We should first clarify the meaning of “interpretation”. Suppose I’m traveling from Berkeley to Timbuktu. At some point I no longer know the way, but I see a signpost reading, “Timbuktu → ”. The signpost expresses a rule – that is, the signpost is an expression of a rule, namely a rule regarding how to get to Timbuktu. I see that sign and think, “Ah, I ought to proceed East to Timbuktu.” This thought, which represents what I take the signpost to be expressing, constitutes my “interpretation” of the rule expressed by the signpost. If I had seen the sign and thought “I ought to proceed West to Timbuktu”, that thought would also constitute an interpretation of the expressed rule.
The paradox is: no course of action can be determined by a rule because every course of action can be brought into accord with the rule (87). In what way is every action capable of according with a rule? Consider a teacher expressing a rule (regarding a particular series) to his pupil: “Add two each time.” The pupil proceeds: 2,4,…,998,1000 . We would say that he is following or acting in accord with the rule. But then the pupil proceeds: 1004,1008,… . The teacher sees that the pupil must not understand, though the pupil cannot be made to see that he was not “adding two each time”: he maintains he was following the rule expressed. We might think that the pupil understands the order as “Add 2 up to 1000, 4 up 2000…” and so on; that is, his actions are still governed by his interpretation of the rule expressed. He did not follow the rule (that the teacher expressed) – even for the first 500 terms – but his actions for the first 500 terms were in accord with that rule. And his actions after the 500th term were in accordance with his [the pupil’s] interpretation of the rule “add two each time” (81). In §198, Wittgenstein’s interlocuter says, “…whatever I do can, on some interpretation, be made compatible with the rule.” This is like how the pupil failed to see his own misunderstanding the order – on his interpretation of the sign (expression of a rule) “Add two each time”, his actions were compatible with the rule.
If any action, on some interpretation, is in accord with the rule, then no rule can determine a course of action. For then any course of action is acceptable by some interpretation of the rule, so no particular course of action is determined by the rule. Wittgenstein responds to the paradox, “if every course of action can be [compatible] with the rule, then it can also be [incompatible] with it. And so there would be neither accord nor conflict here.” (87) This is to say that there is no fact of the matter as to whether a course of action is compatible with a rule, because there exist interpretations of the rule which conflict with the action and those which do not.
That is, here is an expression of a rule: “Add two each time”. The pupil’s interpretation may be “ f(x)=x+2 ”. But this, too, is a sign to be interpreted. And based on the pupil’s behavior, we can say that the pupil interprets the expression “ f(x)=x+2 ” as “ ∀ x( x<1000 → f(x)=x+2 ) ∧ ∀ x( x ≥ 1000 → f(x)=x+4 ) ”. But now how are we to say the pupil interprets this sign? Prima facie, it looks like we need a rule which tells us how an expression of any given rule is to be interpreted. But this cannot be possible, for it leads to regress. Why? Because a rule saying how an expression of rule is to be interpreted must itself be expressed and interpreted. We would need a rule which says how to interpret that rule, and so on and so forth, never bottoming out.
Wittgenstein asserts that there is a “way of grasping a rule which is not an interpretation” (87). This must be so, lest we end up with the regress problem. Recall §154, where Wittgenstein argues that understanding should not be thought of as a mental process. To say, “Now I understand the series” is not to say “the formula occurs to me” – where the formula occuring is something like the interpretation of the rule expressed by the series. We say “Now I understand” when we can continue the series correctly. So when we say that “the pupil understands the rule,” we are saying that has the ability to apply it correctly. What does it mean to apply a rule correctly? From §201, this way of grasping a rule is “exhibited in what we call `following the rule’ and `going against it’” in each particular case with its particular circumstances (87). This means that the correct application of a rule does not have to do with an occurrence or given interpretation in one’s head. The deviant pupil did not grasp the rule, but not because his interpretation of the rule differs from ours. Rather, his actions did not conform to what we call “following the rule”. His actions deviated from the actions the “add two each time” order is supposed to provoke in this kind of circumstance (given the effects the teacher was trying to produce, what sign was used, etc.).
There is an inclination to say that every action according to a rule is an interpretation (87) – this is why we concocted elaborate formulas in the deviant pupil’s head to account for his misunderstanding. But this isn’t right, for understanding isn’t “in the head”, so to speak. Actions either follow the rule or go against it – this being judged externally, case by case – but actions themselves are not interpretations (though it seems they can be interpreted). Indeed, Wittgenstein says, “one should speak of interpretation only when one expression of a rule is substituted for another.” (87) That is, an interpretation is an expression of a rule; the substitution of one expression for another constitutes an interpretation of the original expression. An interpretation is not to be confused with a rule, nor is it to be confused with a given action.