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If I understand Wittgensteins Philosophical Investigations correctly, it is not possible to follow a rule in private. First of all, I am not quite sure if my understanding of rule-following is correct but wouldn't it be possible to create a game and play it in private and afterwards explain it?

What do I mean by that:

Say I create a "semi-digital" dice game. I have a little computer which is almost invisible and I stick it into my ear. The computer is controlled purely with my mind. Every time I say "Stop" in my mind the computer will tell me a number. The algorithm is not known to me and I have to find by skill the best timespan between each "Stop" so that I get the highest number to get as fast as possible to the number 301.

While saying "Stop" in my mind I write down the numbers which the computer tells me.

From an outsiders perspective, I seemingly have no rule that I follow. The "Stop" doesn't correlate to a specific number and I look like I am writing down random numbers. Even if a person could hear me say "Stop" the algorithm would look like this:

Say Stop 
Write number 1 
Say Stop  
writer number 10 
Say Stop 
Write number 8 
Say Stop 
Writer number 12

I must seem to follow no rule. A person who watches me could not know the rule that I am following and so would say, that I am not following any rule.

If I explain to that person afterwards what my rule was, the person would normally agree that I was following a rule. So in retrospective, I was following a private rule.

How would Wittgenstein argue against such an idea of private-rule-following or is my idea of rule-following totally wrong?

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    See Wittgenstein : Rule-following and Private Language : W's critique is aimed at defeating the point of view about meaning as an "abstract" entity. In the same way, alos rules are not abstract entities, transcending their particular applications. This is the gist of W's concept as Language games an of "meaning is use". – Mauro ALLEGRANZA Mar 18 at 13:39
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    "Directly following the rule-following sections in PI, and therefore easily thought to be the upshot of the discussion, are those sections called by interpreters “the private-language argument”. Whether it be a veritable argument or not (and Wittgenstein never labeled it as such), these sections point out that for an utterance to be meaningful it must be possible in principle to subject it to public standards and criteria of correctness. 1/2 – Mauro ALLEGRANZA Mar 18 at 13:40
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    For this reason, a private-language, in which “words … are to refer to what only the speaker can know—to his immediate private sensations …” (PI 243), is not a genuine, meaningful, rule-governed language. The signs in language can only function when there is a possibility of judging the correctness of their use, “so the use of [a] word stands in need of a justification which everybody understands” (PI 261). 2/2 – Mauro ALLEGRANZA Mar 18 at 13:40
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    If so, your example is not a counter-example, because " in principle it is possible to subject it to public standards and criteria of correctness." – Mauro ALLEGRANZA Mar 18 at 13:41
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The interpretation of Wittgenstein on rule-following is by no means straightforward. It will probably be most helpful if I present Norman Malcolm's version since this offers the or 'a' - standard case against private rule-following. The basic idea is that rule-following is inherently public since without public scrutiny, correction and control there is no way in which one can tell (certainly the putative private rule-follower can't) whether s/he is really following a particular rule or merely thinks they are :

The standard account - Norman Malcolm

In 'Wittgenstein on Language and Rules', Norman Malcolm argues that following a rule must be, and according to Wittgenstein is, a social practice. This social practice requires 'a human community in which there is agreement as to whether doing such-and-such is or is not following a particular rule' (p. 5). Malcolm contends, against the individualist position of Gordon Baker and P. M. S. Hacker, that a person's following a rule in complete separation from other human beings for his whole life is impossible. Malcolm also contends that this was Wittgenstein's own view. ... My main concern is Malcolm's argument for the view that rule-following requires a social community.

Let us begin with Malcolm's interpretive contention. He holds that Wittgenstein's remarks on solitary language users concern only individuals who were trained in a linguistic community. Malcolm denies that Wittgenstein ever suggested the possibility of a 'forever solitary person with a language', i.e., someone who follows linguistic rules but has not been trained to do so by the members of a linguistic community. Malcolm contends that Wittgenstein's solitary rule-followers all share a key feature of Defoe's Robinson Crusoe: they have acquired language via training in a social context. On this reading, Wittgenstein holds that all language must be socially established, but grants that Crusoe can speak a socially established language to himself.

...

Malcolm believes PI 202 supports his contention that social training is essential to rule-following and language. The presumed support comes from 'the necessary distinction' between one's following a rule and one's thinking one is following a rule. Malcolm explains: If you try to imagine someone who had never participated in human society, inventing a rule for himself and undertaking to follow it, you will realize that there would be no foothold there for that necessary distinction. . .. There must be a use of a sign that is independent of what an individual speaker does with it, in order for the latter's use of the sign to be correct or incorrect' (p. 28).

Social training is necessary, on Malcolm's account, because it provides the independent basis for correctness and incorrectness of language use. More specifically, 'what a rule requires and what following it is, presupposes the background of a social setting in which there is quiet agreement as to what "going on in the same way" is; [and] this is an agreement in acting, not in opinions' (p. 21). In sum, then, social training is necessary because it provides the background of 'agreement in acting', which provides the independent basis for the 'necessary distinction' between correctness and incorrectness of language use. We now have Malcolm's answer to my crucial question raised above.

Problem with the standard account

But his answer faces a troublesome question: Why should we think that 'agreement in acting' with respect to the actions of a group of people can provide for the distinction between correctness and incorrectness, whereas 'agreement in acting' with respect to various actions of an individual cannot? This is the crucial question that Malcolm must answer, but does not answer.

My own view is that I can see how it is impossible for a private rule-follower to know whether they are really following a rule or merely (mistakenly) thinking they are. But I see no incorrigibility in a public dimension unless one takes an extreme conventionalist position that the rule and following it just are what other language users decide or judge that it is, so that (from this perspective) they cannot be mistaken.

References

Norman Malcolm, 'Wittgenstein on Language and Rules', Philosophy 64 (1989), 5-28.

Gordon Baker & P.M.S. Hacker, 2 Wittgenstein: Rules, Grammar and Necessity (Oxford: Blackwell, 1986), chapter 4.

Paul K. Moser, 'Malcolm on Wittgenstein on Rules', Philosophy, Vol. 66, No. 255 (Jan., 1991), pp. 101-105.

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