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Mereological nihilism
Posted: 29 May 2010 02:42 AM   [ Ignore ]   [ # 31 ]
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Another thought:

There is ontic vagueness there (nice phrase, BTW!  I’ll have to remember it!) but I only see it in whether or not it’s permissible to have a negative number of grains of sand in the heap, because it’s not completely clear whether or not the action being performed is purely a mathematical subtraction (where negative numbers are permissible), or element removal as if from a set of grains of sand like I described (where negative numbers are not permissible).  Observe, even though this is ontologically undefined given Premises 1&2, that in both cases, there is no reason within those two premises why a “heap” suddenly stops being a “heap.”  To create the paradox, one STILL has to change the definition of heap midway through, which is a logical argumentative error.

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Posted: 29 May 2010 06:25 AM   [ Ignore ]   [ # 32 ]
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TromboneAndrew - 29 May 2010 02:22 AM

Aaaaah, the fun of philisophical argument. smile

From your wiki reference on the Sorites paradox:

The word “sorites” derives from the Greek word for heap. The paradox is so named because of its original characterization, attributed to Eubulides of Miletus.[1] The paradox goes as follows: consider a heap of sand from which grains are individually removed. One might construct the argument, using premises, as follows:

1,000,000 grains of sand is a heap of sand (Premise 1)
A heap of sand minus one grain is still a heap. (Premise 2)

Repeated applications of Premise 2 (each time starting with one less grain), eventually forces one to accept the conclusion that a heap may be composed of just one grain of sand (and consequently, if one grain of sand is still a heap, then removing that one grain of sand to leave no grains at all still leaves a heap of sand, and indeed a negative number of grains also form a heap[2]).

On the face of it, there are some ways to avoid this conclusion. One may object to the first premise by denying 1,000,000 grains of sand makes a heap. But 1,000,000 is just an arbitrarily large number, and the argument will go through with any such number. So the response must deny outright that there are such things as heaps. Peter Unger defends this solution. Alternatively, one may object to the second premise by stating that it is not true for all collections of grains that removing one grain from it still makes a heap. Or one may accept the conclusion by insisting that a heap of sand can be composed of just one grain.

I still think that this shows that someone who is trying to show a paradox here is changing their definition of “heap” midway through the process.  If, literally, a “heap” is defined ONLY by those two premises, then there is no reason why a “heap” cannot have 0 or 1 grains of sand in it.  There is no reason why one would need to avoid this conclusion as that last paragraph implies there must be.  It looks to me very much like an example of a semantic “bait and switch,” where the reasoner only starts with Premises 1&2, but then without specifically stating so, adds the common usage definition of “heap” to the mix - without justification.  Those two solutions that you list would still be adding another premise at the outset, essentially coming to a different definition of “heap,” with different conclusions to be drawn.

You presume that the premises are definitions. That is not just a groundless assumption but an incredible one. A statement like “1,000,000 grains of sand is a heap of sand” does not even have the right logical form to be a definition. Of what term is it supposed to be a definition? It certainly can’t be a definition of “1,000,000 grains of sand.” But it can’t be a definition of heap either, because it says only that 1,000,000 grains of sand (piled up, presumably) make a heap of sand, not that a heap of sand consists of 1,000,000 grains. Further, if that were the definition of “heap of sand,” then premise two would have to be false: if a heap of sand consists of 1,000,000 grains, then a collection of 999,999 grains is not a heap of sand—end of argument.

It is impossible to construe either of the first two premises as a definition without making the other false, and to construe both as definitions is logically incoherent. “Heap” functions in the argument as an undefined term. There is no plausible definition of “heap of sand” in terms of a specific number of grains of sand. That is why the argument is so stubbornly problematic.

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Posted: 29 May 2010 06:28 AM   [ Ignore ]   [ # 33 ]
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By the way, both premises are false, and the idea that a heap of sand may be a “set” of grains of sand is utter nonsense. A grain of sand here, another one on another beach, another one someplace else, etc., to a million grains, make up a set of grains of sand, but they do not make a heap.

