The sorites paradox arises frequently when the property in question appeals to some
manner of degree. Above, it was demonstrated that properties that fall under the sorites paradox cannot be natural. This is because such a property, if it is to be coherent at all, is vague. Since there is no vagueness in the world, such a property does not, in the final metaphysical picture, exist.
(1) If a property falls under the sorites paradox, then it is not a natural property.
There are, of course, a plethora of answers to the Special Composition Question. The aim of this section is only to eliminate one type of answer. Under compatibilst answers to SCQ, there are those that appeal to some degree of contact between parts. Prima facie, this type of answer is promising. Consider again our ham sandwich. Isn't it just when we take the ham and cheese and place them between the bread that a ham sandwich is formed?
This, certaintly, is a case in which the distance between the parts is relevent. There is no sandwich when the parts are scattered across the kitchen counter. But what about the parts getting closer allows for a sandwich to form? Perhaps they need to be touching. But if that were the answer to SCQ, then every time two people shake hands, they form a new object. Surely this is not the case.
1) If CONTACT is true, then every time two people shake hands, an object is formed.
2) It is not the case that every time two people shake hands, an object is formed.
3) CONTACT is not true.
A similar line of reasoning denies all answers that appeal to some connectedness between the parts (call these fusion-type answers).
Now what of an appeal direct to distance of parts? Our ham sandwich does not come into existence until all the parts are some distance away from each other. But recall the argument given against baldness. Where would one mark the distinction between ham sandwich and no ham sandwich? One meter? One centimeter? One micrometer? Surely my sandwich is allowed some measure of shifting without falling out of existence.
(1) If two parts n units apart form an object, then two parts n+1 units apart form an object.
Unfortunetly, this is sufficient to run a sorites paradox.
(1) If two parts n units apart form an object, then two parts n+1 units apart form an object.
(2) Two parts 0 units apart form an object. (If there is any distance that permits composition, it is this)
(3) Two parts 1 unit apart form an object. (From 1 and 2)
(4) Two parts 2 units apart form an object. (From 1 and 3)
...
(100,001) Two parts 99,999 units apart form an object. (From 1 and 100,000)
(100,002) Two parts 99,999 units apart do not form an object. (From common sense)
(100,003) Contradiction!
Thus, any answer to the Special Composition question cannot appeal only to the spacial relations that hold between objects. The argument can be run analogously to time, and likely any other quantitative relation. Note, however, that this argument is vulnerable if space is discrete. More on this will be said later.
Friday, July 30, 2010
Saturday, July 24, 2010
An Introduction to the Sorites Paradox
Meet Charles. Charles has graciously volunteered himself for a demonstration. Now, Charles is a middle-aged man and, unfortunately, has started to bald. Now, he isn't quite bald yet; he still has well-groomed mane. But there is a bald spot that for the past few weeks has been growing. Let n be the number of individual hairs on Charles' head. Suffice for our purposes, n is a pretty big number. For those that work better without variables, feel free to plug in 100,000.
(1) Charles has n hairs.
(2) Someone with n hairs is not bald.
(3) Therefore Charles, with n hairs is not bald.
Now comes the experiment. We sit Charles down on a comfortable chair and give him a big, juicy rib eye steak for his troubles. We then take a pair of tweezers and pluck out one of Charles' hairs. It seems obvious that we did not do much to change Charles' being bald or not. Sure, we may have put him one hair closer to complete baldness. But we did not make him bald by removing a single hair.
(4) Charles has n-1 hairs.
(5) Someone with n-1 hairs is not bald.
(6) Therefore charles, with n-1 hairs, is not bald.
What if we kept plucking out a hair of Charles, one by one, and asking ourselves at that juncture if he were bald? Surely, at some point he must become bald. After all, a man with no hair on his head is most certainly bald. But where is that point? A man with only a single hair would still, presumably, be bald. So, too, would a man with two hairs - as evidenced think Homer Simpson. Furthermore, the following principle seems to hold:
(7) If someone with n hairs is bald, then someone with n+1 hairs is bald.
This is grounded in the implausibility that a single hair makes the difference between baldness and non-baldness. Imagine two men standing next to each other, one bald and one not bald. Would you expect there to be only a single hair to separate the two? Is that even possible? Even more troublesome is that it appears we can also reason in the opposite direction.
(8) If someone with n hairs is not bald, then someone with n-1 is not bald.
