Yes, I think that the very first post I ever communicated with you on was you speaking about the idea of levels, when I began referring to the dance track, by Avicii, 'Levels.'

ould your larger interest example be similar to a reference to an analogy of a "grain of sand or a pile of sand"? If so, I would like to offer a solution of how to view the larger picture without reference to vagueness. — Don Wade

Not sure what you mean. But what I wanted to emphasise was how a developmental view of logic leads to Peirce's pragmatism and even Aristotelean finality.

The usual view of logic is Platonic. Truth just obtains. Facts are just facts. Vagueness is just a variety of epistemic ignorance or confusion. Etc.

But admitting vagueness to the fold says truth is contextual. It serves some larger purpose that happens to be operating. The purposes of some inquiry must be taken into account.

So if you are talking about Sorites Paradoxes, it does become an active matter of "who cares?". A heap or the characteristic of baldness are higher level constraints that can be imposed on acts of measurement. And the "paradox" is that in ordinary conversation, it is fine that the numerical precision is rough. We can recognise a group or a bunch or a collection at a glance. For our pragmatic purposes, there is a heap or a bald person. And we can be looser or more precise about the matter to the degree we might agree that a less vague, or even more vague, definition is useful.

And this would be a positive feature. Language would seize up if it had to be exact beyond the point that exactitude is useful. In semiotics, meaningfulness is measured as the differences that make a difference. So the more differences that we can definitely ignore - treat as the meaningless and vague backdrop - the more meaningful is whatever it is that we choose to note.

So everything depends on these kinds of reciprocal relations. More of one means less of the other. And "more or less" is then the Goldilocks balance point where you have struck some kind of useful and stable equilibrium balance in terms of knowledge, truth, whatever?

"Is that man bald?" "Is that a heap of wheat?" Given a logic of vagueness, more or less becomes the best possible answer. It's precision is contexually-based as it relies on the larger circumstance of "who needs to know".

Truth is no longer Platonic but dependent on some collective and purpose-imbued point of view. And this larger view can change its mind. It can insist on a sharper dividing line as to a definition of baldness, or relax it as well. A community of inquirers will settle on the habit of thought that provides the most meaningful-possible boundary line.

But what kind of larger interest are you thinking about that does not rely on the vagueness of a "for all practical purposes" more or less answer?

For our pragmatic purposes, there is a heap or a bald person. And we can be looser or more precise about the matter to the degree we might agree that a less vague, or even more vague, definition is useful.

And this would be a positive feature. Language would seize up if it had to be exact beyond the point that exactitude is useful. In semiotics, meaningfulness is measured as the differences that make a difference. — apokrisis

:up: :up: :up:

Language would seize up if it had to be exact beyond the point that exactitude is useful. — apokrisis

Don't you think it would seize up for the opposite reason, too? If it didn't have a syntax, and in many cases a semantics, based on clarity and the consequent possibility of potentially endless digital reproduction based on sameness of "spelling" (in the widest sense)?

The interesting (and paradoxical) thing is that the clarity is so easily achieved, by choosing obvious counter-examples. Which is what the sorites puzzle reminds us of. Occasionally. When it pumps absolutist zeal, so that the game gets started:

[1] Tell me, do you think that a single grain of wheat is a heap?
[2] No of course not, and I know I'm a long way from the smallest number of grains that could possibly be the smallest heap! Far enough that a single grain is an obvious case of a non-heap!

Of course, later on, the same player may feel differently...

[1] Tell me, do you think that a single grain of wheat is a heap?
[2] Well, certainly, it's the very smallest size of heap.

Game over. People often finish up claiming 2 had been their position all along. Perhaps it should have been, and the puzzle is a fraud. — bongo fury

Which seems to be your position. Oh well.

But what kind of larger interest are you thinking about that does not rely on the vagueness of a "for all practical purposes" more or less answer? — apokrisis

Would you believe: "I'm glad you asked".

In one sentence: Levels is the study of the hierarchy of property groupings.
An example would be a 3-dimensional book index (thought experiment).

