TL;DR: If you remember about Popper's asymmetry between the contrast of your proposition, and the termination upon being proven false, for your given proposition [this procedure is called falsification], you will find by duality that the complementary proposition will be subject to some sort of verification instead.
Verification is not falsification but the opposite, so this question can be answered: exactly in the case that your non-falsable statement is verifiable. This is mostly the case of existence questions (e.g. Higgs Boson). Once you verify the Higgs boson exists, you are right.
Another example: For atheists, God's existence is a matter of verification rather than falsification: I will not accept that as true until you bring God to me, or bring me to God.
Let me explain this:
FSM does not exist Is not falsifiable (it is not even well defined) until you define what kind of existence are you talking about.
On the other side, "There exists a non-white swan" or "Not (all swans are white)" is falsable. This means that you can pick a method of experimentation (which may not seem practical, although possible).
Okay, I will change the sentence. I will say there is a tortoise of the species that can be found on Galapagos Islands (instead of swans) which is black. How could I verify that? Going to Galapagos (and the authorized zoos around the world that breed those species of tortoises) and checking the color.
This effort is huge, yet possible. Even if you don't know how to get there, currently available knowledge would let you develop or find the means of performing your observational experiment (I think that, right now, chinese government could ask this question in their country and actually experiment to track all swans' colors!!!).
In partcular, this statement you asked for is quite stationary: Take a snapshot right now, and you will answer yes or no for both of them.
However, although these two cases are not good examples (since both of them are falsifiable by the same mean, despite the negation taking a huge time), you could find examples when you may say the complementary propositions are not falsifiable.
Let's start by stating Popper's asymmetry in this matter. Popper talks about contrast (falsification) in this way:
- You need to have a method of experimentation / observation that could determine whether your proposition is true or false in such experiment.
- If false, your proposition is plain false and you should state another one (which could be similar, avoiding the false cases by some sort of comprehension or enumeration, but still being a different proposition).
- If true, and your proposition pretends to live across time and is not a proposition scoped to the current instant and circumstances, your proposition is good so far. However it cannot be proven definitely true, but just so far each time. The mostly known example is the study of gravity across all our history of physics, which involved a lot of different theories across the time, when the latter ones superseeded the former ones.
- If true, and your proposition does not pretend to live across time (i.e. is not that... general) but is just a one-time proposition who pretends to be valid just now or during the experiment's lapse, you can safely consider it true.
Most of the scientifically useful propositions satisfy the 1st (They have a method to test the truth) and 3rd point (they want to be generic across time, and want to be... true). So they can be expressed in a shorter way:
- If, by the appropriate experimental mean, you find that your proposition is false, then it is false.
- Otherwise, the proposition is good so far.
Now take the complementary proposition. You will have two analogous points:
- If you found your proposition was false during experimentation, it is false, definitely. This makes the complementary proposition... true.
- Otherwise, since your proposition is good(apparently true) so far, the complementary is bad(apparently false) so far.
Said this, you could somehow remind yourself about Adolph J. Ayer who talks about verification instead of falsification. The concept of verification is rarely used to generate a strong and useful concept on science (Popper himself argues a lot against verification), but you can quickly grasp here (after all this is the dual proposition!!) that falsification of a proposition is exactly as powerful as verification of its complementary, since it is the dual procedure.