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Posted: 29 May 2010 06:48 AM   [ Ignore ]   [ # 34 ]
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Kritikos - 29 May 2010 06:25 AM

You presume that the premises are definitions. That is not just a groundless assumption but an incredible one. A statement like “1,000,000 grains of sand is a heap of sand” does not even have the right logical form to be a definition. Of what term is it supposed to be a definition? It certainly can’t be a definition of “1,000,000 grains of sand.” But it can’t be a definition of heap either, because it says only that 1,000,000 grains of sand (piled up, presumably) make a heap of sand, not that a heap of sand consists of 1,000,000 grains. Further, if that were the definition of “heap of sand,” then premise two would have to be false: if a heap of sand consists of 1,000,000 grains, then a collection of 999,999 grains is not a heap of sand—end of argument.

It is impossible to construe either of the first two premises as a definition without making the other false, and to construe both as definitions is logically incoherent. “Heap” functions in the argument as an undefined term. There is no plausible definition of “heap of sand” in terms of a specific number of grains of sand. That is why the argument is so stubbornly problematic.

You’re defeating your own argument.  If those premises do not form a definition of “heap”, then the whole chain of reasoning is pointless, being based on something which is undefined.  You can’t draw ANY conclusions in such an argument, much less say anything about paradoxes.

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Posted: 29 May 2010 06:49 AM   [ Ignore ]   [ # 35 ]
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Kritikos - 29 May 2010 06:28 AM

By the way, both premises are false, and the idea that a heap of sand may be a “set” of grains of sand is utter nonsense. A grain of sand here, another one on another beach, another one someplace else, etc., to a million grains, make up a set of grains of sand, but they do not make a heap.

Fine.  What is a heap?

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Posted: 29 May 2010 07:02 AM   [ Ignore ]   [ # 36 ]
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TromboneAndrew - 29 May 2010 06:48 AM

You’re defeating your own argument.  If those premises do not form a definition of “heap”, then the whole chain of reasoning is pointless, being based on something which is undefined.  You can’t draw ANY conclusions in such an argument, much less say anything about paradoxes.

I don’t understand why having an undefined term undermines the argument. Why does the argument have to include a definition of “heap” in order to establish the paradox?

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Posted: 29 May 2010 07:15 AM   [ Ignore ]   [ # 37 ]
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Kritikos - 29 May 2010 07:02 AM
TromboneAndrew - 29 May 2010 06:48 AM

You’re defeating your own argument.  If those premises do not form a definition of “heap”, then the whole chain of reasoning is pointless, being based on something which is undefined.  You can’t draw ANY conclusions in such an argument, much less say anything about paradoxes.

I don’t understand why having an undefined term undermines the argument. Why does the argument have to include a definition of “heap” in order to establish the paradox?

Because your paradox depends on what a heap is and is not.  If “heap” is undefined, how can you possibly say anything about what it is or is not?

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Posted: 29 May 2010 07:25 AM   [ Ignore ]   [ # 38 ]
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TromboneAndrew - 29 May 2010 07:15 AM

Because your paradox depends on what a heap is and is not.  If “heap” is undefined, how can you possibly say anything about what it is or is not?

How about “because I understand English”? By your reasoning, we could not establish anything by argument until we had defined all our terms—which we could never do, of course, because we would have to use undefined terms to do so.

The argument neither implies nor depends on any definition of “heap.”

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Posted: 29 May 2010 07:44 AM   [ Ignore ]   [ # 39 ]
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. . . so you say “heap” does not need any definition, yet you rely on the common English definition of “heap”?

smile

(I hate that smiley thing.  It looks like I’m rolling my eyes, when that’s NOT the meaning I want to convey.  It’s just supposed to be a friendly smile, dammit!)

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Posted: 29 May 2010 07:55 AM   [ Ignore ]   [ # 40 ]
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TromboneAndrew - 29 May 2010 07:44 AM

. . . so you say “heap” does not need any definition, yet you rely on the common English definition of “heap”?

I did not propose any definition. I was talking about understanding the word. Your preoccupation with definition seems to me excessive. You seem to imagine that no one can use or understand a word without relying on a definition of it. If that were true, it would be impossible for anyone ever to learn to talk.

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Posted: 29 May 2010 08:10 AM   [ Ignore ]   [ # 41 ]
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Kritikos - 29 May 2010 07:55 AM
TromboneAndrew - 29 May 2010 07:44 AM

. . . so you say “heap” does not need any definition, yet you rely on the common English definition of “heap”?

