Really? That would make a difference in the case of a single grain or hair?

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- May 12th 2012, 11:44 PM #16

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- May 14th 2012, 03:54 PM #17

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## Re: Sorites only joking...

my thinking runs as follows:

if we accept a bivalent truth-system, then a number of wheat either is, or is not a "heap". the question then becomes: if N, why not N-1? it's rather hard to make a good argument for any value of N besides N = 1, or N = 2.

fuzzy logic allows us to establish a range of "semi-heapness", to which N = 1 may, or may not belong.

so yes, it does make a difference.

- May 15th 2012, 01:06 PM #18

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## Re: Sorites only joking...

Ah, right. Thought I ought to check, to see if we could play any further.

Apparently not.

You, from my point of view, fail to be impressed by the absolutist force of the puzzle's opening question, second time around at least.

The round we just played kind of took for granted the first two. The first time, you probably would have indulged me by answering no... a single grain certainly isn't a heap.

But second time, having endorsed the relativist/gradualist claim with noticeably more enthusiasm than did Mathhead200 at first, you are pretty much resolved to sacrifice absolutism.

Mathhead200 forced me to end his first game when he unapologetically diagnosed the relativist/gradualist claim as a fallacy. Thankfully, he was persuaded to try endorsing the second (r-g) question in a form he was more inclined to defend. My problem then was to try to make sure his absolutist conscience didn't desert him.

We started at a similar point, you defending a form of the r-g claim with what seemed to me a worrying lack of regard for absolutism. So, whether or not you would even have indulged absolutism the first time round, as I have speculated, you now were perfectly content to bury it.

My re-posing (twice) of the first question failed to prick any absolutist conscience, though. You seem unable to view the idea of a one- or two-grained heap as an absurd enough thing to motivate further rounds. So the game must end, probably with some sense of mutual incredulity.

Thanks for playing, though, and I'd be grateful to get your perspective on the game. (By the way, I'm rather curious as to why you need to talk of degrees of truth of heap-ness, and not just degrees of heap-ness?)

- May 16th 2012, 12:16 AM #19

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## Re: Sorites only joking...

there are two things at play, here: the notion of MEANING (essentially a linguistic notion), and the notion of TRUTH (essentially a logical notion). the thing is, we only have one (or two, if you want to allow mathematics as "a language unto itself") vehicle to carry both these burdens: namely, the discourse we are undertaking.

for example: if we agree that "a heap" should mean "a definite something" (although we have not yet determined what that "something" actually IS), we are implicitly accepting a bivalent logic: an amount X of wheat either is, or isn't, a heap. this is the principle of the excluded middle, which only allows 2 truth-values for a statement: true, or false.

common sense dictates a "heap" should be "some amount" of wheat (one would never call an empty container with no wheat in it at all, a "heap of wheat"). so we are forced to conclude that whatever amount determines "heapness", it must be positive. so we can establish a minimum of at least 1. "common sense and experience" similarly leads us to reject "a single grain" as a "heap", unless we want to abandon all pretense of using the words we are using in their usual sense. i believe we could quickly come to an agreement that a "heap" is a PLURAL turm, meaning our new minimum of "heapness" is 2.

now 1,000,000 seconds is around 11-1/2 days (without sleep!) so we could probably agree that 1,000,000 is "sufficiently large" that counting the individual grains is impractical, and could serve as a "gross" upper bound for the "threshold of heapness" (see what i did there?). but the negotiation between 2 and 1,000,000 is fraught with peril. there is bound to be a degree of "arbitrariness" in whatever compromise we come to reach. and reach one we must, or banish the term from our speech forever. for in an "absolutist" (bivalent logical) view, "heap" must mean "something", or it means "nothing".

the appeal of the "relativist" (or many-valued) approach, is that it allows us to be vague about the details. we can have a single shade of indeterminacy, or a continuum of such:

(a heap, maybe/maybe not a heap, not a heap)

(definitely a heap, perhaps a heap, mostly a heap, sort of heapish, a bit heapish, not very heapish, perhaps not a heap, definitely not a heap)

(...etc.....)

while this view is easier to defend, it has the disadvantage of giving us no idea what a heap actually may be, and if the in-between stages are finite in number (as they will be if we decide on definite demarcations for "definitely/definitely not"), they are subject to the same suspect negotiation as the bivalent case.

