Sorites paradox and an aspect of objectivity

TheMadFool

I hope everyone is familiar with the sorities/heap paradox. If not here's a Link:The Sorites paradox

I'll offer a slightly different, actually the paradox in reverse, version to say what's on my mind.

One grain of sand doesn't a heap make. Adding another will still not be a heap but carry this on for some time and we arrive at a heap of sand. The paradox is basically about how one grain of sand doesn't count and yet continue this for an adequate length of time and we have a heap of sand. Mathematically I think it can be stated as how 0+0+0+0+0+0+0+0 > 0? Each step doesn't count and yet after a certain time we have something that matters. I think it's about vagueness primarily because a heap is vague term. Any way what you should keep in mind about the paradox is simply that many nothings add up to something.

Coming to objectivity we can consider it a method for arriving at truth. There may be many definitions of objectivity but what I want to stress on is the requirment that there be an adequate number of observations. A single person's testimony amounts to very little these days. Each claim , whatever it may be, needs corroboration if it's to fly in any epistemological setup. What is notable is just like one grain of sand, a single person or observation fails to be objective. Yet, just like many grains of sand in a heap, multiple people or observations make them objective.

An important area of difference between the sorites paradox and objectivity is the former is physical and sand grains have volume and becoming a heap isn't that difficult to imagine. However, in objectivity, specifically the requirement that there be mutliple observations, things are different. Each single observation carries a not-true truth value. However mutliple observations add up to true. In logical terms how can a conjuction of non-truths (subjective) leave us with a truth (objective)?

Comments.

fresco

This so_called 'paradox' is similar to that of 'the Ship of Theseus,' in that they both illustrate the inability of classical logic to deal with the dynamics of shifting set membership as a function of transient perceptual states. See 'fuzzy logic' for possible alternative formalisms.
.

god must be atheist

A heap is what I call a heap; a non-heap is what I call a non-heap. The transition is therefore up to me, since the ultimate judge for me who decides between heaps and non-heaps is myself. At least I, for one, accept this judge's judgement.

The rest follows automatically.

To wit, my criteria for a heap is for it to look like a heap. To have the shape of a heap. That can be achieved by no fewer than 4 sand elements (if they form a tetrahedron) but they don't necessarily form it, so minimum 4 grains of sand can or can not form a heap. Similarly, any number of sand more than four in number can form a heap or a non-heap. Three or fewer can only form a non-heap.

TheMadFool

See 'fuzzy logic' for possible alternative formalisms. — fresco

So, the necessity of multiple observations for objective is in the domain of fuzzy logic?

Can you explain further? My problem is how can a bunch of lies (multiple corroborative observations each by itself not-true/false) add up to the truth (objectivity)?

I guess if we look at individual observations as carrying an "uncertain" truth value (multi-valued logic) until all observations agree we could accept this requirement for objectivity. Actually even this doesn't solve the problem because how can many "uncertains" yield a "certainty" of truth value?

TheMadFool

A heap is what I call a heap; a non-heap is what I call a non-heap. The transition is therefore up to me, since the ultimate judge for me who decides between heaps and non-heaps is myself. At least I, for one, accept this judge's judgement. — god must be atheist

That makes sense but the definition of "heap" in this case would be private and others will probably disagree with you.

god must be atheist

Can you explain further? My problem is how can a bunch of lies (multiple corroborative observations each by itself not-true/false) add up to the truth (objectivity)? — TheMadFool

All lies about facts contain an element of truth.

If a mind is capable of distilling the common elements in the lies, then chances are that the mind is touching on the truth.

This is the process what I tried to expound on in the only discussion I started so far on the forums, to show that the cave images in Plato's/Socrates' "Republic" can be assembled into Forms. My idea encountered dismal reception by the populus on this forum.

This is also how science works. A hundred people take measurements of a length of rope. They each come up with slightly different values. Therefore they don't just draw a consensus of how long the rope is, by computing the average length measured; they state a mean, and a deviance. For instance, they may say that the length of the rope is 5 feet, plus or minus three inches. That means that out of 100 measurements, approx. 66 will be between 4'9" and 5'3".

Exact numbers only exist for mathematicians. To a physicist, every measurement is expressed as it falls in a range.

god must be atheist

That makes sense but the definition of "heap" in this case would be private and others will probably disagree with you. — TheMadFool

You're absolutely right. That's why I made no bones about it, and did not declare that my proposition is the ultimate answer. I came out straight away and said under what circumstances my opinion holds.

bongo fury

That makes sense but the definition of "heap" in this case would be private and others will probably disagree with you.
— TheMadFool

You're absolutely right. That's why I made no bones about it, and did not declare that my proposition is the ultimate answer. I came out straight away and said under what circumstances my opinion holds. — god must be atheist

But what about the big picture, a poll of judgements, or of individual thresholds? What if the tail end of such a distribution (of thresholds) reaches back to a single grain? From your observations about means, we guess that it will.