Property-Groupings comes from the research and writings of David Hume (Scottish), and gestalt research during the early 1900's. David believed that what we perceived as vision of objects was actually visual-inputs of the properties of an object. His thinking was the brain organized the properties into what we define as objects. The gestalt researchers worked to define how the brain used properties.

Modern research shows the brain can only handle one group of properties at any specific time. An example of this is aother gestalt finding called the "Rubin Vase". One can visualize the properties of the face, or the vases, but not at the same time. Another example is the sorites paradox. One can only visualize the properties of a grain of sand (up to nine), or a pile of sand, but only one group of properties at any specific time. We have knowledge that both groups of properties can exist at the same time, but both groups cannot be visualized (by the brain) at the same time. The paradox is introduced by attempting to visualize the two groups. Knowing that both groups can exist at the same time is where the concept of levels is introduced.

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.

Yes, I think that the very first post I ever communicated with you on was you speaking about the idea of levels, when I began referring to the dance track, by Avicii, 'Levels.' — Jack Cummins

Jack, Please read my most recent post to: apokrisis. It gives a little more detail on the concept of levels.

Don't you think it would seize up for the opposite reason, too? — bongo fury

I thought I was clear that fruitful oppositions are what it is always about. So you can be too vague, and also too pernickety, in your language. As any good artist knows, what you leave out matters as much as what you put down.

The interesting (and paradoxical) thing is that the clarity is so easily achieved. — bongo fury

Or that it is always relative ... to some larger purpose.

The sorites paradox is a sharp example that should bring attention to the essential ambiguity of language. The "paradox" is to read this as evidence that language fails in its logicist ambitions - that you can produce falsehoods from apparently impeccable reasoning.

But it is logic that builds in its own problems by traditionally seeking to exclude context from judgement. It fails - if you follow it strictly - to take advantage of the fundamental resource that is vagueness.

[1] Tell me, do you think that a single grain is a heap?
[2] No of course not, and I know I'm a long way from the smallest number of grains that could possibly be the smallest heap! Far enough that a single grain is an obvious case of a non-heap!

Of course, later on, the same player may feel differently...

[1] Tell me, do you think that a single grain of wheat is a heap?
[2] Well, certainly, it's the very smallest size of heap.

Game over. People often finish up claiming 2 had been their position all along. Perhaps it should have been, and the puzzle is a fraud. — bongo fury

How is that my position?

My own response would be to question your claims of being certain that a single grain is a single grain. One can't exclude uncertainty on that score either. Maybe another grain is hidden behind it, or it is in fact a swarm of grain-lets, or a hologram, etc, etc.

To conclude that anything is not a heap is a matter of deliberative judgement as much as deciding a heap exists. You can call a single grain of wheat a non-heap "for all practical purposes", yet there is still residual vagueness or doubt in such a claim as there always must be.

You can't take one side of the paradox for granted as "a fact" and the other as always "a judgement". That would make life far too easy.

Vagueness also seems to be an integral part of our thinking even though we believe we are being precise. So, is vagueness itself a philosophy? — Don Wade

Philosophy is information about the philosophers consciousness. So it is not so much that philosophy is vague, but that the consciousness of the philosopher is. Consciousness is not a fixed quantity. It is an ever evolving process of self organization, so somewhat vague! What is not understood today may well be understood tomorrow. In the meantime there is a transition period where information is integrated and understanding adjusted such that the object of understanding fits into total understanding, this may take minutes, hours, days or years, and whilst it proceeds there exists a vagueness - that disappears once the information is integrated into total understanding, and persists if this never occurs. I think there would be an element of this gong on for all of us always, in some respect.

Consciousness is also anticipative. This has a main thrust to it, but then it allows for a certain amount of variation, such as to allow for a probabilistic emergent future. If it was always exacting in what it anticipates, then when it was wrong, would suffer a major breakdown. So understanding necessarily needs to be flexible in anticipation of various probabilistic future possibilities. Hence, in this respect, vagueness is wise and normal, and certainty of understanding is rare, and risky, imo. :smile:

I thought I was clear that fruitful oppositions are what it is always about. So you can be too vague, and also too pernickety, in your language. — apokrisis

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.

