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Metaphysical/ontological vagueness, classically understood, can be summed up as the problem of borderline cases of some thing x. Clouds are the standard examples of vague objects. If posit that our minds meet up with some things in the world that we call "clouds" that have something akin to definite boundaries beyond which an existential quantifier is rendered false, then what are we to say about the quantities of material that constitute clouds (i.e. water droplets) that fall outside the boundaries? Some (e.g. Williamson) argue that there is something like an epistemic cap to the question: clouds do exist and there is a real boundary between a cloud and not-a cloud, but we will never be able to find it because it is simply beyond our perceptual or reasoning capacities. Others argue for what is called "supervaluationism" or super truth – the idea that truth values are not characterized by the "T or F" dichotomy, but that there is some metaphysical commitment of philosophers to some sort of truth gradient. Still others (e.g. Morreau) argue that there are real vague objects.

Let's work from the "there are real vague objects" camp.

Vagueness is generally seen as an problem wherein predicates can be said to satisfy their descriptions. Borderline cases are observable for all of the following: "George is bald", "Ship A is not the same ship as ship B", and "The table is made of wood". Great. Note though that certain descriptions which sit in the functional "place" of the predicate are not themselves predicates, such as "exists". This is observed by Kant, among others. What this means, for our purposes here, is that existence is not a predicate. What this means, in turn, is that there is something about the logical operation of existential quantification that is somehow more "basic" or "atomic" or metaphysically "true" about positing the existence of something, such that that quantification does not count as a description. Now, let us look at the sorites paradox, a.k.a the paradox of the heap. Take a grain of sand. Is this a mound? Probably, the answer is no. Add another grain of sand. Is this combination a heap? Again, the answer is no. From this, it can be logically deduced (via modus ponens and the logical operation "cut") that adding a single grain of sand will never make the difference between a given non-heap collection of sand and a real heap. The conclusion to be drawn then is that there are no such x's that are heaps of sand. This is patently false. If you don't believe me, then pour a cup of sand onto your desk and tell me that there is no heap on your desk. Thus the paradox.

One way to get around this would to say that it is only a heap because we say it is. But the question I wish to ask is then how, if the material the constitutes the heap exists, and existence is not a predicate, can that material only be a heap when we say it is? If this is true, which is to say, if there is only a heap when we say there is a heap, then it seems that there must be borderline cases of existence, i.e., collections of sand that are possible but not actual candidates for extant heaps of sand, no? But if there are such cases, then existence is a predicate, and there would be no reason to posit the question. However, what are the metaphysical and ontological ramifications, if existence really is a predicate? Is existence to be understood as something like "red" or "bald", which is to say, only true as specifiable by human beings? Is this a deflationary take on truth? I don't know where to go next.

  • 'Vagueness' I should say is a classificatory term. 'Bald' is a vague term - the logic of 'baldness' leaves it vague whether X is bald or at least it can do so. But this is not a borderline case of existence : both X and the state of her or his hair exist. It is simply uncertain whether the category of baldness applies, is properly predicable. Apologies if I have missed your point. – Geoffrey Thomas Dec 30 '17 at 14:31
  • My suspicion here is that the problem will reduce to semantics, after sorting out which theory(s) of truth is applicable/preferable: en.wikipedia.org/wiki/Truth For instance a "vague" concept could not be evaluated in a correspondence theory, but would certainly be acceptable in a consensual theory – christo183 Oct 5 '18 at 12:20
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But the question I wish to ask is then how, if the material the constitutes the heap exists, and existence is not a predicate, can that material only be a heap when we say it is?

The easy answer here is to say that the material exists, but "heap-ness" is a predicate which may or may not be present.

If this is true, which is to say, if there is only a heap when we say there is a heap, then it seems that there must be borderline cases of existence, i.e., collections of sand that are possible but not actual candidates for extant heaps of sand, no?

Not borderline cases of existence, but borderline cases of definition. Something exists; the question, in this context, is whether or not it should be classified as a "heap."

Is existence to be understood as something like "red" or "bald", which is to say, only true as specifiable by human beings? Is this a deflationary take on truth? I don't know where to go next.

No, heapness is to be understood as something like "red" or "bald"; even in classical cases of the Sorites Paradox, the existence of the possibly-bald-man (or the would-be-heap) is not in doubt.

