The Wikipedia summary of the problem: if I have a bale of hay and remove one piece of straw, is it still a bale of hay? What if you remove another? When does it cease to be a bale of hay?
Another version of the problem: a museum displays the ax with which George Washington chopped down the cherry tree. Over time, of course, the wood decayed so they replaced the handle. And a few years ago, the head rusted and so they replaced that.
The word "paradox" probably stretches a bit when it is applied to this problem. And that gets at the heart of the two pieces of the answer I see to this problem -- imprecise use of language.
Let me introduce two concepts that will form my answer to the Sorites Paradox.
1. Partial Identity -- Let's start with Plato's idea of the 'ideal' form of things. Plato suggested that there is an 'ideal' Triangle, one perfect in form and proportion. All triangles in our experience are shadows of this idea. When we call something a triangle, we mean more specifically that it resembles the ideal of a Triangle and varies in one or more ways. What we typically call a square is also a triangle, if we specify that it is a triangle with one side bent into two, shorter sides.
So a specific bale of hay that you might have in a field is a shadow of the Bale of Hay that would ideally define the class. Let's say in the thought experiment we begin with a perfect bale of hay. (We can't, even in a thought experiement, begin with the ideal Bale of Hay, because ours is fixed in time and space and has all manner of attributes specific to that bale of hay.) Then we remove one straw from it. Is it still a bale of hay? Not exactly. It is now a bale of hay with one straw removed. For convenience in discourse, we'll leave out the details that don't affect the material of our discussion.
But when those details begin to affect the substance of the discussion -- once the bale of hay has less commercial value because the rancher who wants to buy it notices that it is much smaller than the other bales of hay, for instance -- we can no longer use the convenient, inaccurate labels. It's not a paradox, it's just plain inaccurate.
2. Functional Duplicates -- So what if every time I remove a piece of straw from the bale of hay, I add one from somewhere else? Once I've replaced every single piece of straw, is it sill the same bale of hay?
No. It's a functional duplicate. In the case of hay, my rancher customer probably doesn't mind that the hay has all been replaced. The value to him did not come from the original composition of the bale -- merely the utility of its end state. George Washington's hammer, on the other hand, is valuable for its original composition far more than its end state. It's the fact that it was George Washington's axe -- meaning George Washington's handle and blade -- that carries the real value.
When the original handle rotted and was replaced, the label George Washington's Axe reflected a partial identity. George Washington's axe handle was gone, and his blade was now paired with a new handle. When the blade was replaced, George Washington's Axe no longer existed. If the museum continues to use the label "George Washington's Axe," it's only because the new axe hanging up there one the wall has some value as a placeholder. But if the axe went up for sale at auction, it would be very important to be much more precise about its identity.
Practical Applications
Abortion is one of the areas where the 'heap paradox' comes into play. When does the growing fetus become a person worthy of protection by the government? While the framework outlined above doesn't directly answer the question, it helps us frame a few more specific questions.
1. Does the government only protect "people?"
2. To what degree is the fetus a "person?"
3. Why does the government protect people?
4. To what extent do those answers apply to the developing fetus?
Personally, I think the government would have serious issues with me killing the unborn or immature young of a protected species, even if they were inside my house without permission. The partial identity of a developing young member of a 'species' does not exclude it from consideration as a full member of that species. ("It wasn't a protected California red-legged frog, officer! It was a tadpole. Very different." Obviously that's not a functional difference in this case.)
What do you think? Useful framework? At least as a high-level response to the "paradox?"
I think accurate communication resolves 90% of problems. (Or, bad communication causes 90% of problems. Once they're caused, it can take a lot more than precise communication to solve them.)