How is that my position? — apokrisis

Well,

And we can be looser or more precise about the matter to the degree we might agree that a less vague, or even more vague, definition is useful. — apokrisis

So if push comes to shove, just specify the precise (possibly unitary) size of heap. Everything is on a spectrum.

Language would seize up if it had to be exact beyond the point that exactitude is useful. — apokrisis

Still, if push comes to shove, give the exact point on the spectrum.

"Is that man bald?" "Is that a heap of wheat?" Given a logic of vagueness, more or less becomes the best possible answer. — apokrisis

Ditto.

And this larger view can change its mind. It can insist on a sharper dividing line as to a definition of baldness, or relax it as well. — apokrisis

Ditto. These all suggest,

[1] Tell me, do you think that a single grain of wheat is a heap?
[2] Well, certainly, [when pressed for details we must admit] it's the very smallest size of heap. — bongo fury

My own response would be to question your claims of being certain that a single grain is a single grain. — apokrisis

Well, that's a different game.

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.

Vagueness also seems to be an integral part of our thinking even though we believe we are being precise. So, is vagueness itself a philosophy? — Don Wade

I agree that vagueness (and/or ambiguity) is integral to our thinking. Look up a word in the dictionary and you get other words, which you can then look up, and get still other words. Without a rough sense of what basic words mean (including words like 'mean') you can't get anywhere. And this point ignores the intrinsic limitations of dictionaries. A market is perhaps a good metaphor for language. The sounds and scribbles have various somewhat predictable effects when used skillfully, without, however, even becoming perfectly clear.

Heidegger, Peirce, and Wittgenstein seem relevant here.

No communication of one person to another can be entirely definite i.e. non-vague… [W]herever degree or any other possibility of continuous variation subsists, absolute precision is impossible. Much else must be vague because no man’s interpretation of words is based on exactly the same experience as any other man’s. Even in our most intellectual conceptions, the more we strive to be precise, the more unattainable precision seems. It should never be forgotten that our own thinking is carried on as a dialogue and thought mostly in a lesser degree, is subject to almost every imperfection of language.

( from “Critical Philosophy and the Philosophy of Common-Sense”)
— C S P

Yet the obviousness and self-assurance of the average ways in which things have been interpreted, are such that while the particular Dasein drifts along towards an ever-increasing groundlessness as it floats, the uncanniness of this floating remains hidden from it under their protecting shelter. — Heidegger

I came upon this link too, which I recommend to those interested in Heidegger:

https://epochemagazine.org/15/pulling-the-normative-threads-of-heideggers-das-man/

I also found another relevant quote:

Most of human sentences are in fact aimed at getting rid of the ambiguity which one has unfortunately left trailing in the previous sentence. Now I believe this to be absolutely inherent in the relation between the symbolism of language (that is, an exact symbolism) and the brain processes that it stands for. It is not possible to get rid of ambiguity in our statements, because that would press symbolism beyond its capabilities. And it is not possible to get rid of ambiguity because the number of responses that the brain could make never has a sharp edge because the thing is not a digital machine. So we have to work with the ambiguities. And nearly all discussions about Turing’s theorem or about poetry always come back to the central point about ambiguity. One of my fellow mathematicians, William Empson, who did mathematics with me at Cambridge, turned to poetry and at once published a book called Seven Types of Ambiguity–it is still a kind of minor bible, but a bible written by a mathematician, never forget that.

Ambiguity, multivalence, the fact that language simply cannot be regarded as a clear and final exposition of what it says, is central both to science, and, of course, to literature. — Brownowski

https://www.waggish.org/2011/jacob-bronowski-william-empson-wittgenstein-and-ambiguity/

So as Don says, — apokrisis

... and then, spectrum, spectrum, spectrum.

[1] Tell me, do you think that a single grain of wheat is a heap?
[2]Well certainly, a single grain is the very smallest size of heap. — bongo fury

You might at least now see how that is your position.

