Doesn't trying to demonstrate how we know anything beg the question?

It needn't. Like Descartes, you might try to demonstrate a priori that you possess perceptual (i.e., external-world) knowledge. Your demonstration needn't presume perceptual knowledge in the course of demonstrating that you possess perceptual knowledge. Therefore, your demonstration needn't beg the question of whether you have perceptual knowledge in the first place. Most philosophers, I think, regard all such demonstrations (including Descartes's) as failures, but I don't see any reason to think that all such demonstrations must fail because they beg the question.

Consider a more interesting case. Suppose I analyze knowledge as true belief produced by a reliable mechanism, i.e., a mechanism that yields far more true than false beliefs in the conditions in which it's typically used. A skeptic then challenges me to show that some perceptual belief I regard as knowledge, such as my belief that I have hands, was in fact produced by a reliable mechanism. In response, I offer empirical evidence in favor of my belief: others verify that I have hands; I clap my hands; I shake the skeptic's hand; I cite other outputs of my perceptual belief-forming mechanism, etc. The skeptic then protests that I'm begging the question because my method of verification simply assumes that I have perceptual knowledge: I simply assume that the evidence I offer was reliably obtained.

I think the skeptic's protest is unfair. It's unfair to ask me to show that my perceptual belief-forming mechanism is reliable while disallowing me the very means I would need to show it, namely, data that I obtain by perception. Alternatively, the skeptic could try to show that my belief-forming mechanisms are not reliable, but skeptics seldom if ever try to show that. Or the skeptic could reject my analysis of the concept of knowledge, but then the skeptic would have to offer grounds for rejecting it, which skeptics seldom if ever do.

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