# Sorites only joking...

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• Jun 25th 2012, 05:00 AM
tom@ballooncalculus
Re: Sorites only joking...
Quote:

Quote:

Originally Posted by tom@ballooncalculus
My impression...

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.

Very accurate impression

Great. If you do embrace the compromise, though, I shall try to sow doubt.

E.g., try to embarass you by requiring a precise location for the boundary of pittance. As in graph number 4.

(So maybe it's an unknown... but then... is it unknown whether a single grain is a pittance? ... etc.)

Not that you won't have a good defence.

Quote:

Quote:

Originally Posted by tom@ballooncalculus
Quote:

$P(n) \equiv NH(n) \wedge NH(n) \equiv ENH(n)$

Typo here?

(I did not see what you thought might be a typo.)

This is too bizarre, though. Are you seriously thinking of defining,

it's raining & it's raining $\equiv$ it's pouring

... ?!

Actually, some versions of that would be worth exploring. (See Game 5.) But this particular one suggests you would want to deny

it's raining & it's raining $\equiv$ it's raining

... denying which you would have to admit looks bizarre! No? Please explain...
• Jun 25th 2012, 06:18 PM
Re: Sorites only joking...
(Doh) Whoops: $( P(n) \equiv NH(n) ) \land ( NH(n) \equiv ENH(n) )$
• Jun 25th 2012, 06:53 PM
Re: Sorites only joking...
Hm...
Quote:

Originally Posted by tom@ballooncalculus
... is it unknown whether a single grain is a pittance?

Well... I suppose it's all in the definition. First I'll assume "pittance" to mean an amount of wheat which could never be called a heap of wheat, and "heap" to mean an amount of wheat which would always be called a heap of wheat.
Now if we also assume (as I have) wheat grain collections can only be counted in whole numbers (i.e. $\mathbb{N} - 1$), then (I suppose) there must be a sharp cut, where $P(k) \land \lnot P(k + 1)$. But the numbers $M$ and $N$ are only determined by the vernacular and/or possibly the context of "a heap of wheat". Plus, $M$ and $N$ are not unique. If $P(0) \land P(1) \land P(2) \land \lnot P(3)$, then $\forall M \ge 2$ work. (Perhaps I need to re-think the logical axioms I made... (Wondering))
• Jun 26th 2012, 01:03 PM
tom@ballooncalculus
Re: Sorites only joking...
Quote:

(Doh) Whoops: $( P(n) \equiv NH(n) ) \land ( NH(n) \equiv ENH(n) )$

Oh, ok. And the formula was also headed:

Quote:

Just so we're clear (in my previous uses): ...

... so we can put it to one side. It doesn't fit with your current understanding of P and H...

Quote:

now re-denoted:
Define $H(n)$ to mean wheat is a heap of grains of wheat.
Define $P(n)$ to mean 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): $P(n) \implies \lnot H(n)$
(2): $\exists M,N$ with such that

So, just to continue: if $\exists k$ with $M < k < N$, then $\lnot P(k) \land \lnot H(n)$
This would be our "semi-heap" region.

... which is exactly mine.

Quote:

Hm...

Quote:

Originally Posted by tom@ballooncalculus
... is it unknown whether a single grain is a pittance?

Well... I suppose it's all in the definition. First I'll assume "pittance" to mean an amount of wheat which could never be called a heap of wheat, and "heap" to mean an amount of wheat which would always be called a heap of wheat.

So, it looks to me as though Game 3 has finished as it seemed likely to, at least by the time you had endorsed my 'impression'.

That is, it finishes much like Game 2, with you accepting a sharp boundary in principle, and thus rejecting the implication of the second question.

But also, just as with Game 2, I'm lucky enough to find that the puzzle still works for you, and you return with ideas for a refined version of the relativist/gradualist tenet, i.e. a good answer agreeing to the second question that doesn't undermine the first.

