Question: **Agrippa's 5 tropes**

1. *Dissent*: For every thesis an anithesis
2. *Relation*: Point of view matters.
3. *Infinite Regress*: Justifications need justifications need justifications ... *ad infinitum*
4. *Axioms*: Assume something to be true *sans* justification.
5. *Circularity*: The conclusion appears in the premises.

3, 4, 5 (above), together, constitute the well-known *Agrippa's trilemma*.

One *seemingly* strong argument against skepticism has been to point out that it's a *peritrope* (self-refuting). However, for it to be so, it *must be a sound argument* and if that's the case the argument that skepticism is a peritrope is itself a peritrope and this further argument I just made is also one ... *ad infinitum*. Skepticism is unsound only if skepticism is sound.

We end in a rather mind-boggling contradiction

1. For skeptics, skepticism is a good argument implies it's a *peritrope*.
2. For dogmatists, skepticism is a *peritrope* implies it's a good argument.

Skepticism → Peritrope → Skepticism → ... ad infinitum/ad nauseum

How do we extricate ourselves from this quagmire?

Answer: 
You didn't explain why you think skepticism is self-refuting, but I've heard people say this would be because skepticism would demand skepticism of skepticism itself, so I'll address that.

Yes, you should be skeptical of skepticism, but that doesn't make it self-refuting.

This seems to conflate skepticism and what I might call universal non-belief.

Universal non-belief may indeed be self-refuting, because you'd need to not believe that non-belief is a good idea, at which point you wouldn't subscribe to non-belief (or something like that).

Skepticism is not that. Skepticism is closer to holding that truly proving anything (that's non-trivial, or without assumptions) is impossible, and you should therefore not have absolute certainty about anything.

Crucially, this does not mean you should believe nothing, but rather that you should not assume any belief to be foolproof, and therefore all your beliefs should be tentative and you should question your beliefs as far as possible.

Now, one could hold that truly proving anything is impossible, without being absolutely sure that truly proving anything is impossible and accepting that your belief in this impossibility may not be foolproof. To apply skepticism here, one could try to truly prove something, or find a way to theoretically do so. Failing that, you would stick to the tentative belief that truly proving anything is impossible. No self-refutation here.
As per [my other answer](https://philosophy.stackexchange.com/a/92305/14508), **a skeptical argument is not different than an assertion of the presence or lack of presence of something**.

For example:

> 
> A: I see a boat out there.
> 
> 
> B: How do you know you see a boat out there?
> 
> 
> 

What B is doing is asserting something, even if only indirectly. B asserts that A's sense (at that point) is incorrect and/or doubtful. But **this is an assertion**, and **an assertion needs to be based on something**, even if it is on a self-evident something.

So extracting the assertion out of a skeptical argument, makes it like any other argument and can be attacked or refuted all the same. One can simply ask the other to ground the implicit assertion the person makes on something, even on a self-evident something.

If one denies that an assertion needs to be based on something, then the skeptic *loses the teeth to bite*, since now the person cannot attack any statement made.

PS: That an assertion needs to be based on something holds for person A too. Possibly A can appeal to the self-evident fact of being a boat there, at least as a starting point.

**Is skepticism invincible?**

1. A skeptical argument or challenge, as argued above, is like any other argument or statement and can be attacked or refuted all the same. Nothing invincible in that.
2. Skepticism as a whole is nothing but the set of potential skeptical arguments or challenges. Nothing invincible in that either.

Note: you may be interested in [this post](https://philosophy.stackexchange.com/questions/81795/is-there-any-counterexample-given-against-radical-skepticism/92321#92321) about *radical* skepticism.
Also you may be interested in the [solution(s) to the problem of infinite regress in epistemology](https://philosophy.stackexchange.com/a/93954/14508)