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.
Skeptism is self-refuting as noted by al-Ghazali which is why no major philosopher has been a sceptic. To do philosophy requires faith ... faith in truth as a category, faith also in our senses that they report the world faithfully.