Sure, a spectrum has extremes. What the puzzle often reminds us, though, is that, pervading language, there is a subtly (puzzlingly) different way of looking at it. Bald and hairy, black and white, on and off, heap and whatever its potential antonym (pittance?)... they all operate perfectly well as alphabets (or conceptual schemes) of two characters (concepts) separated by a comfortable no-mans-land. The puzzle is how to look closely at that without it reverting (under however much cover of mystical pazazz) to a mere spectrum.

The delightful thing about the sorites is that it can spring up again from the rubble...

Or rather, the right predicate value is "vague". — apokrisis

Ah, so maybe we have a new game?

[1] Tell me, do you think that a single grain of wheat is vaguely a heap?

Granted, we may not. You may not be inclined, as I am, to respond,

[2] No of course not, and I know I'm a long way from the smallest number of grains that could possibly be even vaguely the smallest heap! Far enough that a single grain is an obvious and non-vague case of a non-heap!

... leading in turn to the intrigue of,

[3] And would you agree that adding a single grain could never turn a definite case of non-heap into a vague case?

... and so on.

No communication of one person to another can be entirely definite i.e. non-vague… — C S P

But yes it can in the sense that we can reproduce digital or alphabet-based text or speech or music indefinitely. Puzzling, certainly, when we look closer at the fuzzy boundaries of the characters, phonemes, notes and tones.

And now, my spam: How to look closer.

I looked at the post you referred to and it seems that the philosophy of levels is about viewing from a closer level in contrast to seeing from the larger perspective. I came across an associate idea when I was studying English literature at school, which was the idea of the microcosm and macrocosm as perspectives. This distinction has a history going back to Aristotle, but you are quite possibly familiar with it, and perhaps it is part of your own philosophy.

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. — apokrisis

Ah, the beauty of Levels.

A point we have not discussed is: What happens, in our minds, to one group of properties when we delibrately switch to another group? For instance: If we are focused on the grain of sand, and switch to the pile of sand - what happens (in our mind) to the grain of sand that was our original point of focus? It seems to disappear. We can only have one group of properties in our mind at any specific time. Such as: we can focus on the grain of sand, or the pile of sand - but not both. (That is, not at the same time.) This is similar to the (Rubin Vase) analogy. We will be aware of the other group - but the mind can't visualize both groups at the same time. This thought experiment demonstrates the basic reason for the sorites paradox. Many philosophers are still not aware of how the mind only visualizes one property-grouping at a time. Levels incorporates this phenomenon.

I looked at the post you referred to and it seems that the philosophy of levels is about viewing from a closer level in contrast to seeing from the larger perspective. I came across an associate idea when I was studying English literature at school, which was the idea of the microcosm and macrocosm as perspectives. This distinction has a history going back to Aristotle, but you are quite possibly familiar with it, and perhaps it is part of your own philosophy. — Jack Cummins

Thanks Jack. Yes, Aristotle - in my opinion - was both good, and bad for philosophy. (Another discussion point.)

The delightful thing about the sorites is that it can spring up again from the rubble... — bongo fury

Yes, I think you're right. I see it in many instances.

I agree that vagueness (and/or ambiguity) is integral to our thinking. Look up a word in the dictionary and you get other words, which you can then look up, and get still other words. Without a rough sense of what basic words mean (including words like 'mean') you can't get anywhere. And this point ignores the intrinsic limitations of dictionaries. A market is perhaps a good metaphor for language. The sounds and scribbles have various somewhat predictable effects when used skillfully, without, however, even becoming perfectly clear. — T H E

I believe "Levels" will clear up a lot of the vagueness.

... they all operate perfectly well as alphabets (or conceptual schemes) of two characters (concepts) separated by a comfortable no-mans-land. The puzzle is how to look closely at that without it reverting (under however much cover of mystical pazazz) to a mere spectrum. — bongo fury

A spectrum suggest unbroken continuity. But the sorites paradox demands discrete acts of addition or subtraction. So we have the two poles of a metaphysical spectrum right there. The discrete~continuous. And the confusion arises in trying to satisfy these two formally antithetical constraints at the same time.