I did not propose any definition. I was talking about understanding the word. Your preoccupation with definition seems to me excessive. You seem to imagine that no one can use or understand a word without relying on a definition of it. If that were true, it would be impossible for anyone ever to learn to talk.

And I don’t understand your insistence that a word doesn’t need a definition.

Concerning babies, babies learn words by learning the sounds and how they relate to a goal - i.e., a definition.  It doesn’t have to be a strict logical definition - words in language vary widely over how precisely they are used.  But some kind of definition must exist, nonetheless.  If there is no purpose for a word, we don’t use it.

When I see you say “I understand what ‘heap’ means,” I understand that to mean that you have a definition for it.  It doesn’t have to be as strict as what we’re talking about with those Premises #1&2; the definition may be much more vague, but you wouldn’t even be using the word if it were completely meaningless.

[ Edited: 29 May 2010 08:13 AM by TromboneAndrew ]
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Posted: 29 May 2010 09:07 AM   [ Ignore ]   [ # 42 ]
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TromboneAndrew - 29 May 2010 02:22 AM

Aaaaah, the fun of philisophical argument. smile

I still think that this shows that someone who is trying to show a paradox here is changing their definition of “heap” midway through the process.  If, literally, a “heap” is defined ONLY by those two premises, then there is no reason why a “heap” cannot have 0 or 1 grains of sand in it.  There is no reason why one would need to avoid this conclusion as that last paragraph implies there must be.  It looks to me very much like an example of a semantic “bait and switch,” where the reasoner only starts with Premises 1&2, but then without specifically stating so, adds the common usage definition of “heap” to the mix - without justification.  Those two solutions that you list would still be adding another premise at the outset, essentially coming to a different definition of “heap,” with different conclusions to be drawn.

From the wiki on the Sorites paradox:

1,000,000 grains of sand is a heap of sand (Premise 1)
A heap of sand minus one grain is still a heap. (Premise 2)

Repeated applications of Premise 2 (each time starting with one less grain), eventually forces one to accept the conclusion that a heap may be composed of just one grain of sand (and consequently, if one grain of sand is still a heap, then removing that one grain of sand to leave no grains at all still leaves a heap of sand, and indeed a negative number of grains also form a heap

Those were 2 premises, not definitions.

From the wiki on premise

In logic, an argument is a set of one or more declarative sentences (or “propositions”) known as the premises along with another declarative sentence (or “proposition”) known as the conclusion.

Premise 1 could state, say 100,000 or any large enough number of grains of sand to form a heap. Of course, 1 or 0 grains of sand cannot be considered as a heap. If one accepts that as true, then it is reasonable to accept premise 2 since 1 grain less seems negligible.

It is “repeated applications of Premise 2 (each time starting with one less grain)” which leads to the paradoxical conclusion, if one accepts Premise 2.

No definitions have been changed in the course of the argument as those sentences were premises. There is no semantic “bait and switch”.

The bone of contention is “heap”. If one agrees that X grains of sand is a heap from the beginning, the argument will proceed to it’s paradoxical conclusion.

[ Edited: 29 May 2010 09:22 AM by kkwan ]
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Posted: 29 May 2010 09:14 AM   [ Ignore ]   [ # 43 ]
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Edited to add: Thanks for in effect backing me up, Kkwan, but if you will look over the exchange between me and TromboneAndrew, you will see that we have been debating this very point.

TromboneAndrew - 29 May 2010 08:10 AM
Kritikos - 29 May 2010 07:55 AM
TromboneAndrew - 29 May 2010 07:44 AM

. . . so you say “heap” does not need any definition, yet you rely on the common English definition of “heap”?

I did not propose any definition. I was talking about understanding the word. Your preoccupation with definition seems to me excessive. You seem to imagine that no one can use or understand a word without relying on a definition of it. If that were true, it would be impossible for anyone ever to learn to talk.

And I don’t understand your insistence that a word doesn’t need a definition.

Concerning babies, babies learn words by learning the sounds and how they relate to a goal - i.e., a definition.  It doesn’t have to be a strict logical definition - words in language vary widely over how precisely they are used.  But some kind of definition must exist, nonetheless.  If there is no purpose for a word, we don’t use it.

