perfect. You remember every calculation you have ever carried out.
2. You have complete access to all facts about your own mind.
3. This is not about knowledge. The question is not how you know what you meant, but what that fact consists in.

The argument is therefore not an epistemological but a constitutive scepticism. What is asked for is the truth-maker of meaning.

And the fact in question must satisfy two conditions:

The normativity condition is the sharper hook. A meaning-fact is supposed to explain why 125 is the right answer. A mere description of my inclinations cannot do that: if I systematically miscalculate, my inclination is wrong – but if the inclination defines what I mean, it cannot be wrong by construction.

Kripke's sceptic concludes that the entire idea of meaning vanishes into thin air – a result Kripke himself calls intolerable.

Part 3: The Candidates and Why They Fall One by One

The middle part of Kripke's book systematically works through every plausible answer. It is worth following in detail, because the failure patterns recur.

Candidate 1: Dispositions

The most obvious proposal: the fact that I mean "plus" consists in my disposition to answer the question "What is x + y?" with the sum. This is the most naturalistically respectable answer, and it fails in three places:

The finitude problem. My dispositions are finite. For sufficiently large numbers I would die, fall asleep or lose track before answering. So the disposition does not single out the addition function, but at best a function that shifts into an undefined zone somewhere – which is structurally exactly what the sceptic claimed.

The error problem. I am disposed to miscalculate. If my dispositions define what I mean, then I mean a function that incorporates my mistakes – and then they are no longer mistakes. The very concept of an arithmetical error becomes impossible.

The normativity problem. Dispositions are descriptive. From "I would answer this way" it does not follow that "I should answer this way".

There are repair attempts. Simon Blackburn has proposed "infinitary dispositions"; other authors work with "normal conditions" – what I would do given unlimited time, paper and concentration. The standard criticism is that the normal conditions single out the right function only if, in formulating them, you already know which function is meant. The circle is then merely well hidden.

Candidate 2: Simplicity

"Addition is simpler than quaddition, so I meant addition."

That is not false, but it answers a different question. Simplicity is a criterion for the best explanation of someone else's behaviour. It turns my knowledge of what I myself mean into a hypothesis I form about myself. That is exactly what it is not: I do not infer what I mean; I know it immediately. Besides, "simple" is itself relative to a language of description – in a language in which quaddition is primitive, addition is the complicated function.

Candidate 3: Qualia, images, introspection

Perhaps there is a special experience that accompanies meaning "plus". Kripke objects: even if such an experience existed, the question would remain what connects that experience with the addition function. A headache singles out no mathematical function. The same holds for inner images: an image of a table has to be read, and every table is finite.

Candidate 4: A sui generis state

One could posit meaning as an independent, unanalysable state: there simply is the fact of meaning "plus", and it cannot be reduced to anything else. Kripke calls this move desperate – it names the explanandum and declares it to be the explanans.

Candidate 5: Platonism

Perhaps in understanding "plus" I grasp an abstract entity – a Fregean sense, the addition function itself. This solves extensionality elegantly: the function is infinite and unique. But it displaces the problem onto the relation of grasping. Which fact about me makes it true that I grasp this abstract entity and not another? Platonism supplies an object, but no connection to it.

Candidate 6: The machine

Perhaps the most instructive chapter for engineers: perhaps I am like a machine that embodies the addition function. Kripke replies that a real machine is a physical object with finite lifetime, finite memory and failure modes. That a machine "embodies" addition is not a physical fact about it, but our description of what we take it to be. A machine that overflows on sufficiently large inputs is physically indistinguishable from a quaddition machine – the difference lies in our specification, and it is precisely the meaning of that specification which is in dispute.

This is the point at which the argument stops being abstract. Anyone who has ever debugged an integer overflow knows the experience of a machine that "obviously" adds – right up to the point where it obviously does not.

Part 4: The Sceptical Solution

The Humean pattern

Kripke accepts that there is no meaning-fact in the sense sought. But he does not conclude that ascriptions of meaning are senseless. His model is David Hume: Hume denied that there is any observable necessary connection between cause and effect in the world – and explained our causal judgement instead from habit and the projection of an expectation. Talk of causes remains legitimate, but its justification lies somewhere other than one thought.

Such an answer is called a sceptical solution: it grants the sceptic his point and rescues the practice on a different foundation. The counterpart would be a straight solution, which rejects one of the sceptic's premises.

Kripke's proposal: instead of asking for the truth conditions of "Sven means addition by 'plus'", one asks for assertibility conditions – when is it warranted to say this, and what role does such talk play in our lives? The answer points to the community: we ascribe understanding of a rule to someone when his answers agree with ours in sufficiently many cases. This ascription has a tangible use – I can send someone to buy apples if I trust him to calculate correctly.

