 Karl Popper was the first philosopher to take the logical problem of truthlikeness seriously enough to make an assay on it. This is not surprising, since Popper was also the first prominent realist to embrace a very radical fallibilism about science while also trumpeting the epistemic superiority of the enterprise. In his early work, he implied that the only kind of progress an inquiry can make consists in falsification of theories. This is a little depressing, to say the least. It is almost as depressing as the pessimistic induction. What it lacks is a positive account of how a succession of falsehoods might constitute positive cognitive progress. Perhaps this is why Popper’s early work received a pretty short shrift from other philosophers. If a miss is as good as a mile, and all we can ever establish with confidence is that our inquiry has missed its target once again, then epistemic pessimism seems inevitable. Popper eventually realized that falsificationism is compatible with optimism provided we have an acceptable notion of verisimilitude (or truthlikeness). If some false hypotheses are closer to the truth than others, then the history of inquiry may turn out to be one of progress towards the goal of truth. Moreover, it may even be reasonable, on the basis of our evidence, to conjecture that our theories are in fact making such progress, even though we know they are all false or highly likely to be false.