It seems that there is a problem with the example that Saul Kripke gives in "Wittgenstein on Rules and Private Language" to explain Wittgenstein's rule-following paradox. I'm not asking about the validity of the rule-following paradox itself, but rather the validity of Kripke's example. This is the example that Kripke gives (paraphrased):

Suppose someone is completing a series of math problems. There are problems such as 4+2=x, 15+30=y, and 56+56=z, to which this someone gives answers of x=6, y=45, and z=112. Now, they come across the problems 57+1=w and 58+58=v. This time, they answer that w=5 and v=5. As it turns out, they were actually using a function different from colloquial addition the entire time. For sake of example, Kripke calls it the 'quus' function, as opposed to the 'plus' function.

The quus function operates as such: (a quus b) = (a plus b), if a & b < 57. (a quus b) = 5 otherwise. If a number equal to or greater than 57 is being operated on by quus, then the value returned is always 5.

Now, the general rule-following paradox as I understand it states that it is impossible to determine whether or not someone is following a specific rule without testing out every instance in which they could deviate from said rule. If I were to say "Raj is following rule x," it would be impossible for me to determine the truth-value of my statement according to a correspondence theory of truth. Now, I don't intend to argue against the rule-following paradox itself with this post, but I have some confusion with Kripke's example.

(Here I'm going to use + for quus) a + b = 5 for a >= 57 or b>= 57. For this case, let's take a = 57. So we have 5 - b = 57 where b is any number. Next, let's look at 5 + 3 = 8, still using '+' for the quus function here. This can be rewritten as 5 - 3 + 6 = 8. Since 5 - b = 57 for any b, we can rewrite this as 57 + 6 = 8. From this, any equation given that involves quusing together some number with 8 by the form a + 8 = x must have x = 5. Obviously this extends to other numbers besides 8 as well.

To give another example, with the quus function, 58 + 2 = 5 should be true. However, since we know that we can arrive at values greater than or equal to 57 with quus, this can be rewritten as 56 + 2 + 2 = 60, since we are no longer quusing with a value a >= 57. Someone using the quus function cannot produce consistent results for their function. Therefore, Kripke's example for the rule-following paradox is not consistent, and furthermore doesn't actually demonstrate the rule-following paradox, since I could determine someone's use of quus with seemingly any problem.

Am I completely wrong in this assertion? These are possible errors I think I might be making:

Assuming that quus subtraction works the same way that everyday subtraction does is incorrect. The paradox is still followed, since an outside party wouldn't know that 57 is the dropoff point between quus and plus (this seems contingent on the subtraction point, since even to an outsider things like 2 + 3 would equal both 5 and n=[57,inf)). The potential for someone to use an inconsistent quus function is simply another example of the paradox. I totally misunderstand what the rule-following paradox actually is.

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    Yes you answered your own question, you're introducing a new rule with the subtraction and you are in the same skeptical trap of not knowing whether or not you are using real subtraction. Your argument only goes through if you assume without trouble that you are using real subtraction, and the skeptic would press you to explain how you know that you are.
    – Not_Here
    Commented Mar 14, 2019 at 0:58

1 Answer 1


The main problem with your suggestion is not philosophical but mathematical. Let's denote quus by # and plus by +. Even without any skeptical thesis, you simply cannot move from

57 # b = 5


57 = 5 - b.

With plus, such a move is made by subtracting b from both sides. For instance, going from a+b=c to (a+b)-b=c-b to a=c-b. But such a move is valid only because (a+b)-b equals a. With quus, as a matter of math and not philosophy, you cannot make such an inference: (a#b)-b does not always equal a. For instance, (57#1)-1 does not equal 57 but 4, since 57#1=5 and 5-1=4.

Another problem, of course, is that the skeptic will just turn their skeptical scenarios towards your newly introduced signs: - and =. Kripke discusses such moves in the book. But, again, the main problem with your argument is the mathematical one described above.

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