13 ms·

Sorites Paradox

59 points by ikeboy 11y ago

n0us 11y ago
I wrote my senior thesis on this topic when I was completing my philosophy major. One of the best texts on the subject is "Vagueness" by Williamson. It's an arcane but super interesting topic that actually has wide implications. It's also IMO one of the most thorny problems in contemporary philosophy. edit: I'm currently busy but can come back later to answer questions if anyone is interested. http://www.amazon.com/Vagueness-Problems-Philosophy-Timothy-Williamson/dp/0415139805 http://www.amazon.com/Vagueness-Problems-Philosophy-Timothy-...
- ikeboy 11y ago
  I bumped into http://philpapers.org/archive/FRAQTV.pdf http://philpapers.org/archive/FRAQTV.pdf, which argues that most answers will trivially fail. I felt the answer should involve degrees of truth, which can roughly be calculated as "percent of people who agree with the classification" for some reference class, and this degree of truth gradually decreases. But you can't assign any arbitrary cutoff number where above it is true and below false. The way to combine two statements would be to likewise ask what portion of people agree with both, which seems to avoid some of the critiques of degrees of truth. This is only for problems of classification over human undefined terms, where the argument from common usage is valid. Does this have any major problems?
  - n0us 11y ago
    The impulse to develop a system of logic that relies on degrees of truth is one that many people have had over the history of philosophy and I can tell you that it is highly problematic. I am struggling to find a succinct way to explain why this is but the following example might be work. 1. It is 50% true that Sally has red hair. 2. It is 50% true that Sally has brown hair. 3. It is 50% true that Sally has not red hair. (see 2.) You end up running into contradictions all over the place and having a non-binary system of logic makes it nearly impossible to develop any formal system of reasoning. I'm not completely clear on the second half of your comment but when you say "portion of people agree" I assume you are trying to use an empirical theory of truth. There are again many problems with this theory but I can think of an example that might illustrate. It's a toy example that I won't fully expand upon but it illustrates the point at hand. There is a box in a room that is painted yellow. A group of people is asked to state the color of the box but they are all colorblind and state that it is grey. The consensus about the color of the box is clearly wrong in this case. Some people will object to this example because the meaning of words is also clearly derived from their use. The epistemic theory of vagueness is not far off from what you are proposing and it is the theory that Williamson posits in his book. Essentially, there is a hard cutoff for all statements that are vague. It is either true that the collection of grains is a heap or it is not. An omniscient speaker could evaluate the truth of a statement about whether the collection of grains is a heap or not with definite certainty because only the omniscient speaker possesses all of the relevant information about the property of being a heap. In other words vagueness is a problem of inexact knowledge. Another example of inexact knowledge is if you were to look out over a stadium and state "there are 23,422 people here." Barring the possibility that you are a savant and actually counted all of the people, if you made that statement and it was true that 23,422 people were in the stadium it isn't also true that you knew that there were that many people. You lack the relevant information about how many people are in the stadium to safely make the claim that there are n number of people in the stadium just as you lack the relevant information about the property of being a heap to safely make the claim that you know it is true that a collection of n grains is a heap. This is a powerful theory which is not as easily defeated as others but keep in mind that it is not without weakness. Hopefully this at least exposes some of the concepts to you even if it's at a superficial level.
    - ikeboy 11y ago
      I don't see how your example leads to a contradiction. If 1 and 2 hold, and no person considers Sally to have more than one color hair, this seems fine. >There is a box in a room that is painted yellow. A group of people is asked to state the color of the box but they are all colorblind and state that it is grey. The consensus about the color of the box is clearly wrong in this case. My proposal doesn't depend on people actually being asked. It's more of "what would people say, if they were asked, and given all relevant objective information". Your scenario is ruled out.
      - coldtea 11y ago
        >My proposal doesn't depend on people actually being asked. It's more of "what would people say, if they were asked, and given all relevant objective information". Your scenario is ruled out. This clarification rules out your proposal as any kind of actually implementable system then, because it collapses to having a single authority who "knows" (how?) what the people would say. If it doesn't involve actually asking people, then it's not really a polling of perceptions, but some person's idea of what those perceptions would be.
        ikeboy 11y ago
        It's supposed to answer "what does claim X mean". It's not meant to be implemented. An approximation is carried out every time someone pulls out a dictionary to resolve a dispute over common usage.
      - n0us 11y ago