Actually, I'm very pleased about the direction you're taking it. I suppose I'll be obliged to play devil's advocate!

So, roll up for...

Game 6.

Quote:

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

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

Mathhead200: Kind of. I think there is a middle ground between 'heap' and an emphatic grade of 'non-heap', which we shall call a 'pittance' of grains. And I do declare that adding a single grain will never turn a pittance into a heap.

Me: Ok. Now, is a single grain anything other than a pittance?

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

Mathhead200: Let me explain. First I'll assume "pittance" to mean an amount of wheat which could never be called a heap of wheat...

Like I say... I'm interested. (Coffee)
• Jun 26th 2012, 03:27 PM
Re: Sorites only joking...
I guess, based on context, it might be possible to go from pittance to non-pittance by adding a single grain of wheat, but then again... Even a single person may change his own definition of heap based on context. The man who's hungry must have a higher (minimum) value of $M$ after all... The farmer surely has a very specific way of determining $N$ (based on need)... And so on.

So although it seems perhaps unlikely, I stand by the fact that although many people may all agree that a certain amount of wheat is a pittance, one more grain might change at least one mind thus moving it to that gray area.

So, I guess in short: I assert that although it is always true that $P(M) \land \lnot P(M + 1)$, the value of M is so volatile* that it may never be determined for more that the most trivial of cases or circumstance.

* Volatile variable - Wikipedia, the free encyclopedia
• Jun 27th 2012, 01:52 PM
tom@ballooncalculus
Re: Sorites only joking...
As I hoped I would, I do very much like these ideas, which seem designed to answer 'kind of' to my second question, in such a way as to satisfy both of the apparently conflicting intuitions.

You don't claim that your answer is likely to please the relativist-gradualist at first glance. On the contrary, you think that your description will seem 'perhaps unlikely'. But I think it very convincingly fleshes out the r-g intuition.

Very like the marriage-joke's participant in Game 5, you are prepared to see the boundary of pittance (and of heap) as a large distribution of individual thresholds. I would say this is a 'classic' r-g response.

I think it is the right way to go. But the puzzle (which I think is right as well) isn't satisfied. It wants us to check our new framework to see if absolutism has been compromised. So, here I go...

Tell me, do you think that the range of M extends up to a billion, or that of N down as far as 1?

(Was ever a man hungry enough to make a billion grains a pittance, or a farmer charming enough to make a single grain a heap?)
• Jun 29th 2012, 12:25 AM
Re: Sorites only joking...
I think that $M$ and $N$ are defined based on context.

Quote:

Originally Posted by tom@ballooncalculus
(Was ever a man hungry enough to make a billion grains a pittance, or a farmer charming enough to make a single grain a heap?)

(I would say that) most likely not.

Quote:

Originally Posted by tom@ballooncalculus
Tell me, do you think that the range of M extends up to a billion, or that of N down as far as 1?

In a way, these are two different (pairs of) questions...
I think that it is conceivable that there could exist a context in which M is extremely (uncharacteristically) high, or one in which N is minimal.
E.g. How much wheat do we need for an entire army, country, planet, ...? How much wheat do we feed a mouse, or something that is deathly allergic to wheat?

However, within each of these contexts (independently), I still believe my previous "axioms", and "definitions" hold.
• Jul 3rd 2012, 03:52 PM
tom@ballooncalculus
Re: Sorites only joking...
Hi Mathhead200! Sorry to have taken an age this time.

You seem (to me, from my point of view) perilously close to wobbling on absolutism. Once lost, this intuition can be hard to recover. And then my best course of action is usually to admit defeat. (Until the next time.)

Quote:

Quote:

Originally Posted by tom@ballooncalculus
(Was ever a man hungry enough to make a billion grains a pittance, or a farmer charming enough to make a single grain a heap?)

(I would say that) most likely not.

this looks very perilous to me! Merely...