Then, for some enthusiasts at least, this play of the game is over. From their point of view, you won't play. You decline to agree that a single grain is absolutely not a heap. You admit that this grain is, in the current idiom, "on the spectrum" of (usage of) heap. Albeit at one far end of that spectrum. You've lost one of the two required (and puzzlingly opposed) intuitions that we are trying to reconcile.

god must be atheist

But what about the big picture, a poll of judgements, or of individual thresholds? What if the tail end of such a distribution (of thresholds) reaches back to a single grain? — bongo fury

That I don't know.

TheMadFool

What if the tail end of such a distribution (of thresholds) reaches back to a single grain? — bongo fury

Can you expand on that?

Then, for some enthusiasts at least, this play of the game is over. From their point of view, you won't play. You decline to agree that a single grain is absolutely not a heap. You admit that this grain is, in the current idiom, "on the spectrum" of (usage of) heap. Albeit at one far end of that spectrum. You've lost one of the two required (and puzzlingly opposed) intuitions that we are trying to reconcile — bongo fury

Kindly rephrase this. I couldn't understand.

Also can you comment on the multiple observations requirement for objectivity vis-a-vis the sorites paradox.

bongo fury

What if the tail end of such a distribution (of thresholds) reaches back to a single grain?
— bongo fury

Can you expand on that? — TheMadFool

By "threshold" I hoped to refer to what you were calling a "private definition of heap".

E.g., GMBA's 4-grains-or-more. From their observations about taking mean averages I guessed they were interested in a larger pattern or distribution of different individual thresholds, in order to reconcile their own sharp threshold with their own intuition of fuzziness. And I wondered whether this was likely to end up compromising their intuition of clarity, in the case of a single grain (your starting point).

Ok so far?

Sorry for not relating my comments directly to the OP. I was interested in that particular exchange between you, as an instance of the heap game, considered as a challenge to reconcile clarity with fuzziness. For which I am an enthusiast.

fresco

Hands up who has been in a situation where the idea of 'a heap of sand' has been an issue !

Surprise...surprise !...I don't see any hands !...Maybe that's because the social dynamics we call context always renders such 'paradoxes' superficial.

bongo fury

a situation where the idea of 'a heap of sand' has been an issue ! — fresco

Any "slippery slope" issue.

... is where, at any rate, 'a heap of sand' has seemed a pertinent analogy.

fresco

↪bongo fury

Give me an example you have come across.

Terrapin Station

One grain of sand doesn't a heap make. Adding another will still not be a heap but carry this on for some time and we arrive at a heap of sand. The paradox is basically about how one grain of sand doesn't count and yet continue this for an adequate length of time and we have a heap of sand. Mathematically I think it can be stated as how 0+0+0+0+0+0+0+0 > 0? Each step doesn't count and yet after a certain time we have something that matters. I think it's about vagueness primarily because a heap is vague term. Any way what you should keep in mind about the paradox is simply that many nothings add up to something. — TheMadFool

A grain isn't nothing, by the way. It's just not a heap. Adding things to get something else isn't that unusual. We do it with things like houses, musical compositions, journeys by foot, etc.--all sorts of things.

Coming to objectivity we can consider it a method for arriving at truth. There may be many definitions of objectivity but what I want to stress on is the requirment that there be an adequate number of observations. A single person's testimony amounts to very little these days. Each claim , whatever it may be, needs corroboration if it's to fly in any epistemological setup. What is notable is just like one grain of sand, a single person or observation fails to be objective. Yet, just like many grains of sand in a heap, multiple people or observations make them objective. — TheMadFool

You're endorsing argumentum ad populums. There's no way around argumentum ad populums being a fallacy. What lots of people say only tells you what lots of people say.

T Clark

I think it's about vagueness primarily because a heap is vague term. — TheMadFool

Yes, and the word "heap" is intentionally vague. It's a messy word for a messy pile of stuff. If we wanted to be more specific, we could tighten it up. How about this:

Heap (rev 1) - An untidy collection of things piled up haphazardly such that the minimum slope of the sides intersect the ground surface at an angle of 7 degrees or more.

bongo fury

What lots of people say only tells you what lots of people say. — Terrapin Station

Which, when you want to know about usage, is what you want to know.

And, otherwise of course, not. But I am interested in usage, so I am.

bongo fury

If we wanted to be more specific, we could tighten it up. — T Clark

But otherwise, we can use it as it is. With certain embarrassing difficulties on slippery slopes, admittedly.