the thing is: language is not quite a function from reality to expression. parts of it are not "well-defined" (is a few two? or five? or both? or neither? or all of the preceding?). there's "fuzziness" at the edges, so even if i say perfectly well what i mean, and you understand perfectly well what i said, what you understand may still not be what i meant. poets exploit these vagaries, and grammarians debate them. somewhere, some group of people (perhaps a small group of only two, such as you and i) is no doubt discussing a similar conundrum, with the unfortunate result, perhaps, of intending to have the department of agriculture (or perhaps of weights, standards and measures) "officially" declare that a heap is no less than 10,000 grains.

we ACT as if words have a "definite" meaning, but we USE them as if their meaning was "elastic". over time, their "actual" meaning (in terms of consensual agreement) tends to evolve, so that a quantity thought of as "immeasurably large" in prehistoric times, may be thought of as only "middling large" in a modern context. this kind of dynamic tension seems unavoidable, as new situations which have never occurred before are always cropping up, and language "stretches" to accommodate this.

in terms of the game, my perception is you want to see if you can find "the knife-edge" at which the negotiation either succeeds, or blows up. i see pitfalls in any "reasonable" approach.

- May 16th 2012, 01:47 PM #20

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## Re: Sorites only joking...

Good. That's how I see it. I think the puzzle challenges us to give a consistent picture of a vague boundary. We could say, one that avoids logical blow-up.

I hope 'knife-edge' refers to the entertainment value of the game, though, and not some perceived insensitivity on my part towards relativist-gradualist feelings. I shall try not to condone any depicting of a knife-edge boundary for a vague category like*heap*.

i see pitfalls in any "reasonable" approach.

common sense dictates...... unless we want to abandon all pretense of using the words we are using in their usual sense.

Not that we should shrink from refining and adjusting our self-evident truths (axioms, premises, values, whatever) so they fit together.

But the puzzle invites us to test the adjustments by having to play, in a realistic manner, the role of competent user of the language.

If you can't answer a straight no to my various versions of the first question then many observers (a good proportion, I hope) will judge that you have forgotten how to use the word 'heap'.

Saying...

well, are we going to be working in a bivalent truth system, or something fuzzy?

To me, (please forgive the tone of exasperation, probably inevitable at this stage of the game) that's as good as saying

I feel a tiny bit of doubt, because I can't help but admire a 'wrong' answer to the mother of all rhetorical questions!

common sense dictates a "heap" should be "some amount" of wheat (one would never call an empty container with no wheat in it at all, a "heap of wheat"). so we are forced to conclude that whatever amount determines "heapness", it must be positive. so we can establish a minimum of at least 1. "common sense and experience" similarly leads us to reject "a single grain" as a "heap", unless we want to abandon all pretense of using the words we are using in their usual sense.

i believe we could quickly come to an agreement that a ... minimum of "heapness" is 2. now ... we could probably agree that 1,000,000 ... could serve as a "gross" upper bound for the "threshold of heapness" ... but the negotiation between 2 and 1,000,000 is fraught with peril.

the appeal of the "relativist" (or many-valued) approach, is that it allows us to be vague about the details. we can have a single shade of indeterminacy, or a continuum of such:

(a heap, maybe/maybe not a heap, not a heap)

(definitely a heap, perhaps a heap, mostly a heap, sort of heapish, a bit heapish, not very heapish, perhaps not a heap, definitely not a heap)

(...etc.....)

while this view is easier to defend, it has the disadvantage of giving us no idea what a heap actually may be, and if the in-between stages are finite in number (as they will be if we decide on definite demarcations for "definitely/definitely not"), they are subject to the same suspect negotiation as the bivalent case.

somewhere, some group of people (perhaps a small group of only two, such as you and i) is no doubt discussing a similar conundrum, with the unfortunate result, perhaps, of intending to have the department of agriculture (or perhaps of weights, standards and measures) "officially" declare that a heap is no less than 10,000 grains.

- May 23rd 2012, 05:42 PM #21

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## Re: Sorites only joking...

A recap from my point of view:

__Game 1.__

**Me:**Tell me, do you think that a single grain of wheat is a heap?

**Questioner:**Certainly.

**Me:**And do you agree that adding a single grain could never turn a non-heap into a heap?

**Questioner:**Not at all. This question may cause some people to wonder how any particular number could deserve, in preference over its neighbours, the privilege of defining the smallest heap. But I suggest that 3 is quite special enough.

Questioner's position:

Further play was difficult, as we were apparently unable to feel the same intuitive force from the second question, or from Questioner's answer to it.

I also felt that absolutism is confounded by a threshold of 3 grains, though, and I hoped that the hairy man version of the puzzle might better rekindle that intuition for Questioner.

