In much of the philosophy of language and mind coming out of the late Wittgenstein and/or early Heidegger, a distinction is made between merely following a norm versus also being able to correctly assess whether others are following that norm. Note that the Brandom of "Dasein, the Being that Thematizes" (in Tales of the Mighty Dead) and the Mark Okrent of "On Layer Cakes" both mark this distinction, though they disagree on whether the latter ability requires language. Okrent (whose objects that Brandom's view entails that human aphaisics and non-linguistic deaf adults have no minds) writes:
Because all tool use is embedded in a context of instrumental rationality, there is more to using a hammer correctly than using it as others do. Sometimes it is possible to use a hammer better than the others do, even if no one else has ever done it in that way, and no one else recognizes that one is doing so, because the norm that defines this use as ‘better’ is independent of what is actually recognized within the community. That norm is the norm of instrumental rationality: it is good to do that which would achieve one’s ends most completely and most efficiently, were anyone to do it in that way. For the same reason, it is sometimes possible for a member of a society to improve a hammer, or repair it, by giving it a structure that no hammer has previously had in that society.
Okrent continues:
When an agent acts so as to improve or repair a tool she of course does so in light of an understanding of what that tool is for, its equipmental role. And making those improvements amounts to appealing directly to that role in the face of social pressure to use the previously constituted tool as it is to be used, according to community standards. That is, improvement has the structure of appealing to the thing itself in the face of what has already been articulated socially regarding the structure of the equipmental role of the tool.
While I agree with Okrent, I worry about the common presupposition that the ability of an individual to detect faulty behavior according to some set of norms (here involving the telos of hammering) is a more difficult accomplishment than the ability to follow those norms.
For some sentences in mathematical logic (those involving the claim that a given proof is valid) it is provably the other way around. There are algorithms for generating proofs in propositional logic as well as algorithms for checking whether the proofs are correct. If you have the former (equivalent to the recursivity the language's logical truths) you have the latter (which is equivalent to the recursive enumerability of those truths). But not the other way around. Notably, for first-order logic you have the latter without the former. There is an algorithm to check whether first-order proofs are correct, but not one to discover such proofs.
To better appreciate this, consider two character types which (in addition to the scold, the huckster, and the psychopath) are discussed in a paper that Mark Ohm and I just presented at the 2014 International Society for the Study of Narrative, though what I'm saying here isn't in the paper. The beautiful soul is full of empathy in a way that gives her a wonderful ability to intuit the virtuous thing to do in a wide variety of situations. But this ability alone is consistent with her being a Dostoevskyan idiot, unable to see the malice in other people, unable to correct others when they do the wrong thing. On the other hand the hypocrite is a master at picking out other people's moral flaws, yet disastrously bad at following the moral norms herself.
If this is at all analogous to the relevant sentences about proofs in mathematical logic, then it's much easier to be a hypocrite than a beautiful soul, and maybe impossible to be a beautiful soul as I've described it. On the other hand, I do think it's possible to be a beautiful soul. I think there might be human mathematical analogues too. I know that Krisiel was reputed to be great at discovering theorems (supposedly he would just conjecture away at graduate students and he was right about 80% of the time), but not so good at proving them. One can imagine someone very good at discovering theorems who is also very bad at recognizing correct or incorrect proofs. Would Fermat have recognized the proof of the theorem named after him?
I don't know what all of this says about the Heideggerian/Wittgensteinian idea. One might look at the above quote by Okrent and respond that Gerede requires not just assessing whether norms are being fulfilled, but also improving the behavior to better fulfill the norms. And the mathematical analogue to this would not be algorithmically simpler than a program that just followed the norms.
I don't think that's correct though. The Heideggerian desiderata as Brandom and Okrent present it was that a distinction be made between parroting what is said and having an appreciation of the independent reality that is being talked about. At the level of language use, the distinction between being able to say the correct things and being able to discern when incorrect things are said strikes me as doing this work just fine. What would be the linguistic analogue to building a better hammer? And why would that be relevant to determining that the speaker is sensitive to the independent reality that is being talked about? Just as Okrent wants to defend the aphaisic from Brandom's requirements, one wants to defend the tetraplegic from Okrent's requirements.
But, at least with regard to sentences that claim whether or not a purported proof is a valid proof, the ability to follow the norms and assert correct sentences is provably more complex than the ability to assess whether one is following those norms.
Which strikes me as so weird that I wonder if the analogy breaks down somewhere. But I'm not seeing it.
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