One unsolved (or at least, unanswered) problem in Critical Rationalism, is the issue that putative falsifications are themselves fallible, because you are relying on fallible prior knowledge to logically judge that the falsification is legitimate.
Critical Rationalists seem not to take Critical Rationalism to its full logical conclusions. If the Critical Rationalist’s falsification attempts are themselves fallible, shouldn’t he subject his falsification attempts to criticism as well? And shouldn’t he then further criticise the criticisms of his criticisms?
Why does the Critical Rationalist take the first step of insisting on criticism, but fail to continue criticising his own criticisms? Isn’t that just being hypocritical (or at best, lazy)?
Justification requires ways to solve the regress problem (recursive questioning):
The regress problem
“… to justify a belief one must appeal to a further justified belief. This means that one of two things can be the case. Either there are some [epistemologically basic] beliefs that we can be justified for holding, without being able to justify them on the basis of any other belief, or else for each justified belief there is an infinite regress of (potential) justification [the nebula theory]. On this theory there is no rock bottom of justification. Justification just meanders in and out through our network of beliefs, stopping nowhere.” The apparent impossibility of completing an infinite chain of reasoning is thought by some to support skepticism. Socrates said, “The only true wisdom is in knowing you know nothing.”
The regress argument (also known as the diallelus (Latin < Greek di allelon “through or by means of one another”)) is a problem in epistemology and, in general, a problem in any situation where a statement has to be justified.
According to this argument, any proposition requires a justification. However, any justification itself requires support. This means that any proposition whatsoever can be endlessly (infinitely) questioned, like a child who asks “why?” over and over again.
Similarly, falsification is not immune to the regress problem (recursive questioning). Instead of taking the problem seriously, Critical Rationalists seem happy to pretend that the problem does not exist. And if they actually do believe that the problem does not exist, then their beliefs are inconsistent with Critical Rationalism’s demand for criticism in the first place.
There is no universally agreed upon body of prior knowledge. We start from data, and epistemological methods are ways by which we interpret our data. Interpretation is necessarily a subjective endeavour because you cannot escape from making assumptions.
How fallible you deem your own assumptions is a related but separate matter altogether. The problem remains that recursion cannot go on indefinitely. Assumptions surface wherever you arbitrarily (subjectively) decide to stop your recursive falsificatory (or justificatory) questioning.