are you saying that because the definition of a heap is vague, there are multiple feasible definitions of a heap, so some people (using one definition) would call a candidate heap a heap and other people (using another definition) would say it isn’t a heap?
Not that they are necessarily using multiple definitions, but the common definitions themselves do not specify an exact range in quantity in which a cluster of things could be considered a heap. Two people could disagree about whether something is a heap despite using the same vague definition of “heap”. They may be comparing the candidate heap relative to things that they have experienced being called heaps in the past. I suppose you could still treat this as being a difference in their personal definitions of “heap”. However, I don’t think that if pressed to define “heap”, that people would be likely to state an explicit quantity range. They would most likely give vague qualitative definitions. The same person may even use inconsistent definitions at different times or forget to include certain aspects that they would consider to be important defining characteristics. People don’t normally think in terms of definitions when classifying things. They usually just classify based on what feels correct, and definitions are after-the-fact attempted explanations of their classifications.
Almost all terms introduce a simplification, not just the vague ones.
Doesn’t the introduction of a simplification itself give a term some vagueness, because then you don’t know the details of the relevant characteristic, just a qualitative judgment? For “heaps”, the simplification is one that keeps the exact quantity obscured while providing a qualitative description instead. Even if you had a different quantity term that, unlike “heap”, didn’t have fuzzy boundaries, it could still be considered vague in a different sense if multiple quantities could fulfill its definition (assuming an exact quantity really did exist in reality). It would certainly at least be considered somewhat ambiguous. For example, the category “integers” has seemingly clear boundaries, but calling an unknown number an integer is still vague if it doesn’t express all the relevant information.
vague terms have a wider array of meanings than non-vague terms
Not that they are necessarily using multiple definitions, but the common definitions themselves do not specify an exact range in quantity in which a cluster of things could be considered a heap. Two people could disagree about whether something is a heap despite using the same vague definition of “heap”. [...]
Ahhh, I think I understand. Yes, that makes sense.
Doesn’t the introduction of a simplification itself give a term some vagueness, because then you don’t know the details of the relevant characteristic, just a qualitative judgment?
Yes, although (as you suggest) that’s a different kind of vagueness: vagueness as a definition’s inclusiveness vs. vagueness as uncertainty about the definition itself.
Not that they are necessarily using multiple definitions, but the common definitions themselves do not specify an exact range in quantity in which a cluster of things could be considered a heap. Two people could disagree about whether something is a heap despite using the same vague definition of “heap”. They may be comparing the candidate heap relative to things that they have experienced being called heaps in the past. I suppose you could still treat this as being a difference in their personal definitions of “heap”. However, I don’t think that if pressed to define “heap”, that people would be likely to state an explicit quantity range. They would most likely give vague qualitative definitions. The same person may even use inconsistent definitions at different times or forget to include certain aspects that they would consider to be important defining characteristics. People don’t normally think in terms of definitions when classifying things. They usually just classify based on what feels correct, and definitions are after-the-fact attempted explanations of their classifications.
Doesn’t the introduction of a simplification itself give a term some vagueness, because then you don’t know the details of the relevant characteristic, just a qualitative judgment? For “heaps”, the simplification is one that keeps the exact quantity obscured while providing a qualitative description instead. Even if you had a different quantity term that, unlike “heap”, didn’t have fuzzy boundaries, it could still be considered vague in a different sense if multiple quantities could fulfill its definition (assuming an exact quantity really did exist in reality). It would certainly at least be considered somewhat ambiguous. For example, the category “integers” has seemingly clear boundaries, but calling an unknown number an integer is still vague if it doesn’t express all the relevant information.
Yes, terms can be vague in that way too.
Ahhh, I think I understand. Yes, that makes sense.
Yes, although (as you suggest) that’s a different kind of vagueness: vagueness as a definition’s inclusiveness vs. vagueness as uncertainty about the definition itself.