This is justified in exactly the same way as (7). Losing a single hair cannot move anyone into a state of baldness. Thus, fully presented, the argument runs:
(1) If someone with n hairs is not bald, then someone with n-1 hairs is not bald.
(2) Someone with 100,000 hairs is not bald.
(3) Someone with 99,999 hairs is not bald. (From 1 and 2)
(3) Someone with 99,998 hairs is not bald. (From 1 and 3)
...
(100,001) Someone with 0 hairs is not bald. (From 1 and 100,000)
(100,002) Someone with 0 hairs is bald. (From common sense!)
(100,003) Contradiction!
As noted above, the argument could be run in reverse, with the intuition at (100,002) reading, "Someone with 100,000 hairs is not bald."
This type of puzzle is known as the sorites paradox. It is not restricted, of course, to baldness. Parallel arguments can be made for things like heaps of sand (or garbage). There are a variety of replies to the paradox. We will first, however, apply the paradox specifically to material composition.
(1) Charles has n hairs.
(2) Someone with n hairs is not bald.
(3) Therefore Charles, with n hairs is not bald.
Now comes the experiment. We sit Charles down on a comfortable chair and give him a big, juicy rib eye steak for his troubles. We then take a pair of tweezers and pluck out one of Charles' hairs. It seems obvious that we did not do much to change Charles' being bald or not. Sure, we may have put him one hair closer to complete baldness. But we did not make him bald by removing a single hair.
(4) Charles has n-1 hairs.
(5) Someone with n-1 hairs is not bald.
(6) Therefore charles, with n-1 hairs, is not bald.
What if we kept plucking out a hair of Charles, one by one, and asking ourselves at that juncture if he were bald? Surely, at some point he must become bald. After all, a man with no hair on his head is most certainly bald. But where is that point? A man with only a single hair would still, presumably, be bald. So, too, would a man with two hairs - as evidenced think Homer Simpson. Furthermore, the following principle seems to hold:
(7) If someone with n hairs is bald, then someone with n+1 hairs is bald.
This is grounded in the implausibility that a single hair makes the difference between baldness and non-baldness. Imagine two men standing next to each other, one bald and one not bald. Would you expect there to be only a single hair to separate the two? Is that even possible? Even more troublesome is that it appears we can also reason in the opposite direction.
(8) If someone with n hairs is not bald, then someone with n-1 is not bald.
This is justified in exactly the same way as (7). Losing a single hair cannot move anyone into a state of baldness. Thus, fully presented, the argument runs:
(1) If someone with n hairs is not bald, then someone with n-1 hairs is not bald.
(2) Someone with 100,000 hairs is not bald.
(3) Someone with 99,999 hairs is not bald. (From 1 and 2)
(3) Someone with 99,998 hairs is not bald. (From 1 and 3)
...
(100,001) Someone with 0 hairs is not bald. (From 1 and 100,000)
(100,002) Someone with 0 hairs is bald. (From common sense!)
(100,003) Contradiction!
As noted above, the argument could be run in reverse, with the intuition at (100,002) reading, "Someone with 100,000 hairs is not bald."
This type of puzzle is known as the sorites paradox. It is not restricted, of course, to baldness. Parallel arguments can be made for things like heaps of sand (or garbage). There are a variety of replies to the paradox. We will first, however, apply the paradox specifically to material composition.
Thursday, July 22, 2010
On the Types of Answers to the Special Composition Question
There are three broad types of answers that can be given to the Special Composition Question. They are divided by their position on the number of composite objects relative to objects in general:
(N) [∀x: x ∈ M] ~∃y(Pxy ^ x =/= y)
(U) [∀x: x ∈ M] [∀y: y ∈ M] ∃z(Pxz ^ Pyz)
(C) ~(U) ^ ~(N), or [∃x: x ∈ M] ∃y(Pxy ^ x =/= y) ^ [∃x: x ∈ M][∃y: y ∈ M] ~∃z(Pxz ^ Pyz)
(N), or nihilism, claims that there are no cases of composition. All material objects that exist have no proper parts (proper parts satisfy Pxy and x=/=y). Thus, no matter how I arrange my bread, meat, and cheese, they will never compose another object. All material objects are simple.