Relocating the private language argument

Kripke's second major thesis is exegetical but influential: the famous private language argument does not begin only at §243 of the Investigations, where the diary with the sensation "E" is discussed, but is already complete at §202. If correctness can be determined only through agreement with a community, then a subject isolated in principle cannot meaningfully distinguish between "I am following the rule" and "I believe I am following the rule" – and where that distinction collapses, there is no rule.

This turns a specialised discussion about sensation terms into a general argument about meaning as such.

The weak point

The sceptical solution has a well-known point of attack, and it is serious. It can be spelled out in two ways, and both are uncomfortable:

• As an error theory: all sentences of the form "X means Y" are false. Then it remains unexplained why it is useful to say systematically false things – the explanation of usefulness itself requires facts about norm compliance.
• As non-factualism: such sentences state no facts but have some other function. Paul Boghossian, Anandi Hattiangadi and Alexander Miller have raised the charge of incoherence against this: even a non-descriptive sentence has correctness conditions, and so the position presupposes exactly what it denies.

I am of the opinion that this is the strongest objection to Kripke's constructive part – and that the lasting achievement of the book therefore lies rather in the negative part: in the clean demonstration of where all the obvious answers break down.

Part 5: The Responses

The reactions to Kripke's book fall roughly into four camps.

A common denominator of the criticism: Kripke demands of a meaning-fact something no fact can deliver – namely to immunise itself against every reinterpretation. If one rejects that demand as excessive, the paradox loses its edge. The price is that one then has to explain what understanding is, without falling back on content in the head.

After more than forty years the debate counts as open. There is no standard solution that has prevailed.

Part 6: Why This Is Not a Seminar Problem

Here the argument becomes interesting for everyone who works with specifications, models and rule sets.

Machine learning: the same problem, made measurable

A model sees finitely many examples and is supposed to "learn" from them a function defined on infinitely many unseen inputs. Structurally this is exactly the situation of Kripke's sceptic: the training data radically underdetermine the target function.

Learning theory has not only described this problem but made it precise. David Wolpert showed in The Lack of a Priori Distinctions Between Learning Algorithms (Neural Computation 8(7), 1996, 1341–1390) that, averaged over all possible target functions and measured on off-training-set error, no learning algorithm is a priori better than any other. For any two algorithms A and B there are as many target functions on which A does better as the reverse – and this holds even for cross-validation against "anti-cross-validation".

The consequence is the direct technical counterpart to Wittgenstein's point: generalisation is impossible without bias. A learning procedure works not because it derives the right continuation from the data – it cannot – but because its architecture, its regulariser and its optimisation prefer certain continuations. Inductive bias is not a defect to be eliminated, but the condition of anything being learned at all.

Anyone who wants to know how a concrete system arrives at its continuations thereby ends up at the question of internal representations – the subject of The Ghost in the Machine: How to Read a Neural Network From the Inside. And anyone who wants to know how much bias can be replaced by sheer data ends up at the scaling laws of How Big Is Big Enough? Scaling Laws, Chinchilla, and the Measurement of AI.

I am of the opinion that the rule-following paradox is the cleanest available description of why out-of-distribution behaviour is not a bug that can be trained away. A model that behaves unexpectedly outside its training range has not "forgotten the rule". It never had the rule – it had a hypothesis class and a preference within it.

Specifications and compliance

The same structure appears wherever a finite prescription is meant to govern infinitely many cases:

• Requirements. A specification never fully determines an implementation. What it does is constrain the space of admissible implementations. The common practice of settling disputed points by examples is an admission: examples narrow, but they do not close.
• Legal norms. Lawyers have known the problem for a long time under a different name. That a statutory element has to be interpreted, and that the interpretation does not follow from the text itself, is the normal case, not the exception. The solution is institutional – appellate hierarchy, precedent, prevailing opinion – that is, a community solution in Kripke's sense.
• Code reviews. Why are style guides never sufficient? Because as text they are finite, and every new construction raises a question of interpretation. What actually carries a team is an established practice that new members grow into – which in turn explains why a new colleague can read the style guide and still need months.
• AI alignment. The idea that a target behaviour can be fully encoded in an objective function is precisely the idea the paradox attacks. Every objective function is finitely specified and leaves infinitely many continuations open; "specification gaming" is the empirical name for a continuation that is compatible with the specification but was not meant.

The practical lesson

None of this entails defeatism, but it does change what one should expect. Anyone who believes that a sufficiently precise rule excludes misinterpretation will be surprised again and again. Anyone who accepts that rules leave room for interpretation invests instead in the things that actually carry: feedback, correcting instances, shared practice, explicit edge cases – and observation of behaviour precisely where previous experience runs out.

Part 7: A Map of the Positions

The Central Takeaway

The rule-following paradox has a practical point that is surprisingly concrete: a rule does not contain its own application.

What a rule achieves is not to fix the future, but to constrain the space of admissible continuations – and even that only relative to a shared practice of reading. What closes the gap is never the text, but always something external: training, correction, community, institutionalised decision or, in the case of machines, built-in inductive bias.