Terrapin Station

Which, when you want to know about usage, is what you want to know. — bongo fury

Well, or if you want to know what people like (their preferences), or what their opinions about something are, etc., sure.

bongo fury

Well, or if you want to know what people like (their preferences), or what their opinions about something are, etc., sure. — Terrapin Station

And very often (any slippery slope ethical dilemma, any artistic play with discrete perceptual categories, e.g. musical pitches), you want to work with the usage as it is, not precisified. So you need both intuitions, clarity and fuzziness. The heap game, and other natural incursions by logical thought, can make you doubt this is possible, so you abandon one or the other. Then you come to, and realise you are in a mess without both. E.g. IMO fuzzy logic: analog-digital interfacing in wolf's clothing. Or e.g. Brexit.

T Clark

But otherwise, we can use it as it is. With certain embarrassing difficulties on slippery slopes, admittedly. — bongo fury

Yes. I think the vagueness of the word matches the vagueness of what it describes.

bongo fury

Yes. I think the vagueness of the word matches the vagueness of what it describes. — T Clark

Not sure about that. If it at least means we agree that vague words are useful in all their vagueness, then cool.

But does your conception of vagueness allow you to deny absolutely that a single grain is a heap?

TheMadFool

↪bongo fury

Oh I get it. As you rightly pointed out we could study the issue statistically and average the answer but what is unambiguous to ALL is the starting point itself - one single grain of sand is definitely not a heap. This isn't true for the other end of the spectrum viz. people will vary on what counts as a heap but here too a truck load of sand will definitely be a heap of sand.

EDIT: It seems it all depends on your starting point. Given a collection of sand grains that's definitely a heap and you carry out the procedure it becomes impossible to deny that a single sand grain isn't a heap.

However, begin by denying that a single grain of sand is a heap you eventually must deny even a truckload of sand is a heap.

What this illustrates is that some concepts are simply vague and didn't require precise definitions because despite their vagueness conversation/discourse wasn't hampered.

TheMadFool

So you need both intuitions, clarity and fuzziness. The heap game, and other natural incursions by logical thought, can make you doubt this is possible, so you abandon one or the other. Then you come to, and realise you are in a mess without both. E.g. IMO fuzzy logic: analog-digital interfacing in wolf's clothing. Or e.g. Brexit. — bongo fury

This is a good point. Clarity or precision may ease the pain of argumentation but it may be impossible or even undesirable to remove vagueness from discussions. I can't think of a situation where vagueness is a crucial aspect. Do you have an example?

bongo fury

What this illustrates is that some concepts are simply vague and didn't require precise definitions because despite their vagueness conversation/discourse wasn't hampered. — TheMadFool

Except that you do want your conversation/discourse to withstand the pressure of logical clarification. The puzzle suggests that any clarification renders the clarity at one point (e.g. a single grain) incompatible with vagueness/tolerance further along. E.g.,

what is unambiguous to ALL is the starting point itself - one single grain of sand is definitely not a heap. — TheMadFool

I agree, but try setting a limit (higher than zero grains) on your projection of larger and larger samples of usage. E.g. on your statistical "support" for projected possible usage. It doesn't look very scientific to say no one could ever call a single grain a heap, after all. The intuition of clarity is lost rather easily. The heap game usually pumps it, though, which is fun, and gratifying if (like me) you think the intuition of clarity at some point is important.

I can't think of a situation where vagueness is a crucial aspect. Do you have an example? — TheMadFool

When does an abortion become unacceptably late?
How big an overdraft deserves a charge? My bank boasts that it doesn't charge for trifling amounts. Presumably it knows if it sets an exact limit for a free overdraft I will use that as my new zero credit. (Too right.)
Musical intonation, and timing. You want a performance to push the envelopes (preferably in a good way), but not play wrong notes.

ssu

An important area of difference between the sorites paradox and objectivity is the former is physical and sand grains have volume and becoming a heap isn't that difficult to imagine. — TheMadFool

I think one of the reasons why the Sorites Paradox is important and comes up so frequently isn't not only that we have a problem with vagueness. It's also that mathematics, as we understand it today, is built upon or founded on the practical need for counting. Hence we start with counting natural numbers. Now math has developed from this practical need, but it's logical foundations might not be good to be chained to counting. Now a heap confuses this thinking that "Let's start with counting" and we tend to just think of it as problem of mixing math with definitions from a spoken language.

I think it isn't just a problem of vagueness as it isn't so even in Mathematics. The problems what we have with infinity and what many mathematicians and philosophers had with what is now termed limits just shows that everything doesn't start from natural numbers and counting. The obvious reasoning what has puzzled people for long is that there cannot be a largest number and there cannot be a smallest number. Yet infinity and the infinitesimal, or limits are very useful. Both they have a different logic to them. And so does a heap.

Basically you have incommensurability between a heap and an exact number of grains. The paradox rises when we don't take into account the incommensurability between the two.

