I'm of the opinion that paradoxes are useful for understanding how our understanding of the world differs from how the world actually works. They are a kind of optical illusion but for our reasoning abilities, not our vision. If we can resolve a paradox in our minds then we have deepened our understanding of other issues.
An especially simple one that packs a lot of punch is the sand heap problem or sorites paradox. The argument goes like this: Start with a heap of sand. Now take away one grain. What remains is still a heap because a difference of one grain is immaterial. Now repeat that step over and over until you're left with a single remaining grain of sand. One grain of sand is not a heap, and yet we can't identify any step in the process when the heap went away. So we have a paradox: no step of removing a single grain could destroy the heap, but the heap has been destroyed.
Does that mean a single grain of sand must actually be a heap? Of course not; that's clearly not what we mean by "heap." A heap of sand is a vague object but it has some expected features. We expect the vast majority of the grains of sand in the heap will be supported only by other grains lower in the heap. If we spread the sand out as a single layer on the ground it would no longer be a heap. Likewise we expect the heap to be supported mostly from the bottom. If we pour the sand into a container it's not a heap. Opinions might differ, but mostly I think a heap of sand implies a large volume. A small amount of sand might be more like a "pile" than a "heap." A single grain can have none of those properties.
So as the heap is reduced in volume at some point it stops being a heap well before you get down to a single grain. People might consider a few cupfuls of sand as a heap (in the right shape of course), but it's unlikely you'd find anyone who would think that a half gram of sand could ever rightly be considered a heap. In fact this "consensus opinion" standard is one avenue for resolving the paradox. If you could poll a reasonably large sample of people and show them different sizes and configurations of collections of sand, you could plot their responses to the question "is this a heap?" The result would be a continuum, where more heap-like means more people agree and less heap-like means fewer people agree. So there's no single number of grains where the sand stops being a heap, but it gradually gets less heap-like as more is removed. This works, but the underlying truth is more interesting than just measuring opinions.
The deep lesson we learn from this paradox is about emergent properties. An emergent property is a feature of a collection of items that is not found in any of the individual items alone. One example is sound. Individual atoms or small collections of them don't carry sound, but sound waves can start to exist when there are huge enough numbers of interacting atoms, such as in a gas. Likewise one neuron isn't intelligent, but clearly intelligence can be found in the right type of collection of many neurons. Emergent properties are continuous and gradual. We're used to thinking about an object's properties as binaries. Bachelor or not. Pregnant or not. In contrast to this intuition, the emergent properties of collections tend to scale in effect relative to the size of the collection. This is reflected in language when we use adjectives to indicate size: a huge sand heap, a complex nervous system, a thick gas.
With this in mind we can easily spot the flaw in the argument. Because "heap-ness" is an emergent property of collections of sand, and therefore something that scales numerically, we can no longer claim that removing one grain leaves the heap exactly the same as before. We have to agree instead that removing one grain of sand leaves slightly less of a heap than was there before. It's fairly obvious that diminishing some property -- however slightly -- over and over could eventually obliterate that property altogether. Because emergent properties are real things, but are a matter of degree, there is no paradox.