__Game 2.__

**Me:**Tell me, do you think that a single grain of wheat is a heap?

**Mathhead200:**Certainly.

**Me:**And do you agree that adding a single grain could never turn a non-heap into a heap?

**Mathhead200:**Provisionally, yes. But since I find that both premises taken together prevent a million grains from being a heap, I have to reject this second premise as a fallacy. I don't necessarily accept Questioner's threshold of 3, but I can see there must be such a threshold.

__Game 3.__

**Me:**Tell me, do you think that a single grain of wheat is a heap?

**Mathhead200:**Certainly.

**Me:**And do you agree that adding a single grain could never turn a non-heap into a heap?

**Mathhead200:**Kind of. Perhaps there is a (at least one) state that wheat can be in that is both not a heap and not a non-heap. What shall we call this (one of these) state? A half-heap, a part-heap, a maybe-heap, a vague-heap, ...?

**Me:**I like your idea of recognising 'shades of grey' between non-heap and heap, but I worry that in answering to the challenge of the puzzle's second question (the relativist/gradualist one) you have compromised your commitment to the first (the absolutist one). So, tell me, do you think that a single grain of wheat is even minimally a heap?

**Mathhead200:**Hold on there! I didn't quite commit to a great many shades of grey. Anyway, I would want to look carefully at the situation of heap, non-heap and the ground in between, no matter if that ground were a long succesion of shades.

**Me:**Ok, let me re-ask my question? Let's call the non-heap that isn't even in the middle ground an 'extreme-non-heap'. Do you think that a single grain of wheat is anything other than an extreme-non-heap?

Me slightly jumpoing the gun:

Or else:

When I think this kind of picture is envisioned, I want to use a version of the first question to ignite absolutist doubts.

But Mathhead200 seemed to be heading instead towards something like:

So I had to re-re-ask... expecting to debate about the threshold to the middle ground.

__Game 4.__

**Me:**Tell me, do you think that a single grain of wheat is a heap?

**Deveno:**Well, surely that depends on whether the system you are working with looks like graph number one or like graph number 3.

Further play was difficult, as we were apparently unable to feel the same intuitive force from the first question.

Anyway, thanks for those games! I hope I haven't mis-represented any of them too badly.

I sense that there are plenty of further places for the game to go (and not necessarily ones that I have thought about). So, once again...

Tell me, do you think that a single grain of wheat is a heap?

- May 29th 2012, 02:38 PM #22

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## Re: Sorites only joking...

this reminds me of an old joke (made somewhat "family-friendly" to avoid breaking forum rules):

a man asks a woman: "will you marry me for a penny?" the woman, clearly insulted, turns him down curtly.

he then asks her: "will you marry me for a million dollars?" her eyes light up, and she says "oh yes!"

again, he asks her: "well, then, will you marry me for a penny?" she looks puzzled.

he explains: "we've already established that you will marry me for some price. now, we're negotiating."

******

i am also reminded of the "surprise quiz puzzle". a math teacher announces that there will be a surprise quiz the following week. the students reason the test cannot be on Friday (for then it will no longer be a surprise), so the test can only take place on Monday though Thursday. But then, it cannot take place on Thursday, either, since by the time Wednesday finished, everyone would know that it would be on Thursday (since Friday is out). and one by one, every day but Monday is ruled out, so the test must occur on Monday, which will surprise no one. the students therefore conclude, that there can be no surprise test at all. so, when the teacher gave the test on Tuesday, everyone was surprised!

- Jun 6th 2012, 01:51 PM #23

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## Re: Sorites only joking...

__Game 5.__

- Tell me, am I worth marrying for a penny?

- Certainly not.

- But am I worth marrying for a million dollars?

- Certainly. For the chance of raising such an amount for charity, anyone might be prepared to accept your proposal.

- And do you agree that adding a single penny to my dowry could never make me worth marrying?

- Kind of. Some particular additional penny would have to make the difference for each potential suitor, but the distribution of these individual sharp thresholds will look increasingly smooth, the larger the sample of potential suitors. And we could say, at least, that in any reasonably-sized sample, adding a single penny could never turn you from a specimen with no suitors at all into an absolute must-have.

I don't know whether this compromise is likely to satisfy your relativist/gradualist zeal. Perhaps not, as market forces are going to identify one particular threshold as special. If that is a problem for you, though, then your puzzle probably isn't going to get much traction using the theme of marriageability. Because letting the market find a price is obviously a perfectly intuitive means of 'drawing a line'. (Which is what we generally recognise is something we 'have to do somewhere'.)

You might disconcert me by continuing a bargaining or haggling process with bids that I think would invite or project ridicule. But not by suggesting a threshold price.

I suspect the original (heap) version of the puzzle will suffer a similar deflation under a similar kind of treatment (not necessarily monetary, but based on negotiation, and surveying of opinion or demand).