        There is a difference between saying that "it is 50% true that Sally's hair is red" and "it is true that Sally's hair is 50% red". It isn't a scenario, it's a thought experiment. "given all relevant objective information" sounds much like the omniscient speaker which is central to the epistemic theory.
        ikeboy 11y ago
        I still don't see any contradiction. A single omniscient speaker can't work, because people will disagree even with all relevant objective facts agreed upon. When two people dispute whether something is a heap, they aren't arguing over objective facts.
  - mannykannot 11y ago
    That essay ends with this paragraph: "The vagueness paradox is this: although there is an extremely strong argument for sharpism, the truth of sharpism seems to require a linguistic miracle." (my emphasis.) The author concludes that the apparently strong logical argument ultimately fails to address the real issue, which actually concerns usage of the word 'pumpkin’. It is a paradox (i.e. only an apparent problem) because there is neither a practical nor a logical necessity for ‘pumpkin’ to have a singular and absolute definition. The allegedly strong argument is basically begging the question. It is crucially dependent on the assumption that, for any particular ‘situation’, we can unambiguously say whether or not it can be called a pumpkin, but that is the premise that it is attempting to prove. Another way of looking at it is that while we clearly have pumpkin at one end of the truth table and clearly do not at the other, the formalism offers no way to decide where the boundary lies: this representation has its own sorities issue, and models the alleged problem without offering anything that leads to a solution - substituting 'true' or 'false' for the presence or absence of grains of sand does not make the problem any more tractable. A secondary problem is that it depends on there being a total ordering of situations, but this fails when we realize that pumpkin-ness is context-dependent: in some contexts, a set of chromosomes is a pumpkin, while in others, nothing but a whole plant will do. Different contexts will be satisfied with different subsets of the pumpkin's atoms, and, in general, any one context will be satisfied with many different subsets.
    - ikeboy 11y ago
      > It is crucially dependent on the assumption that, for any particular ‘situation’, we can unambiguously say whether or not it can be called a pumpkin, but that is the premise that it is attempting to prove. No. As long as you concede that some pumpkins can be unambiguously called a pumpkin, then you just ask which is the first object on the list that can't be.
      - mannykannot 11y ago
        On reflection, I think I may be wrong to say that he is begging the question, but not because of your counter-argument. If we start from the other end of the list, the first thing that is not unambiguously not a pumpkin is not necessarily the last unambiguous pumpkin you picked in the first pass. You have introduced the ambiguity that the author is trying to insist is not there. My other claims stand, I think. The point is that his linear model is not an adequate fit to the problem (for example, the ambiguity you introduced really is there, but the model excludes it.) Maybe there is something like question-begging in thinking that this is an adequate model.
- random28345 11y ago
  I'm not sure what the big deal here is. This has an obvious solution. "Heap" is a word. Words have meaning to people. Different people ascribe different meanings to the same word. Each grain added to a pile of grain measurably changes the probability that an observer will classify the resulting collection as a heap. While it's not a linear function, it's generally a positive correlation, more grains means a higher probability of any individual calling it a heap. That is a collection of grain of size n+1 has a probability of being classified as a heap that is usually greater than the probability of a collection of grain of size n being classified as a heap. Where's the paradox?
  - n0us 11y ago
    You are conflating probability with truth. A person also may incorrectly classify a collection of grain size n as a heap.
    - dragonwriter 11y ago
      Perhaps he is not conflating them but asserting that, for named categories, there is no truth to membership of an item in them beyond the probability that an observer will ascribe an item to the named category.
  - hderms 11y ago
    It's just interesting when you have discrete entities and you can't show inductively that there is any point where the classification changes from non-heap to heap, but clearly that point must exist. It's interesting from the same perspective as what makes the ship of Theseus interesting. the probability of something being viewed as a heap could only be measured statistically as the probability doesn't derive from some direct process. The issue can be viewed in both an axiomatic and a empirical way and both views lead to interesting conclusions.
    - empath75 11y ago
      That point may not exist though, if it just gains more 'heapness' over time. Even if 99% of people agree that a collection of grains of sand is a 'heap' there might be some people who still insist that it is not.
- danielam 11y ago
  Related: http://edwardfeser.blogspot.com/2011/04/nature-versus-art.html http://edwardfeser.blogspot.com/2011/04/nature-versus-art.ht...
Xcelerate 11y ago
It might not be apparent from the examples provided in the article, but I would argue that this paradox is the primary polarizing force behind most people's morality disagreements. A lot of the disagreements come down to the question "Well where do we draw the line?"
- tunesmith 11y ago