The point on the spectrum marking the division between heap and non-heap shouldn't ever be just a one single grain jump if it is to satisfy the metaphysics of continuity. But a multi-grain leap - say a jump from three to eight - is deemed a failure by a metaphysics of discreteness, as each individual addition or subtraction is taken as separately ennumerable.

So all you seem to be pointing out is this clash of metaphysics. The world must either be discrete or continuous at base. Pick your poison. The PNC and law of the excluded middle leave you no choice but to take a side in traditional logic.

But that is where a logic of vagueness comes in. It can add a third metaphysical-strength ingredient to the story. It says that both poles of any such categorical dichotomy must arise - by reciprocal constraint - out of the common resource which is a vagueness.

This is the developmental view, or symmetry breaking view, of nature. Any sharp disjunctive distinction - like that your two "mutually exclusive and jointly exhaustive" options are either discrete, or continuous – must arise from Peircean Firstness, or a ground of simple unformed potential. A vagueness where the PNC has yet to apply as a dichotomising constraint.

And as I say, the twist is to see ambiguity as a general resource rather than a fundamental problem for logic.

The sorites paradox - in revealing a fundamental clash between the two notions of the continuous and the discrete - suggests something is broken with logic's three conventional laws. But the flip view is that it shows that language only needs to constrain interpretation to the degree it is contextually useful. We can live quite well with inherent ambiguity because metaphysical dichotomies speak of the opposing limits of being, not two actualised states of being. One can approach either limit of being as closely as one likes - by constraints that exclude the other pole of being - but one can't actually reach and exist at that limit.

So neither perfect discreteness, nor perfect continuity, can exist alone. Each quality is always held to be relative to an act of limitation on the presence of its "other".

That gets tricky with the sorites paradox as it asks you to mark a definite transition point on a spectrum. And intuitively - to honour the metaphysics being invoked - you don't want to give privelege to either an answer that is clearly discrete (one more grain does it), nor clearly continuous (eventually and smoothly you have enough).

Some kind of halfway house answer must be the case - one that speaks to the continuous as much as the discrete. And that is where the undecided, unshaped, potential of a vagueness comes in to rescue you - if you are willing to expand your logical system.

But yes it can in the sense that we can reproduce digital or alphabet-based text or speech or music indefinitely. Puzzling, certainly, when we look closer at the fuzzy boundaries of the characters, phonemes, notes and tones. — bongo fury

This touches the issue of the signified versus the signifier. I agree that digital copying is especially impressive. We tend not to lose a single bit. But the meaning of bits (the signified) seems to remain somewhat vague. What we mean by vague in different contexts (however easy it is to encode v-a-g-u-e) is itself vague. One reason for this IMO is that meaning is 'out there' and not 'in here.' There are fuzzy conventions for using words in the contexts of also-conventional actions. No one has to have an exact idea in mind as long as they have the rough skill to get by in the world.

I believe "Levels" will clear up a lot of the vagueness. — Don Wade

I have enjoyed your posts. It seems that they help explain vagueness. I'm personally coming largely from Wittgenstein's analysis of pain. What is pain? We all think we know it intimately. That's part of its grammar. But the meaning of 'pain' can't depend on anything private.

We can only have one group of properties in our mind at any specific time. Such as: we can focus on the grain of sand, or the pile of sand - but not both. (That is, not at the same time.) This is similar to the (Rubin Vase) analogy. We will be aware of the other group - but the mind can't visualize both groups at the same time. — Don Wade

What you are talking about is taking two different attentional points of view.

So in general, the brain is evolved to characterise scenes in terms of dichotomies that symmetry-break reality in its most informational or meaningful way. That is where you get metaphysically-broad operations like lumping vs splitting. The brain is set up – with left vs right asymmetry, for instance - to either group or individualise some clutter of visual elements. And by habit, we will learn to read the world in a way that is most informational in terms of our wants. We will automatically lump and split as appropriate.