So that's what's wrong. Simply that we think every logical system can be reducted to a simple system of arithmetic. Why the paradox is so persistent is that we don't understand that incommensurability is part of the foundations of mathematics, which is extremely important for the whole system to be logical.

T Clark

But does your conception of vagueness allow you to deny absolutely that a single grain is a heap? — bongo fury

Here are some definitions of "heap" I got from the web:

An untidy collection of things piled up haphazardly.
A collection of things thrown one on another
A group of things placed, thrown, or lying one on another.

These are all consistent with my understanding of the meaning of the word. Based on that, I'm willing to state that a single grain is not a heap. Absolutely? Nothing in language, or anything else, is absolute.

bongo fury

Basically you have incommensurability between a heap and an exact number of grains. The paradox rises when we don't take into account the incommensurability between the two. — ssu

Which will be all the time, then, because our usage of heap quite clearly appeals to numerical comparisons to decide cases, and withholds the term from, well, small numbers of grains. It relates in an obvious though not necessarily exact way to usage of large number and, like that sibling concept (which provides its own popular variant on the game), it (heap) inevitably involves some sort of commensuration with the series of natural numbers. Some kind of imperfect correlation, deserving of adequate formulation, why not.

So that's what's wrong. Simply that we think every logical system can be reduced to a simple system of arithmetic. — ssu

(Leaving aside the politics of reducing or not reducing formal systems to arithmetic).
Hooray if that means you want to accord respect to usage of heap, in some way that resists correlating it perfectly with the naturals, in either of the two common but unsatisfactory perfect correlations:

correlation 1: having a different grade of heap for each natural. (Which is, effectively, some people's solution. Boo to that.) Hooray if, for example, you want to resist this correlation because you have a sense of clarity or absolutism about certain cases of heap and of non-heap, and a sense that the same clarity will transmit from these cases to certain others. I.e. a sense that a single grain is far from being any kind of a heap.
correlation 2: an arbitrary individual threshold... a policy with some good PR (e.g. "you have to draw the line somewhere, and that's that"), but which will inevitably deprive the usage of its useful fuzziness / tolerance (boo).

If you are against either of these reductions, then hooray. If your talk of "incommensurability" isn't, after all, about trying to separate usage of heap from the naturals, then even better.

If "incommensurability" means settling for both reductions, to be used according to context, then boo, and the game isn't pumping intuitions as it ought, or at any rate sometimes can.

CaZaNOx

I don't know how relevant my point is but maybe it's worth noting.

I just don't see a paradox. I think this has been pointed out already to a degree. However what hasn't been mentioned is the fact that our human brain is capable of tracking a certain amount of objects accuratley without counting (I think it's around 7-13 but am not sure). This obviously varies between humans. If the amount of elements is to big to track without counting we call it a heap or other simular concepts, that rather seem to be based on size and not ammount. It's worth mentioning that there are primate species (at least one) that seemingly can track more elements without counting. Aswell as a bunch of experiments where rats had to press a button a certain ammount of times to get a food reward. With increasing number the succes rates decrease to blind guesses. So we seem to have species related differences.

I also want to agree with the objection calling this view out for using the ad populum fallacie.

I don't see much legitimacy in using a brain mechanism used for perceptions without specific ammounts be it heaps(visual), noises, smells, touche(s?), tastes to try to establish some kind of quantity for objectivity. Because the quantity of humans beliefing a fact does not change the fact and the paradox exactly arises because one tries to applying quantity (based on countimg) to an area where it does not apply.

bongo fury

Based on that, I'm willing to state that a single grain is not a heap. Absolutely? Nothing in language, or anything else, is absolute. — T Clark

No, but I think when people are inspired to declare that a single grain is absolutely not a heap, they mean it is a safe distance outside the range of potential application of the label "heap".

This sense of distance is destroyed by one of the usual suggested reforms of usage (correlation 1, above), which is to have a grade of heap for each natural number.

Are you comfortable with that reform, for some or all vague predicates?

Wasn't there some point or utility in dividing the domain into relatively few zones? (A slightly different point, I know, but connected with the sense of distance.)

TheMadFool

↪bongo fury

↪ssu

↪CaZaNOx

An interesting aspect of the heap paradox is that, as all you have pointed out, is incommensurability is an issue. The term ''heap'' in common usage doesn't actually mean a ''certain'' number of grains. More accurately a ''heap'' includes in its definition the size of the components, the shape of the collection, in addition to the number of objects in the collection.

Therefore, to isolate one variable, the number of objects in the collection, may be a mistake. Nonetheless this is an issue for the heap paradox specifically and doesn't detract from the problem of vagueness, the central message of the paradox.

Sorites paradox and an aspect of objectivity

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