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Wittgenstein writes in his Philosophical Investigations in paragraph 270:

Let us imagine a use for the entry of the sign "S" in my diary. I discover that whenever I have a particular sensation a manometer shows that my blood-pressure rises. So I shall be able to say that my blood-pressure rises without using any apparatus. This is a useful result.

This seems okay to me. Surely we can establish a correlation (if it exists, and otherwise prove that there is no correlation) between sensation "S" and a rise in blood-pressure. Moreover, it is very well imaginable that if I sense "S" I can prevent better some of the nasty and potentially dangerous side effects of high blood-pressure by taking some medication more timely. In this manner the sensation is useful, beyond the mere information of having high blood-pressure, something you could not care less for if without consequence. But now Wittgenstein continues...

And now it seems quite indifferent whether I have recognized the sensation right or not. Let us suppose I regularly identify it wrong, it does not matter in the least. And that alone shows that the hypothesis that I make a mistake is mere show.

I do not understand how this continuation follows from his preceding remarks. It is rather counterintuitive and calls for an explanation I think. Naturally, if I am not able to distinguish between a multitude of sensations, how can I then possibly observe a correlation (except in the case that all those sensations indicate high blood-pressure, but then it is still useful, since obviously we do not always have high blood pressure)?

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Wittgenstein is not denying that the correlation is useful, he acknowledges as much in the first passage. What he means is that the correlation is established in a way that does not allow for any independent checking of sensation's authenticity. What way is there to tell if the sensation is the 'right one' other than observing the rising blood pressure? We only have sensation's 'word' for it. With a truly private diary the "S" is all there is, by construction there is no possibility of mistakes. And if there is no rising blood pressure then it is not that we identified the sensations 'wrong', there is simply an imperfect correlation or no correlation at all. We can have "S" publicly linked to rising blood pressure with a possibility of mistakes, or we can have private "S" without it, but we can not have the same "S" be both.

This is why in the end he says:"And that alone shows that the hypothesis that I make a mistake is mere show. (We as it were turned a knob which looked as if it could be used to turn on some part of the machine; but it was a mere ornament, not connected with the mechanism at all)". Another example he gives, in PI 266, is that it is pointless to buy several copies of the same newspaper to assure yourself that what it says is true, "justification consists in appealing to something independent" (PI 265).

See Fogelin's commentary in Taking Wittgenstein at His Word and Did Wittgenstein consider the possibility of a private language with public content? for more on the private language argument.

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    "What way is there to tell if the sensation is the 'right one' other than observing the rising blood pressure?" — beautifully put! – nir Feb 11 '16 at 22:29
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    btw, it seems to me that the case of writing the sign "S" in a diary for every day on which Wittgenstein has the sensation (PI, §258) is analogous to modern day talk of neural correlates. Is it not? – nir Feb 12 '16 at 8:20
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    I am not convinced. Wittgenstein says: "And now it seems quite indifferent whether I have recognized the sensation right or not." But that seem untrue to me. Of course I cannot be sure if I made a mistake in recognizing the sensation. Also, I can significantly lower the probability of this possibility when checking my blood-pressure explicitly (with a manometer) if the correlation has already been established, but this --- as you say --- would kill the need for the symbol "S". However, to say that the recognition has no influence whatsoever seems unjust. – Jori Feb 12 '16 at 14:59
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    If we have established a correlation, which we surely can, and I act on this correlation by taking medication, this has a drastic effect. Because if I would have recognized it wrongly the medication might have an reversed --- possibly dangerous --- effect. The possibility of mistake arises here because we have the potential the verify the sensations authenticity in this case. Surely for a sensation that has no interesting correlation with anything no such possibility of mistake exists, but then the whole idea of noting it down does does not make sense. What would it benefit? We live induction. – Jori Feb 12 '16 at 15:04
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    Or is that what Wittgenstein is saying? But then the manometer example would be poorly chosen as it does give a potential for verification... unless Wittgenstein understand this to be part of a public language. A sensation that can be expressed as: "the feeling I have whenever my blood-pressure rises". This would not be intelligible to anyone who does not know this feeling, but there are more of such of course (not everyone has been stoned in their lives, yet they can speak intelligible about "stonedness" and effects of drug use). But what if you are the only person... – Jori Feb 12 '16 at 16:02

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