But both the parts and the wholes get represented because the brain is actually dichotomised in its wiring. And by attentional choice, you can switch between points of view, either focusing on the parts or the wholes.

The contrast can be made stark by a choice of artificial stimuli like letter navons.



So in my view here, the connection between hierarchical levels and the sorites paradox is that the brain is evolved to apply a dichotomising logic on the world as that is the view which always must deliver the maximum salience or useful information. The brain is set up to say if it ain't lumped, it must be split. And vice versa.

And dichotomies are symmetry-breakings taken to the limit - and hence result in the fundamental asymmetry represented by a local~global hierarchical division.

Hierarchies are the (triadic) organisation that represent the final outcome of the (dyadic) act of dichotomisation or symmetry-breaking. And then that leaves vagueness as the (monadic) symmetry - that starting state of cognitive indeterminacy or unconcern - which completes the 1,2,3 that is the Peircean logical system.

What I am trying to point out is that it is no surprise the brain is structured to process the world in this one particular way. It is the only logical way.

Cognition always must start with generalised indeterminacy - anything could be the case. Then it must apply some filtering dichotomy - anything might be the case, but let's see how it might fit some formal opposition of "definitely more this than that". The clarification provided by being able to apply the law of the excluded middle.

And eventually, as the brain evolves to process real scenes in the most efficient and informational ways possible, the asymmetry of a hierarchical organisation will emerge. The proper view of any scene will be divided along local vs global analytical lines.

Lumped vs split, individual vs collective, salient vs peripheral, etc.

So it is not that the mind can't entertain two opposing views at the same time - which makes it sound like some kind of processing shortcoming. Instead, the whole point is to be able to emphasise one view over the other potential view. It is the feature rather than the bug. We get to pick the version of reality that is most functional or informational in light of our perceptual goals.

But that is where a logic of vagueness comes in. It can add a third metaphysical-strength ingredient to the story. It says that both poles of any such categorical dichotomy must arise - by reciprocal constraint - out of the common resource which is a vagueness. — apokrisis

It is my opinion that the solution to the sorites paradox is not in the assumption of "vagueness" - as has been suggested. The solution lies in the method of analyzing the problem.

Try this thought experiment: The brain can visualize (1) grain of sand. The brain can also visualize (2) grains of sand that are (side-by-side), or close together (gestalt). Add another grain in the same manner and the brain can still visualize the image of (3) grains of sand - as long as the distance between them is not too great. The problem comes in by adding just one more grain of sand. The brain cannot visualize (4) grains of sand that are close to each other. In order to visualize four grains of sand the brain must employ a trick - that is, it will visualize two groups of (2) grains each. The brain can actually visualize up to (9) grains in this manner by visualizing three groups of (3) grains each. However, the brain cannot visualize (10) grains, or more. The brain can also visualize (1) pile of sand - in much the same way as visualizing the single grains of sand. However, the brain cannot visualize removing a single grain of sand from a pile and somehow changing the image of the pile. This thought experiment demonstrates why the paradox is not based on "vague predicates", but is based on how the brain visualizes images.

You do raise a nice issue, but let's consider a different example.

'I have a few errands to run first.' Why might I say that instead of 'I have two errands to run.'?

Well the person I'm talking to might have no need for the distinction between 2 or 3 or 4, or whatever we vaguely include in a few, which might vary by context, especially if we allow humor in the mix. Did Jay-Z actually have 99 problems? https://en.wikipedia.org/wiki/99_Problems . An exact number can actually function as a symbol for a large number.

The larger issue is about purpose and social context. What's the right tool for the job?

The brain cannot visualize (4) grains of sand that are close to each other. In order to visualize four grains of sand the brain must employ a trick - that is, it will visualize two groups of (2) grains each. — Don Wade

What? Visualising four grains seems easy. Especially if they are arranged as four corners of a square.

An irregular group of four grains is more of a stretch. But we can also learn that as a pattern.

So the point is that all perceptual judgements rely on both the local and global. We are visualising the relations as well as the parts the entire time.

A single grain has the global feature of zero relations. We are actively suppressing the sense of any connectedness to extremetise our mental state of representation.