Quote:

... most likely...
... a frequency or probability profile, then, something like graph 3 for heap, or this...

http://www.ballooncalculus.org/draw/graph/heap9.png

... for pittance. In such images, one is generally encouraged to see positive probabilities (or degrees, or degrees of truth) of heap-hood (or of the alleging of such) reaching all the way to grain-collections as small as zero or 1. (And with respect to pittance-hood all the way to infinity, or absurdly high numbers.)

I wonder if you can sympathise with my aversion to these particular kinds of pictures of the meaning of heap? And whether you agree that the puzzle's first question requires us to reject them?

The slim chance of this...

Quote:

However, within each of these contexts (independently), I still believe my previous "axioms", and "definitions" hold.

affording me any hope of prolonging the game would depend on a conception of 'contexts' as independent category systems or frames of reference, within at least some of which you might welcome a restart of the questioning.

E.g., for the planet-sized context, Tell me, do you think that a single harvest of wheat is a pittance of harvests?  And do you agree that adding a single harvest... etc.

But I would be hoping that this phase of the game were exactly like the beginning; whereas you would (I guess, but do correct me) reply to the new version of the second question:

No, whether we proceed harvest by harvest or grain by grain, I get a sufficient dose of relativist-gradualist vibes just from viewing the present context (with its sharp pittance and heap boundaries $M$ and $N$ such that $1 < M < k < N < 10^6$ as depicted in graph 4) as part of a whole distribution of such contexts (with different values of $M$ and $N$). That's why I explained to you about particular contexts such as this one in the first place. Only, now, since you asked me again about absolutism, I wanted to point out that within each context we get a hit of that first intuition too, from having pittance, semi-heap and heap always partition the counting numbers in that particular order.

Ok, by this stage (if this is anything like what you're trying to say) I don't like to complain, since you've gone out of your way to answer in my terms.

But I don't get any absolutist satisfaction from $1 < M < k < N < 10^6$ (a graph 4) obtaining within each context, if the overall picture is graphs 3 and 7 superimposed on each other...

And weren't you aiming, here...

Quote:

First I'll assume "pittance" to mean an amount of wheat which could never be called a heap

for something a bit stronger?
• Jul 16th 2012, 04:01 PM
tom@ballooncalculus
Best Abstract ever!
• Jul 31st 2012, 12:43 PM
Re: Sorites only joking...
I suppose I'm on the "absolutist" side of the argument. When I say things like "most likely not", I am being cautious because the question was worded with out a detailed enough context as to provide an absolute answer. Although with some assumed context I think an absolute "no" could have been answered.

Let's move on. We're on earth (as our "planet", and let's say present day). I very much doubt a single harvest of wheat would be considered a heap to anyone (with respect to the planet's population) so we can call it a pittance. However, the next part: adding a single harvest doesn't change a pittance to a non-pittance... Well I'm just not subscribed to that. Surly 2, 3, 4, and maybe more harvests are a pittance, but in general adding a single harvest, or possibly (as unlikely as it seems) just adding a single grain, could change one persons mind, ergo we've moved to a non-pittance. I believe the answer to be unintuitive, but people can be irrational and their behavior can be unintuitive.

• Aug 1st 2012, 12:54 PM
tom@ballooncalculus
Re: Sorites only joking...
A pattern is definitely emerging here...

Once again, you say that you come to bury relativism-gradualism (or what ever we want to call aquiescence to the second question), not to praise it...

But to me, as before, what you're proposing looks to be a classic r-g response, and undermines absolutism, if anything.

And perhaps not anything.

I mean, what you're proposing may not undermine either of the two intuitions that are inspired (in some) by the puzzle. As I said before, I dare to hope that this is the case, and that your direction is roughly mine.

But the game is how I choose to check whether it is the case. By, that is, seeing whether your modification of the second (r-g) tenet has undermined the first (absolutist) one.

So, where you say...

Quote:

... possibly (as unlikely as it seems) just adding a single grain, could change one persons mind, ergo we've moved [from a pittance] to a non-pittance.

