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As you add more and more to something eventually it changes, but at what point does it change? For example, everyday a boy gets older, and eventually he becomes a man. But can you call him a man tomorrow? How about the day after that? If you don't know at what point to call him a man, then he will always be a boy.

People use this false argument sometimes, for example against drinking laws. If it is legal to drink at the age of 18, why couldn't I drink the day before I turned 18, what was different?

Is this is a fallacy? If so what is it called?

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There is an important difference between the paradox of the heaps and the fallacy of the heaps. The former also is known as the continuum paradox or the sorites paradox (from σωρείτης), and the latter also is known as the continuum fallacy or sorites fallacy. (soros, from σωρός, is Greek for "heap".)

The paradox of the heaps is the paradox described in the prompt-question of this thread. A man was once a boy, but the instant the boy became a man is equivocal. The continuum of a lifetime seems to make arbitrary any rigid designations of time distinguishing childhood from adulthood. Nonetheless, it would be fallacious to argue that this, on its own, implies that a lifetime either is entirely childhood or is entirely adulthood.

It is a fallacy of the heaps to argue that the fact of continuum precludes any meaningful distinctions of regions along a continuum. For example, amounts of heat vary on a continuum, and water boils/freezes when it becomes sufficiently hot/cold such that, on a heat-continuum, there are meaningfully distinct regions. Therefore, it is not true that, for every continuum, there are no meaningfully distinct regions. Of course, there are some continuums in which there do not seem to be any meaningfully distinct regions. For example, the real number line, as a continuum of 1-dimensional space, does not seem to have any meaningful distinct regions, although it has meaningfully distinct subsets (e.g. the set of integers, of irrational numbers, and of natural numbers).

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It's not a fallacy per say, but more of a definitional paradox or lack of clarity; it's called Sorite's Paradox and it stems from poorly defined predicates. The paradox basically asks "At what point is a heap of sand not a heap anymore if a grain of sand is removed at a time?" There are several proposed solutions to the paradox such as setting a defining limit or gathering a group consensus.

I think this vagueness is more of a point of dispute rather than a fallacy however. Here's a wikipedia article if you want to know a bit more:

https://en.wikipedia.org/wiki/Sorites_paradox

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    The problem of thresholds is also called the problem of vagueness. – Mitch Nov 4 '16 at 14:35
  • The fallacy based on the Sorites paradox is the continuum fallacy, where one for example argues for the non-existence of the two sides via arguing that the line between them is blurred. See here. – MM8 Jan 31 '17 at 19:32

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