- I suspect it will react vigorously instead! And I look forward to finding out.

But you are right that the current game is about to deflate. Not (at least not directly) because of the threshold price, but because the puzzle needs absolutist assurances regarding the distribution of thresholds. It doesn't approve of a framework whose support reaches all the way to zero or one (like graph number 3).

Tell me, is it conceivable that anyone could ever agree to marry me for a penny?

- Well, you were right to think that the assurances are not forthcoming! Come on now, don't do yourself down! Someone might come along... anyway, how absurd to exclude the possibility! And for the puzzle to work, I thought you needed premises to be so secure intuitively that
**denying**them feels absurd?

- Ok, so game over. But notice how different the outlook were we discussing not some empirical outcome like someone eventually coming along, but an outcome that seems absurd just by definition?

- Jun 23rd 2012, 02:07 AM #24

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- Jun 23rd 2012, 02:57 AM #25
## Re: Sorites only joking...

Let go back. (Let $\displaystyle P(n) = NH(n)$.) I feel we can definitively assert $\displaystyle 0 \le M < N$. Now you had a problem with this, but

*I*don't. Negative numbers just don't apply in this situation (and there's good argument to whether they exist at all), so let's imply safely (now) that $\displaystyle P(0)$ must be true.

Now for the sake of argument and as to not detach ourselves from the "real world" usefulness of the word "heap" I feel I can safely assert $\displaystyle P(1)$. Otherwise all existent amounts of wheat be heaps. However, this is an unproven assertion. I reiterate: I assert $\displaystyle P(1)$ for the sake of argument and usefulness. It is not a logical nessesity (I hope. ) (We are creating a new logical system here anyway... Are we not?)

- Jun 23rd 2012, 03:04 AM #26
## Re: Sorites only joking...

Oh! And just so we're clear. I just want to say: I love what

**Deveno**did to come up with $\displaystyle N \le 1,000,000$. However, I completely refuse to work this problem in a "fuzzy" logical system. First, it somewhat takes the fun out of the paradox; but second, all math (that I know, and I am by no means an expert) is still based on deductive (boolean) logic. Even those percentages between 0 and 1.

- Jun 23rd 2012, 02:24 PM #27

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## Re: Sorites only joking...

Hey, Mathhead200! Thanks for returning. By the look of things, we're able to continue Game 3, as I'd hoped.

First though, please let me try to clear up one big misunderstanding - probably not surprising given my howling typo in line 2 of Games 1. 2 and 3, above. For which, apologies.

But now please re-scan this...

Originally Posted by**tom@ballooncalculus***heap*-ness (as in graphs 2 and 3, above) one must have become so disconnected from ordinary competence with use of the word 'heap' that one was prepared to accept such (hopefully absurd-seeming) notions as a zero-heap or a negative-heap.

I should point out, though, that an achilles' heel of the puzzle is that this loss of absolutist intuition can occur with bewilderingly total and immediate effect. I resorted to sarcasm half in the hope of stimulating recovery, which I fancied might then be indicated by Deveno's indignant disavowal of the more radical (absolutism-free) notions... but also because it seemed better to acknowledge the gulf of incomprehension sooner rather than later. I could then try and re-ignite absolutism through a different example.

After all, I expect that, whatever your philosophical views on various categories of number, you don't have a problem conversing in real life about negative amounts of money. So you will perhaps see that the sorites enthusiast has to be prepared to at least recognise the attraction of a 'spectral' interpretation of*heap*-ness that looks like graphs 2 or 3, or, rather, like the developments of them, here:

Graph 5:

Graph 6:

Thankfully, this analogy supports absolutism as well as notions that might compromise it:*heap*has a useful imitator in*fortune*, which can be negative, but never (not even a 'small' grade of it) remotely close to zero or one.

Anyway, hurray if we can both (and apparently also Deveno) affirm $\displaystyle 0 \le M \le N$... and I did applaud Deveno's version of it, $\displaystyle 2 \le M \le 1,000,000$, above.

So, hey, don't feel you have to defend a statement like,

This was the reply I'd hoped for, in my last question to you (post #12). It gets Game 3 restarted. We have...

**Me:**Tell me, do you think that a single grain of wheat is anything other than an extreme non-heap?

**Mathhead200:**No, on the contrary, I say that P(1).

In particular, I'm alarmed that you seem prepared to set up a category called 'non-heap' that isn't to be taken as completely interchangeable with 'not a heap'. If so, we'd better have that squabble...

