I admit that my knowledge of philosophy is very limited; not advanced, yet it is

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I admit that my knowledge of philosophy is very limited; not advanced, yet it is my overall second favorite subject after science. If one accepts the proposition, "I do not know anything with absolute certainty," then is it actually self-refuting or logically contradictory? The reason, is that, if one accepts it, then one must know something with absolute certainty, which is the proposition itself. Therefore, one knows with absolute certainty that one does not know anything with absolute certainty. However, it seems to become infinitely (pun intended) problematic if one thinks about it deeply enough. For instance, if one knows with absolute certainty that one does not know anything with absolute certainty, then one must also know with absolute certainty that one knows with absolute certainty that one does not know anything with absolute certainty. I think that one knows where I am going with this. It could be extended ad infinitum. If one, however, accepts that one does not know with absolute certainty that one does not know anything with absolute certainty, is it still self-refuting or contradictory. How would the most radical skeptics reply to the above? I hope that this makes sense, because it does not to me.

Nice question! It's one with a long history, as something like what you're saying was one of the main objections to the ancient skeptics and their intellectual decedents. Let me just say a couple of things. First, to answer your question, I don't think your observation would show that skepticism is contradictory or self-refuting, at least not technically. The observation is that some skeptics take themselves to know for certain that nothing can be known with certainty. The view that they take themselves to know with certainty, namely that nothing can be known with certainty, is not thereby shown to be contradictory. It is compatible with that view that some people take themselves to know something for certain. To see this, just notice that it is compatible with the view that no one knows anything that some people, who unlike skeptics don't accept that view, take themselves to know some things. So the mere fact that people take themselves to know (or "accept" that they know) does not show that the view that no one knows is false or contradictory. Nor is the view that nothing can be known for certain thereby shown to be self-refuting, though that may depend on exactly how we understand "self-refuting." A view or position that is self-refuting is one such that, if it is true or correct, then it is false or incorrect. But the view or *position* that nothing can be known for certain does not entail that something can be known for certain. For, that view doesn't say that it (the view) can be known for certain. (Consider the statement "this statement can't be known for certain." That's a weird statement, but it is not self-refuting or a contradiction.) So, it doesn't seem we've found the skeptical view, or claim, to be a contradiction or self-refuting. The key is that what a view, or claim, or position, states need not include that it itself can be known. If the skeptical claim were, instead, that it can be known for certain that nothing can be known for certain, then that would be a contradictory view. However, we must distinguish one's being certain about claim C and the claim that C can be known with certainty. That is, we must distinguish the attitude one has towards a claim or view and the content of that claim or view. As you note, one might also form a further thought: I know for certain that skepticism (the view that nothing can be known for certain) is true. But, the content of that further thought is not skepticism. It is a thought about what one's attitude about skepticism is. This further thought contradicts skepticism, but it is not entailed by skepticism (skepticism, the view, does not imply that this further thought is true). So, that one might have this further thought does not show that skepticism is contradictory, either.
Secondly, now that I've suggested that the view skepticism, is not a contradiction or self-refuting, we should ask whether skeptics, or those who accept skepticism are doing anything irrational. Of course, there does seem to be something incoherent, inconsistent, and perhaps hypocritical with someone who takes herself to know for certain that nothing can be known for certain. Does "acceptance" require taking oneself to know for certain? I doubt it. But let's grant that for a moment. Such a person has contradictory or inconsistent attitudes. Does skepticism, the claim that nothing can be known for certain (at least as you have formulated it) entail that we should accept or take ourselves to know skepticism for certain? I don't see any reason to think that it does. So, even if skeptics are being inconsistent, this doesn't seem to reflect badly on skepticism, the view. Consider a slightly different sort of skepticism though: you should suspend judgment about everything. That view does seem to imply that you should suspend judgment even about it. So anyone who holds that view is doing something that their view says not to do. To hold such a view is, perhaps, irrational, or at least not ideally rational, because such a person has inconsistent or incoherent attitudes. It doesn't, though, seem to show that the view is false. Couldn't it be true, fully coherently, that nothing should be believed at all, not even that nothing should be believed? That could be true, but no one could ever believe it coherently, or while being rational. It seems we accept, generally, that there are truths that are unknowable and not rationally believable (think of all the insanely complex mathematical truths that no one with a human mind ever has good reason to believe). So maybe this skepticism is just another one of those truths. If so, then here is what we can conclude: believing such a skeptical view is irrational.
Ok, where are we? Skepticism, the view, is not contradictory or self-refuting. But, perhaps, being a skeptic, or believing skepticism, may be irrational. This may not be so bad if one specifies how exactly it is irrational: it is irrational only because it is a view that is believed, or a position, not because the content of the view has not been well argued for or has been shown to be false or contradictory. Maybe that takes some sting out of the objection. Skeptics are irrational, but not in the same way that, say, astrologists are irrational. Astrologist believe something on the basis of bad arguments. But nothing of the sort has been shown about skepticism. Skepticism of this sort says that no one should believe anything, not even skepticism. So while skeptics, or those who believe it, are being irrational (though, as I just said, not for the reasons that philosophical or superstitious positions are usually said to be irrational), skepticism as a view may still be true, supported by the best arguments, and so on.
I haven't answered the question, what would the most radical skeptics claim. But maybe they'd point out something like I just did.

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