Well at least we know it's not something on our end, with our accounts, and they need to fix it. Hopefully soon it will be resolved.

Thanks for the reply!

I would like to clarify one thing, the point of my comment (my last sentence in the original comment).

Firstly, I just want to say that agree with your comments about vagueness.

I agree that vagueness exists and isn't like ambiguity in the sense that 'some' or 'many' are ambiguous. Sorites paradoxes exist for vague concepts, like 'heap' or 'bald' and this doesn't apply to ambiguities.

The lesson uses the word 'heap' in contrast to the word 'some' by pointing to one lower boundary as being sharp or not sharp, but the contrast, as its presented, isn't as sharp itself in the following sense.

It's true that I'm defining 'heap' as a set of members that are organized in a given way, but I'm just appealing to what I think 'heap' means through its use by common speakers. In my definition of heap, I quoted a dictionary definition (with the assumption that a dictionary definition is what's commonly used to define words that common speakers use). Through this definition, I take it as an implication that it is a set that excludes one and only one member.

Of course, these vague concepts aren't sharply defined in a logical sense from how people use 'heap' or 'bald.' Yet, common use of vague concepts do have some prima facie sharp boundary intuitively even if there seems to be no logical sharp boundary.

Sorties cases result from a given starting point of F being a predicate and through a series of applying, in some formulations, some method (Fn-1 or Fn+1) results in a conclusion where F remains constant even though that conclusion is counterintuitive (and therefore paradoxical).

I'm pointing at our intuitions, given what I take we mean by a word through its common use, and saying "here's a boundary" where F is or isn't sharp. This reference is in comparison to what the lesson uses with one boundary for one ambiguous word.

In my original comment I take it to be the case that vague concepts are fuzzy (in opening question of my comment), but I don't think the distinction between ambiguity and vagueness is given through the one example of lower boundaries by way of introducing 'heap' and 'some.'

I think the distinction would have been given if the lesson would have taken upper boundaries as examples instead since it would fall outside of what I think is the common definition of 'heap' to mean ( "a disorderly collection of objects placed haphazardly on top of each other").

Great comment! I applaud you for getting into some of the nitty gritty of the sorties paradox!

#philosophicaldigression

Doesn't vagueness have some, or at least one, sharp boundary condition even if it's fuzzy? For example, one grain of sand isn't a heap ("a disorderly collection of objects placed haphazardly on top of each other") while two grains of sand seems unclear if it is. Therefore, one gain of sand is a sharp boundary for 'heap,' namely that it isn't a heap.

The lesson takes lower boundaries of ambiguities like 'some' and says it has a sharp boundary and vague concepts like 'heap' as having a fuzzy boundary, but thinking of what would constitute a lower boundary for 'heap' seems like it would be a sharp boundary in the same way as 'some' has a sharp boundary. For upper boundaries, this distinction between sharp boundaries and fuzzy boundaries might be different and I think probably is. Yet, in this lesson alone, given the two examples, lower boundaries between the two aren't exactly different in the way it's making it out to be .