Intended as interchangeable in anything I've said:

is a heap; $\displaystyle H()$; is a full-heap; is not a non-heap; is a non-non-heap; is not a semi-heap and not a pittance either.

is not a pittance; is a non-pittance; is either a semi-heap or a full-heap; is either a semi-heap or a heap; $\displaystyle \neg P()$; $\displaystyle \neg ENH()$.

is a semi-heap; is a non-heap but not a pittance; is neither full-heap nor extreme non-heap; is neither full-heap nor pittance; $\displaystyle NH()\ \&\ \neg P()$; $\displaystyle NH()\ \&\ \neg ENH()$; $\displaystyle \neg H()\ \&\ \neg P()$.

is not a heap; is not a full-heap; is a non-heap; $\displaystyle \neg H()$; $\displaystyle NH()$; is a semi-heap or an extreme non-heap; is a semi-heap or a pittance.

is an extreme non-heap; is a pittance; $\displaystyle ENH()$; $\displaystyle P()$.

Anyway, if we can get past that, I hope we can resume as follows,

**Me:**Tell me, do you think that a single grain of wheat is anything other than an extreme non-heap?

**Mathhead200:**No, on the contrary, I say that P(1).

**Me:**And do you agree that adding a single grain could never turn a pittance into a non-pittance?

- Jun 23rd 2012, 07:06 PM #28
## Re: Sorites only joking...

Seems like we're back to the original paradox (although with a bit more formalism.)

Just so we're clear (in my previous uses): $\displaystyle P(n) \equiv NH(n) \land NH(n) \equiv ENH(n)$

Let's not use $\displaystyle NH(n)$ or $\displaystyle ENH(n)$ anymore to avoid confusion.

So now re-denoted:

Define $\displaystyle H(n)$ to mean $\displaystyle n$ wheat is a heap of grains of wheat.

Define $\displaystyle P(n)$ to mean $\displaystyle n$ wheat is a pittance of grains of wheat. (A pittance will be what you called an "extreme non-heap". We should cease using the term "extreme non-heap", and let a "non-heap" be synonymous with "not a heap".)

(1): $\displaystyle P(n) \implies \lnot H(n)$

(2): $\displaystyle \exists M,N$ with $\displaystyle 0 \le M < N$ such that $\displaystyle P(n) \implies n \le M \land H(n) \implies n \ge N$

So, just to continue: if $\displaystyle \exists k$ with $\displaystyle M < k < N$, then $\displaystyle \lnot P(k) \land \lnot H(n)$

This would be our "semi-heap" region.

Originally Posted by**tom@ballooncalculus**

- Jun 24th 2012, 05:23 AM #29

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## Re: Sorites only joking...

Originally Posted by**Mathhead200**

Originally Posted by**Mathhead200**

Originally Posted by**Mathhead200**

Originally Posted by**Mathhead200**

Originally Posted by**Mathhead200**

Brackets, though, please: Logical connective - Wikipedia, the free encyclopedia

Originally Posted by**Mathhead200**

For what it's worth, my view of the game so far...

Game 3:

Tell me, do you think that a single grain of wheat is a heap?

Me:

**Mathhead200:**Certainly not.

**Me:**And do you agree that adding a single grain could never turn a non-heap into a heap?

**Mathhead200:**Kind of. Perhaps there is a (at least one) state that wheat can be in that is both not a heap and not a non-heap. What shall we call this (one of these) state? A half-heap, a part-heap, a maybe-heap, a vague-heap, ...?

**Me:**I like your idea of recognising 'shades of grey' between non-heap and heap, but I worry that in answering to the challenge of the puzzle's second question (the relativist/gradualist one) you have compromised your commitment to the first (the absolutist one). So, tell me, do you think that a single grain of wheat is even minimally a heap?

**Mathhead200:**Hold on there! I didn't quite commit to a great many shades of grey. Anyway, I would want to look carefully at the situation of heap, non-heap and the ground in between, no matter if that ground were a long succesion of shades.

**Me:**Ok, let me re-ask my question? Let's call the non-heap that isn't even in the middle ground a 'pittance' of grains. Do you think that a single grain of wheat is anything more than a pittance of grains?

**Mathhead200:**Certainly not.

**Me:**And do you agree that adding a single grain could never turn a pittance into a non-pittance?

**Mathhead200:**Well, I'll get back to you... but I'm minded to reject this proposal as a fallacy, just as I did with respect to*heap*in Game 2. And this time I'm more likely to stick by this outcome, because I can point out to you that a sharply defined buffer zone is better than none at all... it enables us to argue that adding a single grain will never turn a pittance into a heap. And this should be a good enough compromise.

- Jun 24th 2012, 10:33 AM #30