> true, because it doesn't make sense to "know" a falsehoood That's a problem right there. Maybe that made sense to the Greeks, but it definitely doesn't make any sense in the 21st century. "Knowing" falsehoods is something we broadly acknowledge that we all do.
False propositions are not knowledge, only true propositions are knowledge. Therefore you cannot know something true that is actually false, you can only believe something true that is actually false. Precisely describing how one moves from belief to knowledge is exactly what epistemology is about.
This is something that a lot of Greeks would have had issues with, most probably Heraclitus, and Protagoras for sure. Restricting ourselves to Aristotelian logic back in the day has been extremely limiting, so much so that a lot of modern philosophers cannot even comprehends how it is to look outside that logic.