Years ago I had a boss, a mentor really, who had a saying
"Every problem must have a solution."
I liked this; its optimistic note stuck a chord, and I've often trotted it out in the face of a challenging situation. It's worked well for me. Once, an organisation I worked for faced a really serious issue. Everyone who might be able to help avoid the crisis was summoned to a big room. Most people there were, at least metaphorically, clasping their hands to the sides of their heads and rocking slowly from side to side in a despairing manner.
"Every problem must have a solution", I chirped up brightly. "We must be able to solve it." (A bit Bob-the-builder-y. "Can we solve it? Yes we can!")
My colleagues perked up immediately and we got down to at looking what to do. Lots of hard work but we managed and it came right. I've always remembered it.
Luckily there was no logician in the room (I was holding by breath at the time). Otherwise they might have piped up: "Excuse me. It can't be true that every problem must have a solution. What about the problem of finding a problem with no solution? If every problem has a solution there must be a solution to the problem of finding a problem with no solution, so there must be a problem with no solution. It follows that not every problem has a solution, otherwise we reach a contradiction." This would have held things up a bit.
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(Is that akin to Bertrand Russell's set of sets that do not include themselves?)
Hmm. What if the only problem without a solution is that of finding a problem without a solution? Wouldn't the solution be that it is no longer a problem since it is solved, thereby collapsing the paradox? That is, the problem without a solution itself has a solution that is its self. It had only a brief temporal existence between being asked and being solved.
While playing with paradoxes: the barber paradox. "The barber shaves everyone who does not shave himself." Well, surely the barber shaves before he goes to work.