  This is why I feel like I'm transitioning, over time, from value ethics to deontology. Value ethics are really susceptible to this paradox. But if I can induce what appear to be my principles from my value-ethics instincts, and then, as an exercise, try to reason out from my principles and act accordingly, then perhaps I am more consistent. That's if you see being consistent as an axiom to try to always meet. A lot of people don't, and I don't know how to argue with them.
SilasX 11y ago
I love this paradox! It actually transforms into an interesting general heuristic for life[1]: basically, any time reality throws a categorization boundary at you (you observe f(a) = X and f(b) = not-X), gradually transform it (a to b) until you see exactly where the category flips. For example, if you have a bug in case A but not case B, gradually transform A into B and see where it disappears. If a law says that you can do X in condition A but not condition B, see if it's legal midway between A and B. [1] which I call the "continuity heuristic" since it depends on reality obeying a rule that if a is close to b, f(a) is close to f(b).
mrpoptart 11y ago
Why is it defined that a single grain of wheat isn't a very small heap?
- n0us 11y ago
  If a single grain is a heap then any number of grains is a heap and the property of being a heap is meaningless. A single grain as a "small" heap also runs into a problem called higher order vagueness. Where do you draw the line between a small heap and a regular heap or a large heap? Determining the cutoff seems to be another heap problem.
  - empath75 11y ago
    That problem applies with any arbitrary category. If you look at any color, you're going to describe it as whatever color category its closest to. If it's more red than yellow, you might call it orange, I might call it red, someone else might call it orange-red, or I dunno 'pumpkin'. Language is imprecise.
    - n0us 11y ago
      Is it categories themselves that are arbitrary or the set of items belonging to a category are arbitrary? Do you mean that the meaning of a statement is imprecise in that it is vague or ambiguous? Or do you mean that the way that humans employ language to convey concepts is imprecise? You are thinking like a psychologist or a linguist. I'm not denying that many people view this is a tedious and seemingly silly thing to argue about but you can't metaphorically just throw up you hands and claim that it doesn't matter as a valid counter argument when you are doing philosophy.
      - mannykannot 11y ago
        You make philosophy sound like a form of OCD, obsessing about putting everything in its place. The answer to your questions is 'all of the above, to some degree'.
  - 11y ago
chris_va 11y ago
Why is it necessary to reduce language to set theory? I say it is a heap when doing so improves the odds of my preferred outcome in a conversation. Being labeled a heap doesn't have to be a binary truism. Or even a probability. I guess I reject the entire field of study :).
- coldtea 11y ago
  >Being labeled a heap doesn't have to be a binary truism. Or even a probability. That in itself is the whole point of the original observation. That there are things, and not fleeting things like notions and feelings, but actual tangible things, that are nevertheless not in a specific binary state, but in some non-specified intermediate state.
  - chris_va 11y ago
    The semantics are difficult here, apologies. I can't tell if we are agreeing :) Why do things have to be in any state, even a "non-specific intermediate" one? The original observation seems to have devolved into an argument on the fundamental nature of language expressing "state". State here implying some sort of formalism, like sets. Can't we just assert that language doesn't assign or derive from a formal state? Rather, the use of language is highly correlated with outcomes/observations, but not intrinsically meaningful.
    - coldtea 11y ago
      >Why do things have to be in any state, even a "non-specific intermediate" one? Because being in a state is just another name of being a thing (or plain "being"). If you "are", by definition you are in some state (at any given point in time). >Can't we just assert that language doesn't assign or derive from a formal state? Rather, the use of language is highly correlated with outcomes/observations, but not intrinsically meaningful. Well, even without using language, we experience things as being in particular states -- we see them having color, making a noise, being there and not here, etc.
      - 11y ago
zw123456 11y ago
This is related to the Continuum Fallacy https://en.wikipedia.org/wiki/Continuum_fallacy https://en.wikipedia.org/wiki/Continuum_fallacy Or another really fun one, Loki's wager.
stcredzero 11y ago
Given then that one grain of wheat does not make a heap, it would seem to follow that two do not, thus three do not, and so on. The weasel words in this case are "and so on." Four grains of wheat can make a heap. Put 3 on the bottom and one on top. If it's too windy to work where you are, slather them in honey. A sticky heap is still a heap.
- dsg42 11y ago
  You're right, that isn't really the best statement of the paradox. Try this: Imagine you have a heap of sand. Now, if you remove a single grain of sand from a heap, it's still a heap, right? But if you remove every grain of sand, then eventually, you're left without any sand, which is definitely not a heap. So somewhere along the way, when removing individual grains of sand, the heap stopped being a heap. But where, exactly, did that happen?
  - chongli 11y ago
    It happened at the exact moment when none of the grains were supported by any of the others. A flat, single grain layer of sand on the ground is not a heap no matter how many grains are in it.