... I must ask, are you 'zooming in' on the classificatory threshold of one person, so that when we zoom out again, and consider a large distribution of such thresholds, we get a picture like graph number 3?

That would be to praise the second tenet by burying the first.

Then, hopefully, when I re-ask the first question...

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But, is it possible that any person competent with using the word 'heap' could apply it to a single grain?
... you'll feel an urge to answer no, absolutely not, and thereby become dissatisfied with your previous answer.

Of course, you must then be prepared to lose the sympathy of those large numbers of people who are not at all charmed by the puzzle, and are content to answer yes, or a grade of yes, to the very first question. That is a limitation of the puzzle, admittedly.

But that may not, anyway, be what you were proposing. Maybe the perspective you were zooming in on wasn't just any random example, but that of the person with the very lowest threshold?

This person's 'change of mind' is far more crucial to the zoomed-out picture. It represents the lower end of the whole distribution - the smallest non-pittance.

http://www.ballooncalculus.org/draw/graph/heap10b.png

Quite possibly you'll want to disavow any such madness.

(From 4 minutes 50!) (Smirk) (At 4 minutes 50!)

On the other hand, it would fit with...

Quote:

First I'll assume "pittance" to mean an amount of wheat which could never be called a heap

That, as you'll have gathered, was my favourite thing you've said so far.

Quote:

• Aug 2nd 2012, 04:38 AM
Re: Sorites only joking...
Hm... (Thinking)

(I was in fact referring to the person with the minimum threshold, thereby establishing the move from pittance to non-pittance.)

I'm really not sure how to respond yet... I'm somewhat happy with my current stance. There exists a maximum threshold for pittance, a minimum threshold for heap, and adding a single grain won't necessarily change a pittance to a heap. However, on the individual level, there is still a stark enough contrast between heap and pittance to avoid stripping the useful meaning from the word.

(I re-read through all the posts; it has been quite a discussion so far.)
• Aug 2nd 2012, 04:54 AM
tom@ballooncalculus
Re: Sorites only joking...
Quote:

(I was in fact referring to the person with the minimum threshold, thereby establishing the move from pittance to non-pittance.)

Hey, that's great! You could be a 'proper' (in my book) absolutist.

But I'm not sure yet.

Do you mean to deny, even, that anyone ever could have a personal threshold at 0/1, or 1/2?

You seemed to deny it with...

Quote:

First I'll assume "pittance" to mean an amount of wheat which could never be called a heap

... but fleshing that out is going to be interesting, to say the least.

Quote:

(I re-read through all the posts; it has been quite a discussion so far.)

Yes, I've been surprised (and delighted) that you so 'get' the puzzle, and are susceptible to having both consciences (absolutist and relativist-gradualist) pricked at any time.
• Aug 2nd 2012, 05:42 AM
tom@ballooncalculus
Re: Sorites only joking...
Quote:

Originally Posted by tom@ballooncalculus
... but fleshing that out is going to be interesting, to say the least.

To be specific, game 6 might shape up like this...

Quote:

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

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

Mathhead200: Kind of. I think there is a middle ground between 'heap' and an emphatic grade of 'non-heap', which we shall call a 'pittance' of grains. And I do declare that adding a single grain will never turn a pittance into a heap.

Me: Ok. Now, is a single grain anything other than a pittance?

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

Mathhead200: No. You see, "pittance" turns out to mean a number of grains which could never be called a heap. So the smallest non-pittance is simply the smallest number of grains which could, conceivably, be called a heap. Don't ask me what that number is, but I don't see that I need to know. This is where I just am not what you are calling a 'relativist-gradualist'. Perhaps we should need an empirical survey of some kind, in order to locate the number in question. But the point is that it's there to be located.

Me: On the other hand, do we need to conduct an opinion poll (or whatever kind of survey you had in mind) to establish whether a single grain could ever be called a heap?