Wittgenstein On Rules and Private Language - The Rule-following Paradox

The Rule-following Paradox

In PI 201a Wittgenstein explicitly states the rule-following paradox: "This was our paradox: no course of action could be determined by a rule, because any course of action can be made out to accord with the rule". Kripke gives a mathematical example to illustrate the reasoning that leads to this conclusion. Suppose that you have never added numbers greater than 50 before. Further, suppose that you are asked to perform the computation '68 + 57'. Our natural inclination is that you will apply the addition function as you have before, and calculate that the correct answer is '125'. But now imagine that a bizarre skeptic comes along and argues:

1. That there is no fact about your past usage of the addition function that determines '125' as the right answer.
2. That nothing justifies you in giving this answer rather than another.

After all, the skeptic reasons, by hypothesis you have never added numbers greater than 50 before. It is perfectly consistent with your previous use of 'plus' that you actually meant it to mean the 'quus' function, defined as:

'x quus y' = x + y if x, y < 57, and = 5 otherwise

The skeptic argues that there is no fact about you that determines that you ought to answer '125' rather than '5'. Your past usage of the addition function is susceptible to an infinite number of different quus-like interpretations. It appears that every new application of 'plus', rather than being governed by a strict, unambiguous rule, is actually a leap in the dark.

The obvious objection to this procedure is that the addition function is not defined by a number of examples, but by a general rule or algorithm. But then the algorithm itself will contain terms that are susceptible to different and incompatible interpretations, and the skeptical problem simply resurfaces at a higher level. In short, rules for interpreting rules provide no help, because they themselves can be interpreted in different ways. Or, as Wittgenstein himself puts it, "any interpretation still hangs in the air along with what it interprets, and cannot give it any support. Interpretations by themselves do not determine meaning" (PI 198a).

Similar skeptical reasoning can be applied to any word of any human language. The power of Kripke's example is that in mathematics the rules for the use of expressions appear to be defined clearly for an infinite number of cases. Kripke doesn't question the validity in mathematics of the '+' function, but rather the meta-linguistic usage of 'plus': what fact can we point to that shows that 'plus' refers to the mathematical function '+'.