In the sorites-paradox example the group of sand-grains is at one level, and the sand-pile is at another level. We can have knowledge that both can exist at the same time but they exist, in the mind, only at different levels - hence the paradox. The concept of levels solves the paradox. — Don Wade
But is it too pernickety to insist that a single grain is absolutely and obviously not a heap? That's what I was trying to get at.
So if push comes to shove, just specify the precise (possibly unitary) size of heap. Everything is on a spectrum. — bongo fury
So as Don says, cognition is a hierarchical modelling of the world. We are psychologically evolved to divide the world according to the contrasting extremes of what might be the case. Either we focus on the sand pile as a group of individuals or as an individual grouping. Either we are lumping or splitting. Either we are seeing signs of larger meaningful order or local random accident.
But then in fact, this categorical division allows us to construct spectrums of possibility. We can see the range of different balances of lumped~split, grouped~scattered, general~individual that lie between the polar extremes.
And likewise, given a spectrum defined by two complementary opposites, there must be the third thing of some exact borderline case - the balancing point where judgement could go either way. That is the point of a Gestalt bistable stimulus. It illustrates how we can be tipped back and forth where two opposite interpretations – grouped or scattered, cohesive or disorderly, lumped or split - are in some exact state of tension.
There is a mid-point on the spectrum where one answer becomes as good as the other. There is a symmetry or inherent ambiguity - a logical vagueness.
As Peirce defined it, vagueness is that to which the principle of non-contradiction fails to apply. You could say that the point at which a scatter becomes a heap, or a heap a scatter, is neither definitely the one nor the other. There is no fact of the matter. Or rather, the right predicate value is "vague".
So with the Sorites paradox, the ambiguity of the transition from (purposeful and collective) heap to (random and individualistic) scatter should be what is expected, not bemoaned.
It is not helped that the set-up of the paradox contains many confusions. Is a stack a heap?
An ordinary language definition of "heap" suggests that a pile is being created in one place in a constrained fashion. But the pile is meant to arrive at its heaped arrangement - that is, grains piled on each other - in random fashion.
So a scatter of grains lacks any grains on top of each other as well as a lack of clumped grouping. Every grain qualifies as a solitary individual by usual standards. (As long as they don't also lie on a hot surface that is melting them to a collective puddle of glass.)
But if we were to pick out the Platonically minimal geometric arrangement of a trihedral stack - one grain balanced on top of three like cannonballs – would this be a heap? Could such a clear lack of random organisation logically meet the definition of a heap?
So in a world of pure Platonic order, there is a smallest heap - a minimal pile of regular spheres. But our ordinary language definition of a heap is based on some key supplementary notions about nature. We see the Humean causes of a heap as a combination of order and chaos. And that introduces plenty of ambiguity.
A stack of cannonballs permits neat and direct counting of the parts. And we get a simple answer because of the extra constraint of being able to order the whole situation. There is only ever the one answer to what is the fewest number of perfect spheres that can form some stack of round objects with more than a single layer.
Well one cannonball could be perfect balanced on another. However that reveals another ordinary language constraint. A stack should be stable. And that normally means wider at the base. And actually held together by friction, so the spheres can't be too smooth, or on too smooth a surface even.
You get the idea. Everywhere you turn, you start to encounter the ambiguous or vague elements in your little logical fables about reality.
But anyway, a stack of four sand grains seems too small to be a randomly accumulating heap. Less than four is always going to be layer at best, a scatter more likely. Yet how many more than four is evidence for a properly random pile? Doesn't this ontological demand for randomness make that answer itself statistically variable? Isn't that perhaps a key, and indeed logically valid, reason why folk don't want to commit to some hard number of sand grains? Intuitively, it would be improper to be able to mark some definite point where the heap is defined by some Platonically fixed number.
I could go on. Science and maths can keep refining our concepts of the world, and hence our capacity to be more pernickety.
One could appeal to
sphere packing theory as that indeed gives a narrow answer. Orderly stacking of cannonballs can achieve a volume-filling density of 74% while a random packing - if you could only shake them about inside a crate - arrives at a 64% density. Or at least that is the statistical average enough shaking would converge on after a reasonable time.
Maybe - psychologically informed by this new information – we might see why a heap of say five or six grains might be enough to qualify as both a pile dropped in the one place, yet with an irregular enough structure to indeed count as an untidy heap rather than an orderly stack.
We can eliminate vagueness in our concepts of nature by adding such constraints to our definition. We can increase our pernicketiness ad infinitum.
But that in turn presumes nature to be counterfactually definite all the way down to its atomistic foundations, not vague, indeterministic, stochastic or random in any meaningful way.
And we know from quantum theory, spontaneous symmetry breaking, and other modern physics that that ain't a true fact any more.
So a logic of vagueness is needed just for epistemology - our conceptual reasoning about the world. And it is needed also for ontology, as ambiguity in the guise of symmetry, tipping points, emergent dynamics, quantum indeterminacy, etc, is now an accepted aspect of reality.