    - cgio 11y ago
      Soros in Greek does not have the structural connotation of heap in English (which I infer from your reply, given English is not my mother tongue).
  - stcredzero 11y ago
    For very blocky, stackable grains of sand, it could've happened at 2. It undoubtedly happened somewhere below 9. That's as exact as this needs to be, unless you are a pedant. In which case, fuggedaboudit!
  - ikeboy 11y ago
    Just use a different example, e.g. baldness, or the pumpkin one from http://philpapers.org/archive/FRAQTV.pdf http://philpapers.org/archive/FRAQTV.pdf
trhway 11y ago
"Vagueness" is one side where the other side of the same coin is abstraction - instead of counting number of sand grains in any given case and using "set of N grains" we abstracted such cases - where N is large enough for precise value of N not being important - into "heap". Abstraction is a convenient tool, and trying to put abstracted out details back in - like the precise number of grains - makes the tool more of a unnecessary liability, i.e "heap" abstraction disappears the moment you ask the precise number of grains, and instead "set of N grains" appears back.
pjdorrell 11y ago
A related question (I think), is what counts as "impossible"? According to quantum mechanics, anything that is not prohibited by a conservation law is possible, with some non-zero probability. So we have to pragmatically redefine "impossible" to mean "the probability is so small that it won't happen". But then where do we draw the line between "too small" and "not too small"? (We can also qualify "won't happen" by saying "cannot have happened in the history of the observable universe", and take into account an upper bound on the number of "events" that can ever happen in the history of the observable universe, eg ~ 2^380. So, for example, an event with probability 2^-500 on one occasion has probability < 2^-120 of ever having happened that we could know about.)
GunboatDiplomat 11y ago
This is simply a failure to sufficiently define a word.
- cbd1984 11y ago
  > This is simply a failure to sufficiently define a word. That's the point: How do you "sufficiently define" the word involved here? Also, how do you "sufficiently define" the word 'ethical'?
Dove 11y ago
So, mathematics has this concept of an expression being "well-defined". It means you can always tell if something meets the definition, exactly, in every case. Almost no definitions in the real world are well-defined. Heaps are just the tip of the iceberg. Any practical thing, a cat, a tree, a table, a car -- if you mutilate it enough, it turns into something else, and the transition point is rarely sharp. And the reason is very simple; it's that in practical life, we don't define things propositionally, we define them inductively. This is a cat, and that's a cat, and that's another cat, and cats are things with this constellation of properties that you learn over time, so when you come across an item where some of those properties are true and some aren't, you say, "Well, that's sort of a weird cat". It is such an incredibly common thing in daily life to run up against fuzzy definition boundaries. "Can you get milk from the store for me if it's open?" "Well, there's some sort of demonstration going on right outside -- how bad do you need it?" We handle it just fine. We recognize when "open" may not be well-defined in every circumstance and valid in the same way for every intent, and we're able to discard the simple definition for something more complex as needed. It's not even weird when you consider that definitions are just models you are using to describe the world, and those models can be made simple or complex. We rarely describe the world as it IS, down to the atom. We simplify. Sure, when you throw inductively defined practical models intended for communication into a rigorous logical framework, they don't have the sharp edges such a framework expects. So? That really shouldn't be unexpected. :) Everything we have to say about the real world is an abstraction. Some ideas are leakier abstractions than others, but it's all simplification and modeling in the end.
jbandela1 11y ago
For the HN population, translating this into security can be useful. Here is a Sorites argument for AES key strength. A key of length 1 bit is not secure A key of length 2 bit is not secure A key of length 3 bit is not secure ------------------------------------ A key of length 1024 bits is not secure We would consider the conclusion absurd. Why? Because we have a definition of security as such: The expense of breaking the key is not worth the value of the secret it protects. For 1 bit, the expense of breaking the key is so low, it is very easy to come up with secrets that are greater than the value. For 256 bits, the expense of breaking the key is so (literally) astronomically high, that it would be very hard to come up with secrets that have greater value than the expense of breaking the key. In addition, note that for a given secret 1 bit can make the difference between secure and not secure. Notice as we increase the key length bit by bit, more and more secrets are having values that are less than the expense of breaking the key. Now let us look at a heap. First let us come up with a working definition. Let us say that we define a heap as an amount of grain that is larger than can be held in a closed fist. Notice how this varies from person to person. Also notice for a given person, there can easily be the case where 1 extra grain makes the difference between heap and not. Also notice, that as we increase the number of grains by one, more and more people would consider the collection of grains a heap.

