Is it rational to believe that some of my beliefs are false? This seems like a reasonable claim. After all, most people have some false beliefs, and I know that I've had plenty of beliefs in the past which I later learned were false. On the other hand, I obviously believe that each of my beliefs is true (otherwise, they wouldn't be my beliefs). So how could I also believe that some unspecified beliefs among them are false?

It certainly looks like the height of rationality for you to believe that at least some of your beliefs are false. Yet, as you point out, there's no particular belief of yours that you regard as false. Any given belief of yours you regard as true; otherwise, it wouldn't be a belief of yours. This pair of attitudes gives rise to what's usually called the "Paradox of the Preface." One place to start looking is the SEP entry on "Epistemic Paradoxes," available here, which contains both discussion and references.

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