For practice this means three things:

1. Do not invest in completeness, invest in feedback. There is no specification that "covers everything". There is a process that makes misinterpretation visible early.
2. Treat the region beyond experience as a question in its own right. With people this means edge-case training; with models it means out-of-distribution evaluation. In both cases it is precisely the region about which past behaviour says nothing.
3. Make the practice explicit, not only the rule. Teams, courts and standards bodies do not solve the problem with better texts, but with procedures that make interpretations binding.

Once you see this pattern, you recognise it in a great many disputes – and you spare yourself the attempt to end them by sharpening the wording.

A Closing Question for Reflection

Think of a rule you follow professionally and consider unambiguous – a coding standard, a compliance requirement, an architectural principle.

Find the case in which two competent, well-meaning colleagues would apply it differently. That case almost certainly exists; the only question is whether you have encountered it yet.

And then the real question: on what basis would you decide which of the two applications is the right one – if not on the wording? Your answer to that question is your personal position on the rule-following paradox. And it says more about how your organisation actually works than any rule book does.

Cross-References in the Vault

• Why the Sun Will Rise Tomorrow: Hume's Problem of Induction, Grue, and the Limits of Learning – the closest relative. Goodman's "grue" is the inductive twin of "quus": finite evidence, infinitely many compatible continuations. And Kripke's sceptical solution is explicitly modelled on Hume.
• Three Pages Against 2,000 Years: The Gettier Problem and What Knowledge Really Is – the same methodological pattern: a short text, a constructed counterexample, and a century-old consensus wobbles.
• The Chinese Room: Searle and the Question of Whether Machines Can Understand – Searle asks whether rule-following suffices for understanding. Kripke asks the prior question of whether rule-following is itself well defined.
• The Ghost in the Machine: How to Read a Neural Network From the Inside – the attempt to actually locate the "rule in the head" of a machine instead of presupposing it.
• How Big Is Big Enough? Scaling Laws, Chinchilla, and the Measurement of AI – how much underdetermination can be compensated by data and parameters, and where the limit lies.
• The Error That Learns Backward: Backpropagation, the Chain Rule, and the Algorithm Behind Deep Learning – the mechanism through which inductive bias concretely takes effect.
• Will I Still Be Me Tomorrow? Personal Identity from the Ship of Theseus to the Teletransporter – a related case of a question that no inner matter of fact settles.
• What Mary Did Not Know: The Knowledge Argument and the Riddle of Consciousness – the counter-test: an argument that rests precisely on a non-public, purely inner fact.

Sources

• Miller, A., Wright, C. et al.: Rule-Following and Intentionality. Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/rule-following/
• Wittgenstein, L. (1953): Philosophical Investigations. Blackwell, Oxford (posthumous). Relevant: §§138–242, especially §201 and §202. German text available at the Ludwig Wittgenstein Project: https://www.wittgensteinproject.org/w/index.php/Philosophische_Untersuchungen
• Kripke, S. A. (1982): Wittgenstein on Rules and Private Language. An Elementary Exposition. Harvard University Press, Cambridge (MA), 150 pp., ISBN 0-674-95401-7. https://philpapers.org/rec/KRIWOR
• Baker, G. P., Hacker, P. M. S. (1984): Scepticism, Rules and Language. Basil Blackwell, Oxford, xiii + 140 pp. https://philpapers.org/rec/BAKSRA
• McDowell, J. (1984): Wittgenstein on Following a Rule. Synthese 58, 325–363. https://doi.org/10.1007/BF00485246
• Boghossian, P. A. (1989): The Rule-Following Considerations. Mind 98(392), 507–549. https://philpapers.org/rec/BOGTRC
• Soames, S. (1997): Skepticism about Meaning: Indeterminacy, Normativity, and the Rule-Following Paradox. Canadian Journal of Philosophy, Supplementary Volume 23, 211–249.
• Wright, C. (2001): Rails to Infinity. Essays on Themes from Wittgenstein's Philosophical Investigations. Harvard University Press, Cambridge (MA).
• Wolpert, D. H. (1996): The Lack of a Priori Distinctions Between Learning Algorithms. Neural Computation 8(7), 1341–1390. https://doi.org/10.1162/neco.1996.8.7.1341

Note: The rule-following paradox is an open philosophical debate; none of the positions presented counts as decisively established, and even the question of whether Kripke's reading captures Wittgenstein is contested – Kripke himself presented it only as "Wittgenstein as the argument struck him". What is reliably documented are the bibliographical details and the learning-theoretic result cited (Wolpert 1996), which is a mathematical theorem under precise assumptions – in particular averaging over all target functions and measuring error off the training set – and therefore does not mean that all algorithms perform equally well in practice. Where this article goes beyond the established state of knowledge, this is marked with "I am of the opinion that ...".

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