Seems contradictory but you can clarify. If a proposition can be falsified then that is knowledge that said proposition is false, and the negation is true. If nothing can be proven or known then it must follow that nothing can be falsified.
"All swans are white."
This statement cannot be proven because it's not possible to observe all swans. There may be some swan in some hidden corner of the earth (or universe) that I did not see.
If I see one black swan, I have falsified that statement.
When you refer to "Not all swans are white" This statement can be proven true but why? This is because the original statement is a universal claim and the negation is a particular claim.
The key distinction between universal claims and particular claims explains why you can "prove" the statement "Not all swans are white." Universal claims, like "All swans are white," attempt to generalize about all instances of a phenomenon. These kinds of statements can never be definitively proven true because they rely on inductive reasoning—no matter how many white swans are observed, there’s always the possibility that a counterexample (a non-white swan) will eventually be found.
In contrast, particular claims are much more specific. The statement "Not all swans are white" is a particular claim because it is based on falsification—it only takes the observation of one black swan to disprove the universal claim "All swans are white." Since black swans have been observed, we can confidently say "Not all swans are white" is true.
Popper's philosophy focuses on how universal claims can never be fully verified (proven true) through evidence, because future observations could always contradict them. However, universal claims can be falsified (proven false) with a single counterexample. Once a universal claim is falsified, it leads to a particular claim like "Not all swans are white," which can be verified by specific evidence.
In essence, universal claims cannot be proven true because they generalize across all cases, while particular claims can be proven once a falsifying counterexample is found. That's why you can "prove" the statement "Not all swans are white"—it’s based on specific evidence from reality, in contrast to the uncertain generality of universal claims.
To sum it up. When I say nothing can be proven and things can only be falsified... it is isomorphic to saying universal claims can't be proven, particular claims can.