The value of a paradox is diagnostic. When an argument you cannot fault reaches a conclusion you cannot accept, something in your concepts is broken, and the paradox shows you roughly where. Several of the most important developments in modern logic began as attempts to defuse one.
The liar paradox"This sentence is false." If true, it's false; if false, it's true. Attacks the concept of truth itself, and drove much of twentieth-century work in logic and semantics.
Zeno: Achilles and the tortoiseAchilles must first reach where the tortoise was, by which time it has moved. Infinitely many steps, so he never overtakes — yet obviously he does. Attacks our concepts of infinity and continuity.
Zeno: the arrowAt any instant, a flying arrow occupies exactly its own length and is therefore motionless. If it's motionless at every instant, when does it move?
The sorites (the heap)One grain isn't a heap. Adding one grain never turns a non-heap into a heap. So no number of grains makes a heap. Attacks vague predicates — and most of our vocabulary is vague.
The ship of TheseusReplace every plank, then rebuild the originals. Two ships, one history — which is the original? Attacks identity through change.
The unexpected hangingA prisoner is told he'll be hanged one weekday and won't know which. He reasons it's impossible — then is hanged Wednesday, entirely surprised. Attacks self-referential prediction.
Newcomb's problemTwo boxes, a reliable predictor. Two solid decision principles give opposite answers, and there's no agreement on which to abandon. Attacks decision theory itself.
The paradox of the stoneCan an omnipotent being create a stone it cannot lift? Either answer seems to limit omnipotence. Attacks the coherence of unlimited power.
Russell's paradoxDoes the set of all sets that don't contain themselves contain itself? Either answer contradicts. It broke naive set theory and forced its rebuilding.
The paradox of toleranceUnlimited tolerance permits the intolerant to destroy tolerance. Should a tolerant society tolerate intolerance? Attacks a principle many hold absolutely.
Buridan's assAn animal exactly between two identical food piles, with no reason to prefer either, starves. Attacks the idea that action always requires a determining reason.
The preface paradoxAn author reasonably believes each claim in their book, and reasonably believes the book contains at least one error. Both attitudes seem rational and jointly inconsistent.
Every premise true, every step valid, the conclusion impossible.What makes it a paradox
There are exactly three moves available. You can reject a premise that looked innocent — the usual resolution of Zeno's paradoxes involves denying that infinitely many intervals must sum to an infinite quantity, which the mathematics of convergent series supports. You can reject the reasoning, showing a step is invalid despite appearances, which is how restrictions on self-reference address the liar. Or you can accept the conclusion, deciding the unacceptable result was actually correct — the standard response to Russell's paradox was to conclude that naive set theory was simply false and rebuild it.
Some resist all three. The sorites has no consensus solution: denying that vague terms have sharp boundaries seems obviously right and yet leaves the argument standing. Newcomb's problem divides decision theorists who agree on everything else. That these remain open after decades of concentrated attention suggests they are pointing at something real about the limits of the concepts involved — which is precisely what makes them worth studying rather than merely enjoying.