(U), or universalism, claims that for any two distinct objects (x =/= y), they compose an object. Thus, the two slices of bread compose an object. That object and the meat compose an object. And that object and the cheese compose another object. This theory still allows for simple objects. Note, however, that an object cannot be a part of a composite object if another of the objects parts already contains the first as a part. That is, all objects can be parts only once; there is no double dipping of parthood. Consider objects A, B, and 3. Object 3 is the composite of objects A and B. It is impossible for there to be a 4th object that is composed of object 3 and object B. This is because object B is a part of object 3.
(C), compatiblism, claims that composition sometimes occurs. This answer is logically incompatible with either (N) or (U). Nihilism and compatibilism disagree on there being at least one composite object. Universalism and compatibilism disagree on there being an instance of failed composition. All three disagree on the number of material objects (or at least the number of potential material objects).
(N) [∀x: x ∈ M] ~∃y(Pxy ^ x =/= y)
(U) [∀x: x ∈ M] [∀y: y ∈ M] ∃z(Pxz ^ Pyz)
(C) ~(U) ^ ~(N), or [∃x: x ∈ M] ∃y(Pxy ^ x =/= y) ^ [∃x: x ∈ M][∃y: y ∈ M] ~∃z(Pxz ^ Pyz)
(N), or nihilism, claims that there are no cases of composition. All material objects that exist have no proper parts (proper parts satisfy Pxy and x=/=y). Thus, no matter how I arrange my bread, meat, and cheese, they will never compose another object. All material objects are simple.
(U), or universalism, claims that for any two distinct objects (x =/= y), they compose an object. Thus, the two slices of bread compose an object. That object and the meat compose an object. And that object and the cheese compose another object. This theory still allows for simple objects. Note, however, that an object cannot be a part of a composite object if another of the objects parts already contains the first as a part. That is, all objects can be parts only once; there is no double dipping of parthood. Consider objects A, B, and 3. Object 3 is the composite of objects A and B. It is impossible for there to be a 4th object that is composed of object 3 and object B. This is because object B is a part of object 3.
(C), compatiblism, claims that composition sometimes occurs. This answer is logically incompatible with either (N) or (U). Nihilism and compatibilism disagree on there being at least one composite object. Universalism and compatibilism disagree on there being an instance of failed composition. All three disagree on the number of material objects (or at least the number of potential material objects).
Wednesday, July 21, 2010
On the Arcane Musings of Philosophers
In everyday discourse, we make reference to an abundance of objects. In fact, the previous sentence implied at least three with the words 'discourse', 'we', and 'abundance'. Many of these are things we can directly sense, like my cat Gizmo. Some objects, are of an abstract nature. The number three is not an object that one would expect to bump into on his way to work.
This paper will focus on the first category, so-called material objects. A material object is one that is located in the material world. It has extension: length, width, height, and therefore volume. (Note here that, technically, a material object will be defined as an object that has a part that has extension. Recall that all objects have themselves as a part.)
Where M is the set of all material objects, Pxy is the relation of x being a part of y and Ex is the property of extension:
(1) [∀x: x ∈ M] ∃y(Pyx ^ Ey)
The principle target of inquiry will be an answer to van Inwagen's Special Composition Question. That is, when is it the case that two or more objects compose an additional object?
SCQ: ∃y the xs compose y?
All of this probably seems quite obscure to the reader. Let me try to distill the jargon. Consider the food I've laid out on the kitchen counter. There are two pieces of bread, some ham, and a slice of cheese. The Special Composition Question amounts to this: in what circumstances do the bread, the ham, and the cheese compose some further object?
This paper will focus on the first category, so-called material objects. A material object is one that is located in the material world. It has extension: length, width, height, and therefore volume. (Note here that, technically, a material object will be defined as an object that has a part that has extension. Recall that all objects have themselves as a part.)
Where M is the set of all material objects, Pxy is the relation of x being a part of y and Ex is the property of extension:
(1) [∀x: x ∈ M] ∃y(Pyx ^ Ey)
The principle target of inquiry will be an answer to van Inwagen's Special Composition Question. That is, when is it the case that two or more objects compose an additional object?
SCQ: ∃y the xs compose y?
All of this probably seems quite obscure to the reader. Let me try to distill the jargon. Consider the food I've laid out on the kitchen counter. There are two pieces of bread, some ham, and a slice of cheese. The Special Composition Question amounts to this: in what circumstances do the bread, the ham, and the cheese compose some further object?
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