59 points by ikeboy 11y ago

n0us 11y ago

I wrote my senior thesis on this topic when I was completing my philosophy major. One of the best texts on the subject is "Vagueness" by Williamson. It's an arcane but super interesting topic that actually has wide implications. It's also IMO one of the most thorny problems in contemporary philosophy. edit: I'm currently busy but can come back later to answer questions if anyone is interested. http://www.amazon.com/Vagueness-Problems-Philosophy-Timothy-Williamson/dp/0415139805 http://www.amazon.com/Vagueness-Problems-Philosophy-Timothy-...

ikeboy 11y ago
I bumped into http://philpapers.org/archive/FRAQTV.pdf http://philpapers.org/archive/FRAQTV.pdf, which argues that most answers will trivially fail. I felt the answer should involve degrees of truth, which can roughly be calculated as "percent of people who agree with the classification" for some reference class, and this degree of truth gradually decreases. But you can't assign any arbitrary cutoff number where above it is true and below false. The way to combine two statements would be to likewise ask what portion of people agree with both, which seems to avoid some of the critiques of degrees of truth. This is only for problems of classification over human undefined terms, where the argument from common usage is valid. Does this have any major problems?
- n0us 11y ago
  The impulse to develop a system of logic that relies on degrees of truth is one that many people have had over the history of philosophy and I can tell you that it is highly problematic. I am struggling to find a succinct way to explain why this is but the following example might be work. 1. It is 50% true that Sally has red hair. 2. It is 50% true that Sally has brown hair. 3. It is 50% true that Sally has not red hair. (see 2.) You end up running into contradictions all over the place and having a non-binary system of logic makes it nearly impossible to develop any formal system of reasoning. I'm not completely clear on the second half of your comment but when you say "portion of people agree" I assume you are trying to use an empirical theory of truth. There are again many problems with this theory but I can think of an example that might illustrate. It's a toy example that I won't fully expand upon but it illustrates the point at hand. There is a box in a room that is painted yellow. A group of people is asked to state the color of the box but they are all colorblind and state that it is grey. The consensus about the color of the box is clearly wrong in this case. Some people will object to this example because the meaning of words is also clearly derived from their use. The epistemic theory of vagueness is not far off from what you are proposing and it is the theory that Williamson posits in his book. Essentially, there is a hard cutoff for all statements that are vague. It is either true that the collection of grains is a heap or it is not. An omniscient speaker could evaluate the truth of a statement about whether the collection of grains is a heap or not with definite certainty because only the omniscient speaker possesses all of the relevant information about the property of being a heap. In other words vagueness is a problem of inexact knowledge. Another example of inexact knowledge is if you were to look out over a stadium and state "there are 23,422 people here." Barring the possibility that you are a savant and actually counted all of the people, if you made that statement and it was true that 23,422 people were in the stadium it isn't also true that you knew that there were that many people. You lack the relevant information about how many people are in the stadium to safely make the claim that there are n number of people in the stadium just as you lack the relevant information about the property of being a heap to safely make the claim that you know it is true that a collection of n grains is a heap. This is a powerful theory which is not as easily defeated as others but keep in mind that it is not without weakness. Hopefully this at least exposes some of the concepts to you even if it's at a superficial level.
  - ikeboy 11y ago
    I don't see how your example leads to a contradiction. If 1 and 2 hold, and no person considers Sally to have more than one color hair, this seems fine. >There is a box in a room that is painted yellow. A group of people is asked to state the color of the box but they are all colorblind and state that it is grey. The consensus about the color of the box is clearly wrong in this case. My proposal doesn't depend on people actually being asked. It's more of "what would people say, if they were asked, and given all relevant objective information". Your scenario is ruled out.
    - coldtea 11y ago
      >My proposal doesn't depend on people actually being asked. It's more of "what would people say, if they were asked, and given all relevant objective information". Your scenario is ruled out. This clarification rules out your proposal as any kind of actually implementable system then, because it collapses to having a single authority who "knows" (how?) what the people would say. If it doesn't involve actually asking people, then it's not really a polling of perceptions, but some person's idea of what those perceptions would be.
      - ikeboy 11y ago
        It's supposed to answer "what does claim X mean". It's not meant to be implemented. An approximation is carried out every time someone pulls out a dictionary to resolve a dispute over common usage.
