The rule-following paradox : where is it?

The example mostly given to show evidence of this paradox is the following: assume you've never added any numbers greater than 50, how do you know that 60 +74 = 134 and not 26 or even "blue" for that matter ? But I find myself absolutely unconvinced by this example for two reasons: first of all it's just an example, and I've been struggling to find some sort of general instance of this paradox; moreover, it seems to suggest that a rule can only be explained through the use of previous applications of that rule, which is a claim I find unconvincing as well, as I haven't seen any justification for it.

Could anyone explain to me how these issues are dealt with, and why (if there is a why) this is considered a paradox ?

• Can you supply references for where you are getting this? I believe this in Philosophical Investigations but it's been a while (you can edit your question to do so -- also you can make it look prettier on your readers by adding two CRs between paragraphs so they show up correctly). – virmaior Dec 30 '16 at 10:19
• There aren't any particular references, to be honest, I just found this "paradox" mentionned somewhere and then looked it up on Google to find this. As Google isn't usually very good at explaining things, their version of the paradox wasn't compelling at all, so I asked here to get more details and see why it is actually (if it is) a paradox; although I can't really see I'm satisfied with the answers – Maxtimax Dec 30 '16 at 12:21

You know how to apply the rule of addition because it is an algorithm spelled out beforehand. This is not the kind of rule Wittgenstein had in mind.

So Kripke's example is a bit silly. Kripke's claim is that "That there is no fact about your past usage of the addition function that determines you have the right answer." But the rule is stated. You do have evidence that you are meant to use the ordinary rules of addition in this case. You have been told they are correct by authorities in your culture. You know when you are following the rules because you have been inaugurated into the rules with extensive operant conditioning.

A better example is how children acquire grammar. No one sits them down and states the rules, because the rules would have to be stated in some language, and the child is just beginning to acquire their first language.

Wittgenstein used this acquisition of first language as a general model for all learned meaning. He had in mind that the rules of a culture, a science, or a philosophy are really the rules of a 'language game', which are not expressed beforehand by anyone, and are never clearly expressible by anyone, because they are subject to continual change. He imagines that people decide what rules they like and converge together on the right answers, praising the people who guess the 'right' answer first or more often. Then the actual statements of cultural, scientific or philosophical thoughts are mere approximation to this more genuine set of rules no one can formulate completely.

The paradox is that there is no way to discern a rule by simple observation when you cannot state them clearly. In that case, you could never really learn language without Chomsky's 'built-in grammar instinct' that already knows the limits on what is possible valid grammar because grammar itself is evolved to have boundaries.

Kripke says Wittgenstein is wrong (and Chomsky closer to correct), because there is no real way to learn or agree on a broad rule with a finite number of examples, unless you state it. Wittgenstein imagines, for instance, that grammar should be learned by examples alone, and Kripke would protest that no matter how long you spoke a language, unless you actually stated a rule of grammar, you would never be truly sure you are right.

But people have languages with no recorded grammar, and their children do acquire them. At the same time, folks doubt Chomsky, now that we know dogs learn complex language rules like the process of elimination, and apes can learn sign-language. Kripke's argument that it is not possible is logical, but does not seem to matter. There has to be a way learning rules works, that is not just about passively collecting examples.

The way out of this, to my mind is Popper. You can generally be really sure when you are wrong, and you can learn enough about the world from how seriously you are corrected that you can come to follow the rule almost exactly after relatively few tries. This is verified by interactive simulations of evolution that led us to discover genetic algorithms. A good learning algorithm can converge on something very close to a goal after relatively few clear corrections or competitive pairings.

So I would consider Wittgenstein vindicated on this one. It does not matter than induction or abduction does not really work, competition still allows for ways to negotiate rules over time with arbitrary accuracy.

Perhaps it would be of some use to compare the situation we find ourselves in with respect to rules to where we find ourselves when we need to use a map, say a physical map rather than an electronic one. The map may say 'you are here,' but the act of locating oneself on a map necessitates nevertheless something outside of the map itself. So the map is incomplete. Now if the map were complete, it could not be consistent. The same is so for rules, and we see this playing out with Godel's second incompleteness theorem with respect to Hilbert's formalist efforts at trying to ground out the foundations of arithmetic.

• Yes but again, you're just giving an example, and not actually giving the general instance of the paradox, and my issue was that there seemed to be none. – Maxtimax Dec 27 '16 at 10:47
• By the way, a map can be complete and consistent, you shouldn't try to give Gödel's theorem more meaning than it actually has – Maxtimax Dec 27 '16 at 10:48
• @Maxtimax: to give a "general instance" (whatever that means) would be to give a rule. So it should not come as a surprise that there is no such thing. Compare "the only rule is that there are no rules". – user20153 Dec 28 '16 at 6:21
• There is, to my mind, a huge difference between giving a general instance and a rule as meant in the alleged paradox. A general instance isn't something you have to follow, it's simply a conceptualization of a precise example, such as that of addition. – Maxtimax Dec 28 '16 at 10:54

Numerical sequences provide good material for examples because each term of the sequence is usually thought too follow in accordance to a rule. For example, the sequence 2, 4, 6, 8, is likely to be the even natural numbers, and the next term would be 10 according to the rule f(n+1) = f(n) + 2. However, there are other possible rules that could determine some value other than 10:

• Numbers n such that 6^n - 7 is prime gives the sequence: 2, 4, 6, 8, 9...
• The Beatty sequence for sqrt(5) consists of the sequence: 2, 4, 6, 8, 11...
• Numbers whose binary representation has exactly 2 runs: 2, 4, 6, 8, 12...

However, what is important here is not so much that we can come up with such examples, but that they indicate that there is more to the meaning of rules than simply descriptive conformity. Given that Wittgenstein's theory of meaning was based on the idea of community agreement, he considered the nature of rules paradoxical because they seemed to indicate that there is more to meaning than could be easily explained by the agreement of associations. In the case of simple associations, such relations become reinforced by custom and habit. However, in the case of rule, there seems to be an inner principle that determines meaning in a much more definitive manner. As Saul Kripke emphasized, the nature of meaning is normative rather than merely descriptive. He clarified this point by comparing the weaker nature of the descriptive association with that of the Humean notion of causation, as explained here:

"What Kripke means by this comparison with a Humean problem is that Wittgenstein is questioning the nexus between a past act of meaning and subsequent practice in a way analogous to that in which Hume questions the causal nexus between a single past event and a subsequent one. And what he means by a Humean solution is that there is a corresponding analogy between the ways in which Hume and Wittgenstein handle their respective problems. Kripke's view is that, just as Hume says, in effect, 'Unique causation is unintelligible; A's causing B consists in A's being embedded in a pattern of the following of A-type-events by B-type-events which inclines us to assert that A caused B', so, comparably, Wittgenstein says, in effect, 'Unique meaning is unintelligible; rather, someone's meaning the usual addition function when saying "plus" consists in their being considered by a community to have passed the community's test for employing that function.'" ("Private Language", SEP)