  • "The easy answer here is to say that the material exists, but "heap-ness" is a predicate which may or may not be present." Do present things meet the existential criteria of quantification? – Jaime Ravenet Sep 28 '11 at 6:50
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    Sure, if they meet the definition of the predicate in question. Let's take a simpler example first-- a possibly-bald man. He's definitely a man; he may or not be bald, depending on how you come down on the Sorites. So, existential quantification isn't a problem. The same applies, ceteris paribus for a heap. Or am I misunderstanding your question? – Michael Dorfman Sep 28 '11 at 7:41
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    "The easy answer here is to say that the material exists, but "heap-ness" is a predicate which may or may not be present." "No, heapness is to be understood as something like "red" or "bald"" These aren't compatible. If heapness gets understood as something like "red" or "bald", then it comes in degrees. A red firetruck is more red than red hair. Patrick Stewart is more bald then someone with a slight bald spot. So, heapness can both come as present and not present. – Doug Spoonwood Sep 29 '11 at 1:10
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    In the case where X is not a heap, we may choose to no longer refer to it as a single object, but I don't see that as a problem; we are conventionally used to referring to objects that are composed of atoms, after all. – Michael Dorfman Sep 29 '11 at 7:23
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    @JaimeRavenet: Effectively, yes. And, naturally, the same for all nouns. I don't see this as a problem. Because, if we choose not to call the pile of sand "a heap", there is still something there--the same number of atoms as before, in fact-- we're just choosing to group those atoms differently into conventional objects for our convenience. – Michael Dorfman Sep 29 '11 at 15:57
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Viewing these sorts of questions mathematically / computationally as decision theory sheds light on the subject more efficiently, I believe, than traditional philosophical approaches.

Suppose you have a universe, and you have an indicator function f that you can call on subsets of the universe and it will return true or false (guaranteed--no halting problem). Every such indicator function defines a category, and if there exists any subset of the universe for which f returns true, then you can say there is f-existence.

Now, if you have some f that describes "heapness", then presumably if you pass in the (information about the) cup of sand on my desk, it will return "true". On one grain it would return "false". As you add and reorient sand from one grain to a cupful, it will presumably go from false to true at some point (maybe many points--it might oscillate back and forth for a while!). We certainly have f-heapness, then.

The question is then whether the category of f-heap is useful or agrees with human intuition. Here we take a detour through machine learning: in many, many cases we can create classifiers that agree with humans on obvious cases 100% of the time, and where people disagree will tend to come up with a value (if we parameterize from 0 = surely-not-an-X to 1 = surely-an-X) equal to the probability that people will think an object is in that class. Some such classifiers have nice properties of stability as values increase, so you could be sure that two cups of sand would also be a heap. Thus, we could create (in principle) a well-behaved f that would give us an objective definition of f-heapness.

Now, though, we wonder what happens at the decision boundary? How does sand go from being a not-a-heap to a heap when no single grain ought to make the difference? If you look at a classifier of heaps, what would happen is that--if we use one that is not forced to return false or true but can return a value from 0 to 1--at some point, as we start adding more sand, the value creeps upwards from 0 until it reaches 1. (Orientation of grains and context of the question might also be important.) If we force it to make a false-true choice, it could either pick at random, taking the value as a probability, or it could have a sharp threshold. But even if we take a sharp threshold, all we learn is that this is the point where there is maximal disagreement about whether this is a heap of sand or not: 50% of people would take each perspective. Even if we all agree that f-heapness is true heapness, we would be maximally unhappy (on average) about its agreement with our intuitions at that point.

So, in conclusion, logical inferences "one x is not a y, and if n x's is not a y, (n+1) x's is so similar that it could also not be a y" are not really capturing the essence of what needs to happen to classify things. If you take an approach that does usefully classify things (e.g. machine learning), you find that the paradox cannot be constructed.

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The example of clouds is interesting because it is difficult to draw the boundaries between a system we would like to describe, and its environment.

This problem exists in miniature for a great number of things, including apples, tables, and people. Is the air in your lungs a part of you? What if the air is dissolved in your bloodstream? What if that air is absorbed by a cell of your body, or a cell of some micro-organism living inside of you? Is that micro-organism, which is a part of an ecosystem which might be instrumental in your digesting food, a part of you despite being of an identifiable (non-human) species?

The problem, it seems to me, is one of dividing the world into disjoint systems which we then (quite often) imagine interacting with each other. On the macroscopic level, with humans and apples and tables, this mode of reasoning is more-or-less unproblematic, assuming we ignore the (potentially) common provenance of the apple and the table from a single tree, and what happens to the apple once you manage to eat it. With clouds, we're struck with the fact that while there is definitely some difference between the interior of the could and a relatively water-droplet-free point on its exterior, there is no particularly meaningful way to define the boundary of the cloud even once we acknowledge that there can be gradients in water droplet density in the air, at least not on spatial scales of centimeters. (On scales of dozens of meters, we frequently recover the ability to make meaningful statements on such topics as light albedo and opacity.)

So when you want to talk about a cloud as an object, or even merely as a substantially different aggregate system from clear sky around it, you often have to ask to what level of precision that you would like statements you make about the cloud to be evaluated. The very same is true of humans, and apples, and tables; only the precision which we may casually assume, without much reflection, in those cases is often quite extremely high.

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From a different perspective one can say darkness is the absence of light and is not a being. So there are things which exist without being, it's another way to see borderline existence. This consideration is deeper than it appears as in a christian tradition the evil exists but just as an absence of good, the evil exists without being.

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