When I see you say “I understand what ‘heap’ means,” I understand that to mean that you have a definition for it.  It doesn’t have to be as strict as what we’re talking about with those Premises #1&2; the definition may be much more vague, but you wouldn’t even be using the word if it were completely meaningless.

Perhaps I have not understood your point. It has seemed to me that you are making the extravagant claim that nobody understands any word that he uses unless he has a definition, but then you seem to understand—define?—the word “definition” to have so vague an application as to include what babies learn when they react appropriately to the sound of a word. If that is what you mean by “definition,” then your demand that the Wikipedia version of the sorites paradox furnish a definition of “heap” is vacuous: the people are using the word “heap” competently, therefore ipso facto they have (by your standards) a “definition,” which does not need explicit verbal formulation and need not be part of the argument. For my part, I think that you have deprived thereby the word “definition” of all useful meaning. But the main point is that you have undercut your own criticism of the argument.

On the other hand, I think that there is a logical fault in the Wikipedia version of the argument of the Sorites paradox that does have to do with definition. The implied reconstruction is (and here I have to reconstruct the reconstruction, because they only supply two premises and no explicit conclusion):

(1) 1,000,000 grains of sand is a heap of sand. (Premise)
(2) A heap of sand minus one grain is still a heap. (Premise)
(3) 999,999 grains of sand is a heap of sand. (From (1) and (2))
(4) 999,998 grains of sand is a heap of sand. (From (3) and (2))
(5) 999,997 grains of sand is a heap of sand. (From (4) and (2))
. . .
(1,000,002) 0 grains of sand is still a heap. (From (1,000,001) and (2))

Now the soundess of this argument does not require that its premises be definitions (as they clearly are not), but only that they be true statements. My objection is that they are not true statements. (In fact, they are not even grammatical sentences of English: the plural subject “1,000,000 grains of sand” requires a plural verb, not a singular one.) My argument for the claim that they are untrue requires some understanding of the meaning of “heap.” That understanding can be expressed in a remark such as the following: grains of sand, however numerous, do not as such make up a heap unless they are accumulated together. It is the body made of the accumulated grains, not the individual grains themselves or the set that has them as its members, that makes up the heap. (The failure of grammar on this point in the original formulation reflects the underlying failure of logic.)

This statement does not have the right form to be a definition, but a possible definition can be derived from it: “A heap of sand is a body made of grains of sand accumulated together.” Is this a correct and adequate definition of “heap”? I don’t think so, because the definiens would apply to a collection of sand grains that are touching one another but all completely flat on the ground, with none on top of the others. I take it to be obvious that “heap” implies “one (element or part) on top of another.” If we add that, have we got a workable definition—“A heap of sand is a body made of grains of sand accumulated together one on top of another”? Well, maybe. There may be some further troubles to be raised.

Of course, someone may object that such a definition is vague. “What do you mean, ‘one on top of another’? Each one on top of all the others? That makes no sense! Merely some on top of some other or others? Then you could take your one-grain-thick collection and put one grain on top of another and have a heap, by that definition!” And so on. I think such an objection would be silly: the proposed definition is indeed vague, because it is an attempt to make explicit the meaning of a vague term. So even if the definition has some logical weakness that I have not yet thought of, its vagueness is not an objection to it but a virtue.

All right, so now we try to use this proposed definition to correct the argument. Note that the definition is not in the argument as a premise but is merely used in formulating the premises so that they are, or appear to be, true:

(1) A body made of 1,000,000 grains of sand accumulated together one on top of another is a heap. (Premise)
(2) A heap of sand minus one grain is still a heap. (Premise)
(3) A body made of 999,999 grains of sand accumulated together one on top of another is a heap. (From (1) and (2))
. . .
(1,000,002) A body made of 0 grains of sand accumulated together one on top of another is a heap. (From (1,000,001) and (2))

The argument certainly seems to be logically valid, and it certainly seems to have a false conclusion, while the premises still seem compelling; thus we still have a paradox.

Edited to add: I can already think of a reason why my proposed definition will not work. “Heap” implies “disorder” So, e.g., a body of grains of sand forming a sand castle would not be called a “heap of sand,” I don’t think. But I don’t think that this particular deficiency in the definition is pertinent to the paradox, because the inferences in the argument, as far as I can tell, do not depend on whether the grains of sand are organized or not. This does, however, underscore the point that in ordinary speech and thought, we are able to use terms perfectly well even though we do not have definitions of them, in the sense of statements of logically necessary and sufficient conditions for their application, but this lack of definition comes back to bite us when we construct sorites paradoxes using those terms.

[ Edited: 29 May 2010 09:23 AM by Kritikos ]
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Posted: 29 May 2010 09:22 AM   [ Ignore ]   [ # 44 ]
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kkwan - 29 May 2010 09:07 AM

Premise 1 could state, say 100,000 or any large enough number of grains of sand to form a heap. Of course, 1 or 0 grains of sand cannot be considered as a heap. If one accepts that as true, then it is reasonable to accept premise 2 since 1 grain less seems negligible.

It is “repeated applications of Premise 2 (each time starting with one less grain)” which leads to the paradoxical conclusion, if one accepts Premise 2.

No definitions have been changed in the course of the argument as those sentences were premises.
There is no semantic “bait and switch”.

You actually have 3 premises that you listed:

#1: 100,000 or any large enough number of grains of sand to form a heap.
#2: 1 or 0 grains of sand cannot be considered as a heap.
#3: each time starting with one less grain (paraphrased by you).

I agree that under these premises, there is a conflict between #2 and #3, because #3 obviously cannot be applied to such an extent that one gets to 1 or 0 grains.  Observe that under these premises, there is a pretty clear cut-off point at which the conflict occurs - specifically, at there being 1 grain of sand left.

Also, from the wiki page, the very first line:

The sorites paradox (σωρός sōros being Greek for “heap” and σωρείτης sōreitēs a derived adjective meaning “heaped up”) is a paradox that arises from vague predicates.

This is a fancy way of saying exactly what I’ve been saying.  Vague predicates are a language problem.  To tie this back into the subject of this thread, there is no reasonable way to get from a language problem to any inference about the nature of the universe.  These are two different topics.

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Posted: 29 May 2010 10:11 AM   [ Ignore ]   [ # 45 ]
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TromboneAndrew - 29 May 2010 09:22 AM
kkwan - 29 May 2010 09:07 AM

Premise 1 could state, say 100,000 or any large enough number of grains of sand to form a heap. Of course, 1 or 0 grains of sand cannot be considered as a heap. If one accepts that as true, then it is reasonable to accept premise 2 since 1 grain less seems negligible.

It is “repeated applications of Premise 2 (each time starting with one less grain)” which leads to the paradoxical conclusion, if one accepts Premise 2.

No definitions have been changed in the course of the argument as those sentences were premises.
There is no semantic “bait and switch”.

You actually have 3 premises that you listed:

#1: 100,000 or any large enough number of grains of sand to form a heap.
#2: 1 or 0 grains of sand cannot be considered as a heap.
#3: each time starting with one less grain (paraphrased by you).

I agree that under these premises, there is a conflict between #2 and #3, because #3 obviously cannot be applied to such an extent that one gets to 1 or 0 grains.  Observe that under these premises, there is a pretty clear cut-off point at which the conflict occurs - specifically, at there being 1 grain of sand left.

Also, from the wiki page, the very first line:

The sorites paradox (σωρός sōros being Greek for “heap” and σωρείτης sōreitēs a derived adjective meaning “heaped up”) is a paradox that arises from vague predicates.

This is a fancy way of saying exactly what I’ve been saying.  Vague predicates are a language problem.  To tie this back into the subject of this thread, there is no reasonable way to get from a language problem to any inference about the nature of the universe.  These are two different topics.

I wrote that to illustrate what could be modified in Premise 1 of the Sorites paradox argument, not to state premises.

Vague predicates are not purely a language problem. From the wiki on vagueness

The philosophical question of what the best theoretical treatment of vagueness is - which is closely related to the problem of the paradox of the heap - has been the subject of much philosophical debate.

The epistemic view:

They maintain that vague predicates do, in fact, draw sharp boundaries, but that one just does not know where these boundaries lie.

As a property of objects:

One possibility is that one’s words and concepts are perfectly precise, but that objects themselves are vague.

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