Two grains are related in linear fashion, Three grains is triangular. Four as a square. We could go five as a pentagon. But also this simple conceptual geometry is under strain as the relations - in any idealised group - are all members to all members.

So all four are touching, and all five are touching. Eventually, and quite quickly, we cross over from the sense of looking at a scene dominated by parts to a scene that is a mess of relations. Relatedness becomes what we "see" if pushed to give a logically dichotomised reply. We see simply "many grains" with the sense of isolated parthood maximally de-emphasised.

This thought experiment demonstrates why the paradox is not based on "vague predicates", but is based on how the brain visualizes images. — Don Wade

My argument is that the brain visualises by dichotomising scenes. And that in turn relies on hierarchically organised states of constraint. One metaphysical aspect of the scene has to be suppressed at the expense of the other ... so as leave the other as the one being emphasised.

Your choice in a world that is only ever relatively lumped or split is to visualise its degrees of connectedness or division, its degrees of integration or differentiation. And once you get into counting games - mathematical-level semiosis - that runs into cultural misconceptions about the world actually being always definitely one thing or the other. Like either discrete, or continuous, as that is what the formal laws of thought appear to demand.

But language-level semiosis is more relaxed - more tolerant of vagueness or ambiguity. And that better suits the real world of social actions. You don't have to force everything you believe or perceive into rigid or permanent categorisations. A heap is whatever suits a community of speakers in the pursuit of their social purpose. A pseudo-mathematical precision is making a fuss about differences that don't make a sufficient difference.

Then actual brains are evolved to serve an even more relaxed level of judgement - neural semiosis. That is why an animal sees the world largely as one or many. This human level obsession with either linguistic or mathematico-logical clarity is baffling to them. It is not in fact wired into the brain as a habit of visualisation.

So another problem with your analysis is mixing up levels of semiosis. At least three levels of "visualisation" are taking place. Each can try to sharpen the distinctions naturally offered by the one below. But the important question then becomes "for what purpose?"

Brains just want to visualise in a general lumping and splitting fashion. They want to present a world divided into focus and fringe, grouped and scattered, individual and continuous, etc, etc.

Language has its social purpose. Maths and logic move up to the Platonic abstraction of countable numbers - data, or information bits – that represent reality as if it were digitally discrete. Totally unambiguous.

This is just a useful extremetisation of metaphysics. Treating reality as a material machine is the basis of our technological way of life.

But for philosophy, this reduction to the countable, the digital, is a problem. It builds in a big mistake. And that is what a Peircean stepping back fixes. It makes it explicit that logical semiosis is a triadic sign relation. And vagueness is the underpinning to that.

A spectrum suggests unbroken continuity. — apokrisis

I suppose "spectrum" ought to have come with a health warning: it alludes only to the vague non-technical attitude of mind that "everything is on a spectrum". Which nicely describes the attitude of this play-stopping first reply:

[1] Tell me, do you think that a single grain of wheat is a heap?
[2] Well, certainly, it's the very smallest size of heap. — bongo fury

Yes of course "spectrum" might suggest unbroken continuity. The fact that the sorites doesn't have to is for me one of its most attractive features. So, warning not too late, I hope.

That's why I said that doubting that a single grain is a single grain is playing a different game... An interesting variant, quite possibly. And I admit that black vs. white (or red vs. yellow) does raise the question of continuity (or at least density... with terminology potentially confusing there also), and that that question is deeply relevant to the issues that concern us. But the classic heap version, as well as bald vs. hairy, and "small" vs. "large" number, show that the question is removable.

Equally so with the colours. We can simply use a sequence of different (discernibly or indiscernibly, it doesn't really matter) shades and - with or without presenting actual samples of the shades, it doesn't really matter either - ask the same questions about those as about the numbers of grains or hairs.

Note that @Don Wade's version is essentially that one. His sequence of photographs may be either discernibly or indiscernibly different, and need not correspond to each cardinal size of grain-collection. (Except for the first few.) Each step in the sequence might represent a fairly large addition. Just as the actual change in luminance from shade to shade is arbitrary.