    - n0us 11y ago
      There is a difference between saying that "it is 50% true that Sally's hair is red" and "it is true that Sally's hair is 50% red". It isn't a scenario, it's a thought experiment. "given all relevant objective information" sounds much like the omniscient speaker which is central to the epistemic theory.
      - ikeboy 11y ago
        I still don't see any contradiction. A single omniscient speaker can't work, because people will disagree even with all relevant objective facts agreed upon. When two people dispute whether something is a heap, they aren't arguing over objective facts.
- mannykannot 11y ago
  That essay ends with this paragraph: "The vagueness paradox is this: although there is an extremely strong argument for sharpism, the truth of sharpism seems to require a linguistic miracle." (my emphasis.) The author concludes that the apparently strong logical argument ultimately fails to address the real issue, which actually concerns usage of the word 'pumpkin’. It is a paradox (i.e. only an apparent problem) because there is neither a practical nor a logical necessity for ‘pumpkin’ to have a singular and absolute definition. The allegedly strong argument is basically begging the question. It is crucially dependent on the assumption that, for any particular ‘situation’, we can unambiguously say whether or not it can be called a pumpkin, but that is the premise that it is attempting to prove. Another way of looking at it is that while we clearly have pumpkin at one end of the truth table and clearly do not at the other, the formalism offers no way to decide where the boundary lies: this representation has its own sorities issue, and models the alleged problem without offering anything that leads to a solution - substituting 'true' or 'false' for the presence or absence of grains of sand does not make the problem any more tractable. A secondary problem is that it depends on there being a total ordering of situations, but this fails when we realize that pumpkin-ness is context-dependent: in some contexts, a set of chromosomes is a pumpkin, while in others, nothing but a whole plant will do. Different contexts will be satisfied with different subsets of the pumpkin's atoms, and, in general, any one context will be satisfied with many different subsets.
  - ikeboy 11y ago
    > It is crucially dependent on the assumption that, for any particular ‘situation’, we can unambiguously say whether or not it can be called a pumpkin, but that is the premise that it is attempting to prove. No. As long as you concede that some pumpkins can be unambiguously called a pumpkin, then you just ask which is the first object on the list that can't be.
    - mannykannot 11y ago
      On reflection, I think I may be wrong to say that he is begging the question, but not because of your counter-argument. If we start from the other end of the list, the first thing that is not unambiguously not a pumpkin is not necessarily the last unambiguous pumpkin you picked in the first pass. You have introduced the ambiguity that the author is trying to insist is not there. My other claims stand, I think. The point is that his linear model is not an adequate fit to the problem (for example, the ambiguity you introduced really is there, but the model excludes it.) Maybe there is something like question-begging in thinking that this is an adequate model.
random28345 11y ago
I'm not sure what the big deal here is. This has an obvious solution. "Heap" is a word. Words have meaning to people. Different people ascribe different meanings to the same word. Each grain added to a pile of grain measurably changes the probability that an observer will classify the resulting collection as a heap. While it's not a linear function, it's generally a positive correlation, more grains means a higher probability of any individual calling it a heap. That is a collection of grain of size n+1 has a probability of being classified as a heap that is usually greater than the probability of a collection of grain of size n being classified as a heap. Where's the paradox?
- n0us 11y ago
  You are conflating probability with truth. A person also may incorrectly classify a collection of grain size n as a heap.
  - dragonwriter 11y ago
    Perhaps he is not conflating them but asserting that, for named categories, there is no truth to membership of an item in them beyond the probability that an observer will ascribe an item to the named category.
- hderms 11y ago
  It's just interesting when you have discrete entities and you can't show inductively that there is any point where the classification changes from non-heap to heap, but clearly that point must exist. It's interesting from the same perspective as what makes the ship of Theseus interesting. the probability of something being viewed as a heap could only be measured statistically as the probability doesn't derive from some direct process. The issue can be viewed in both an axiomatic and a empirical way and both views lead to interesting conclusions.
  - empath75 11y ago
    That point may not exist though, if it just gains more 'heapness' over time. Even if 99% of people agree that a collection of grains of sand is a 'heap' there might be some people who still insist that it is not.
danielam 11y ago
Related: http://edwardfeser.blogspot.com/2011/04/nature-versus-art.html http://edwardfeser.blogspot.com/2011/04/nature-versus-art.ht...

Xcelerate 11y ago

It might not be apparent from the examples provided in the article, but I would argue that this paradox is the primary polarizing force behind most people's morality disagreements. A lot of the disagreements come down to the question "Well where do we draw the line?"

tunesmith 11y ago
This is why I feel like I'm transitioning, over time, from value ethics to deontology. Value ethics are really susceptible to this paradox. But if I can induce what appear to be my principles from my value-ethics instincts, and then, as an exercise, try to reason out from my principles and act accordingly, then perhaps I am more consistent. That's if you see being consistent as an axiom to try to always meet. A lot of people don't, and I don't know how to argue with them.

SilasX 11y ago

I love this paradox! It actually transforms into an interesting general heuristic for life[1]: basically, any time reality throws a categorization boundary at you (you observe f(a) = X and f(b) = not-X), gradually transform it (a to b) until you see exactly where the category flips. For example, if you have a bug in case A but not case B, gradually transform A into B and see where it disappears. If a law says that you can do X in condition A but not condition B, see if it's legal midway between A and B. [1] which I call the "continuity heuristic" since it depends on reality obeying a rule that if a is close to b, f(a) is close to f(b).

mrpoptart 11y ago

Why is it defined that a single grain of wheat isn't a very small heap?

n0us 11y ago
If a single grain is a heap then any number of grains is a heap and the property of being a heap is meaningless. A single grain as a "small" heap also runs into a problem called higher order vagueness. Where do you draw the line between a small heap and a regular heap or a large heap? Determining the cutoff seems to be another heap problem.
- empath75 11y ago
  That problem applies with any arbitrary category. If you look at any color, you're going to describe it as whatever color category its closest to. If it's more red than yellow, you might call it orange, I might call it red, someone else might call it orange-red, or I dunno 'pumpkin'. Language is imprecise.
  - n0us 11y ago
    Is it categories themselves that are arbitrary or the set of items belonging to a category are arbitrary? Do you mean that the meaning of a statement is imprecise in that it is vague or ambiguous? Or do you mean that the way that humans employ language to convey concepts is imprecise? You are thinking like a psychologist or a linguist. I'm not denying that many people view this is a tedious and seemingly silly thing to argue about but you can't metaphorically just throw up you hands and claim that it doesn't matter as a valid counter argument when you are doing philosophy.
    - mannykannot 11y ago
      You make philosophy sound like a form of OCD, obsessing about putting everything in its place. The answer to your questions is 'all of the above, to some degree'.
- 11y ago

chris_va 11y ago

Why is it necessary to reduce language to set theory? I say it is a heap when doing so improves the odds of my preferred outcome in a conversation. Being labeled a heap doesn't have to be a binary truism. Or even a probability. I guess I reject the entire field of study :).

coldtea 11y ago
>Being labeled a heap doesn't have to be a binary truism. Or even a probability. That in itself is the whole point of the original observation. That there are things, and not fleeting things like notions and feelings, but actual tangible things, that are nevertheless not in a specific binary state, but in some non-specified intermediate state.
- chris_va 11y ago
  The semantics are difficult here, apologies. I can't tell if we are agreeing :) Why do things have to be in any state, even a "non-specific intermediate" one? The original observation seems to have devolved into an argument on the fundamental nature of language expressing "state". State here implying some sort of formalism, like sets. Can't we just assert that language doesn't assign or derive from a formal state? Rather, the use of language is highly correlated with outcomes/observations, but not intrinsically meaningful.
  - coldtea 11y ago
    >Why do things have to be in any state, even a "non-specific intermediate" one? Because being in a state is just another name of being a thing (or plain "being"). If you "are", by definition you are in some state (at any given point in time). >Can't we just assert that language doesn't assign or derive from a formal state? Rather, the use of language is highly correlated with outcomes/observations, but not intrinsically meaningful. Well, even without using language, we experience things as being in particular states -- we see them having color, making a noise, being there and not here, etc.
    - 11y ago

zw123456 11y ago

This is related to the Continuum Fallacy https://en.wikipedia.org/wiki/Continuum_fallacy https://en.wikipedia.org/wiki/Continuum_fallacy Or another really fun one, Loki's wager.

stcredzero 11y ago

Given then that one grain of wheat does not make a heap, it would seem to follow that two do not, thus three do not, and so on. The weasel words in this case are "and so on." Four grains of wheat can make a heap. Put 3 on the bottom and one on top. If it's too windy to work where you are, slather them in honey. A sticky heap is still a heap.

dsg42 11y ago
You're right, that isn't really the best statement of the paradox. Try this: Imagine you have a heap of sand. Now, if you remove a single grain of sand from a heap, it's still a heap, right? But if you remove every grain of sand, then eventually, you're left without any sand, which is definitely not a heap. So somewhere along the way, when removing individual grains of sand, the heap stopped being a heap. But where, exactly, did that happen?
- chongli 11y ago
  It happened at the exact moment when none of the grains were supported by any of the others. A flat, single grain layer of sand on the ground is not a heap no matter how many grains are in it.
  - cgio 11y ago
    Soros in Greek does not have the structural connotation of heap in English (which I infer from your reply, given English is not my mother tongue).
- stcredzero 11y ago
  For very blocky, stackable grains of sand, it could've happened at 2. It undoubtedly happened somewhere below 9. That's as exact as this needs to be, unless you are a pedant. In which case, fuggedaboudit!
- ikeboy 11y ago
  Just use a different example, e.g. baldness, or the pumpkin one from http://philpapers.org/archive/FRAQTV.pdf http://philpapers.org/archive/FRAQTV.pdf

trhway 11y ago

"Vagueness" is one side where the other side of the same coin is abstraction - instead of counting number of sand grains in any given case and using "set of N grains" we abstracted such cases - where N is large enough for precise value of N not being important - into "heap". Abstraction is a convenient tool, and trying to put abstracted out details back in - like the precise number of grains - makes the tool more of a unnecessary liability, i.e "heap" abstraction disappears the moment you ask the precise number of grains, and instead "set of N grains" appears back.

pjdorrell 11y ago

A related question (I think), is what counts as "impossible"? According to quantum mechanics, anything that is not prohibited by a conservation law is possible, with some non-zero probability. So we have to pragmatically redefine "impossible" to mean "the probability is so small that it won't happen". But then where do we draw the line between "too small" and "not too small"? (We can also qualify "won't happen" by saying "cannot have happened in the history of the observable universe", and take into account an upper bound on the number of "events" that can ever happen in the history of the observable universe, eg ~ 2^380. So, for example, an event with probability 2^-500 on one occasion has probability < 2^-120 of ever having happened that we could know about.)

GunboatDiplomat 11y ago

This is simply a failure to sufficiently define a word.

cbd1984 11y ago
> This is simply a failure to sufficiently define a word. That's the point: How do you "sufficiently define" the word involved here? Also, how do you "sufficiently define" the word 'ethical'?

Dove 11y ago

So, mathematics has this concept of an expression being "well-defined". It means you can always tell if something meets the definition, exactly, in every case. Almost no definitions in the real world are well-defined. Heaps are just the tip of the iceberg. Any practical thing, a cat, a tree, a table, a car -- if you mutilate it enough, it turns into something else, and the transition point is rarely sharp. And the reason is very simple; it's that in practical life, we don't define things propositionally, we define them inductively. This is a cat, and that's a cat, and that's another cat, and cats are things with this constellation of properties that you learn over time, so when you come across an item where some of those properties are true and some aren't, you say, "Well, that's sort of a weird cat". It is such an incredibly common thing in daily life to run up against fuzzy definition boundaries. "Can you get milk from the store for me if it's open?" "Well, there's some sort of demonstration going on right outside -- how bad do you need it?" We handle it just fine. We recognize when "open" may not be well-defined in every circumstance and valid in the same way for every intent, and we're able to discard the simple definition for something more complex as needed. It's not even weird when you consider that definitions are just models you are using to describe the world, and those models can be made simple or complex. We rarely describe the world as it IS, down to the atom. We simplify. Sure, when you throw inductively defined practical models intended for communication into a rigorous logical framework, they don't have the sharp edges such a framework expects. So? That really shouldn't be unexpected. :) Everything we have to say about the real world is an abstraction. Some ideas are leakier abstractions than others, but it's all simplification and modeling in the end.

jbandela1 11y ago

For the HN population, translating this into security can be useful. Here is a Sorites argument for AES key strength. A key of length 1 bit is not secure A key of length 2 bit is not secure A key of length 3 bit is not secure ------------------------------------ A key of length 1024 bits is not secure We would consider the conclusion absurd. Why? Because we have a definition of security as such: The expense of breaking the key is not worth the value of the secret it protects. For 1 bit, the expense of breaking the key is so low, it is very easy to come up with secrets that are greater than the value. For 256 bits, the expense of breaking the key is so (literally) astronomically high, that it would be very hard to come up with secrets that have greater value than the expense of breaking the key. In addition, note that for a given secret 1 bit can make the difference between secure and not secure. Notice as we increase the key length bit by bit, more and more secrets are having values that are less than the expense of breaking the key. Now let us look at a heap. First let us come up with a working definition. Let us say that we define a heap as an amount of grain that is larger than can be held in a closed fist. Notice how this varies from person to person. Also notice for a given person, there can easily be the case where 1 extra grain makes the difference between heap and not. Also notice, that as we increase the number of grains by one, more and more people would consider the collection of grains a heap.