The sorites starts from the happy reality of being able to order a sequence of objects (grain collections, heads, photos) in correspondence with the natural numbers and to discuss choices of how to superimpose a dramatically smaller ordering on the same objects.

But the sorites paradox demands discrete acts of addition or subtraction. — apokrisis

Yes, and these create enough of a puzzle.

So we have the two poles of a metaphysical spectrum right there. The discrete~continuous. And the confusion arises in trying to satisfy these two formally antithetical constraints at the same time. — apokrisis

No, not at all, the discrete version is enough.

All of your strenuous metaphysics might be missing the point.

But the meaning of bits (the signified) seems to remain somewhat vague. — T H E

Yes, with the interesting exception of systems of notation, as investigated by Goodman (along with the varieties of vagueness) in Languages of Art.

What? Visualising four grains seems easy. Especially if they are arranged as four corners of a square. — apokrisis

I applaud your effort, but visualizing a square (one shape - or (1) item) is not the same as (4) distinct grains. One can also physically count grains as they are placed, and count many grains. But, that same person cannot "imagine" the four seperate grains without some form of added aid - such as what you just demonstrated. Again, I refer to the example of the Rubin Vase: https://en.wikipedia.org/wiki/Rubin_vase . Another example of something there but not visualized is: "Gorilla in the room" experment: https://gorillaitr.com/2018/08/21/update-to-the-original-gorilla-in-the-room-experiment/#:~:text=Update%20to%20the%20original%20Gorilla%20in%20the%20room,they%20miss%20a%20lot%20of%20intuitive%20%28non-conscious%29%20information.

Yes of course "spectrum" might suggest unbroken continuity. — bongo fury

It both might and usually does....

A spectrum is a condition that is not limited to a specific set of values but can vary, without steps, across a continuum.

No, not at all, the discrete version is enough.

All of your strenuous metaphysics might be missing the point. — bongo fury

I think I was explicit enough. The problem is with the kind of monadic logical system you are championing. A logic of vagueness - a triadic logical system - is needed to situate the sorites paradox in a more intelligible world than that provided by mere counting.

I haven't missed any point. I just supplied a missing meta-logical argument.

I applaud your effort, but visualizing a square (one shape - or (1) item) is not the same as (4) distinct grains. — Don Wade

I can picture four grains without a problem. I merely point out the psychological machinery involved. It helps to have the simplest and most regular global arrangement in mind, even if that geometry of relations is then also suppressed to a large degreed to emphasise the distinctness of each grain.

Again, I refer to the example of the Rubin Vase: — Don Wade

A bistable stimulus is a rigged and artificial cognitive situation. So it shows interpretations of scenes can be pulled two ways - if the scene is designed to have precisely that characteristic. The image is created so that it is literally a black and white, cut and dried, situation. Pick either one vase or two faces as your only legitimate choices. The PNC applies. In fact, that is what the image actually illustrates.

But if we are talking about how perception applies to the real world, then that is where vagueness certainly becomes a valuable expansion of the logicist's impoverished world model.

Yes, with the interesting exception of systems of notation, as investigated by Goodman (along with the varieties of vagueness) in Languages of Art. — bongo fury

I need to get around to Goodman. But if you mean the exception of formal languages, then I agree. Of course there's still the 'problem' of how formal languages connect to the practical world. What do I do with the value of an integral? Perhaps I buy a certain amount of paint, implying some kind of rough connection between moves in a symbolic game and the amount of pain that one should by is one is rational and/or prudent.

It both might and usually does.... — apokrisis

Yes, but as I say, a nice feature of the sorites is how it shows that the vague and non-technical usage "everything is on a spectrum" can be interrogated, with interesting results, even on the discrete interpretation, "everything is on a scale of tiny steps", or "there is only (some large number of) shades of grey".

And it's not clear that a non-discrete interrogation could look anything like the sorites, where a problematic place is reached worryingly soon: it would probably have to be more like one of Zeno's puzzles, where you can't get anywhere.

:up: