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Why paradoxes?

  1. Dec 29, 2003 #1
    Paradoxes exist in our logic. They do not exist in the real world (to our knowledge at least; no one has ever encountered proof of a pardox existing). Logic is sopposed to describe the real world. Could there be some sort of flaw in our logic then :S?
     
  2. jcsd
  3. Dec 30, 2003 #2

    Njorl

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    There are physical analogs of paradoxes in the physical world. Bistable states in logic devices for instance.

    There is more to what you say though. I think the simple, everyday logic we learn is not adequate for all occasions. This is not suprising. By analogy, arithmatic and algebra will let you do your household budget, but they won't put a man on the moon. The simple logic of true/false and epistemology work for most requirements, but modern mathematical logic has other states. I'm no expert, but I believe there are states like unknowable, indeterminate, and immaterial in addition to true and false.

    Njorl
     
  4. Dec 30, 2003 #3
    Sikz,
    One must remember that, as Tom has stressed before, Logic is a prescription for the Universe, not a description of it. One can deduce, from Logic, what the Universe should be like if such-and-such were true...however, the Universe is not always as one deduction says it "should be". Also, if a deduction leads to a paradox, it is pretty well impossible (just my opinion) for it to be expressed in reality...

    Of course, there are those who say that QM requires paradoxes of a sort, but I think QM doesn't so much counter Logic, as it does Common Sense...as such, I think that one must use a slightly refined reasoning method, but it remains in the realm of "logic", and doesn't produce paradoxes.
     
  5. Jan 3, 2004 #4
    Short answer: Logic is not a set of laws which govern the universe.
     
  6. Jan 3, 2004 #5
    Re: Re: Why paradoxes?

    Well, you're right, of course, but what of the fact that logic is supposed to be a prescription for the Universe (anyone who can catch the intended joke here has my admiration)? If Logic shows what the Universe "should be" like, and yet it (the Universe) has no paradoxes, then Logic is doing a poor job of prescribing, isn't it?
     
  7. Jan 3, 2004 #6
    What? Have you never met a teenage girl?
     
  8. Jan 4, 2004 #7
    Paradox means two different things.

    1. Paradox means defiance of accepted thought. The antonym of this kind of paradox is "orthodox". For example: it was long considered axiomatic that a whole entity must be greater in every way than any proper part of the entity. But the theory of transfinite sets produced a paradox that shows that the even part of the whole numbers can be mapped into an exact one-to-one correspondence with the whole set of odd and even numbers. Of course, as sets, the subset of even numbers is still properly smaller than the set of all numbers. But in at least one respect, one-to-one association, the two are equivalent. Since the acceptance of this paradox, it has been understood that this is a defining character of the transfinite.

    2. Paradox means logical self-defeat. This is probably what you have in mind using the word 'paradox". A synonym that specializes this meaning is "antinomy", an outlaw conceptual entity. A famous example in the theory of transfinite sets in the set of all sets. logicians and mathematicians have learned strategies for dealing with these outlaws when they arise and avoiding their antinomous behavior. The set of all sets is a kind of class, but it is not allowed to be a member of another class.

    Not all antinomous paradoxes are from set theory. Many come from troubles in semantics. Here is an ancient example:

    A: Mr. X is bald. This statement should be clearly true or false.
    B: Agreed.
    A: First I present Mr. X with a full, lush head of hair growing out of his scalp. Is he bald?
    B: No.
    A: Now I cut one hair follicle off Mr. X's head. Is he now bald?
    B: No.
    A: Now I cut another hair off Mr. X's head. Is he now bald?
    B: No.
    ......
    A: Each time I cut one hair off Mr. X's head, it does not change the status of his baldness.
    B: Correct.
    A: Therefore, if I continue cutting his hair this way, he will never become bald.

    This is an example of vagueness at work.
     
  9. Jan 4, 2004 #8
    Re: Re: Re: Why paradoxes?

    But that's like saying that since language is supposed to describe the universe, anything that you can state using language must therefore exist within the universe.

    Logic is a set of rules that tell us how reason is supposed to work in the universe. That doesn't mean that just because we can describe something with logic that the "something" must manifest itself within the universe.


    Also, I don't know of any paradoxes of logic itself. The only paradoxes that I know of are ones where you assert something is true and then use logic and the assertion to derive a contradiction (a paradox). So the paradox occurs because of the assertion, rather than just logic itself.
     
  10. Jan 5, 2004 #9
    Re: Re: Re: Why paradoxes?

    Paradoxes certainly exist in the universe. It takes a bit of preperation, but any engineer or statistician is familiar with Simpson's Paradox (that link describes how a datatable can be created from legitimate study to demonstrate how those with no education in Physics perform better on Physics tests... the paradox is obvious, though the conditions which create the paradox tend to fool most people).

    Other paradoxes include the familiar Liar's Paradox: This sentence is false.

    Another paradox is the Birthday Paradox: If there are 23 people in the room, there is a 50/50 chance that two of them have the same birthday.

    Theirs no problem in the logic. Its important to keep in mind the definition of a paradox: A paradox is an apparently true statement that seems to lead to a logical self-contradiction, or to a situation that contradicts common intuition. Not all paradoxes imply a logical contradiction.

    On a note of logical contradiction and paradoxes, here is one paradox which does creates an apparent logical contradiction: Considering the concept of omniscience, you might ask "if something knows all there is to know, then that something must know something which it doesnt know". That statement creates a paradox. In the case of omniscience, the particular "know what you do not know" demonstrates the incoherency of omniscience. (Most other prefixes that start with the word "omni" to describe things - especially deities - create similiar contradictions.)

    (Here is a whole list of popular paradoxes: Wikipedia - Paradox)
     
    Last edited: Jan 5, 2004
  11. Jan 21, 2004 #10
    Re: Re: Re: Re: Why paradoxes?

    So, simply put, thats a statement for which you can not determine whether its true or false?
     
  12. May 29, 2004 #11
    Words only have demonstrable meaning according to their function in a given context. To the layperson, paradox can mean anything from a simple contradiction to an apparently insoluable puzzle. To the mathematician and logician, the word has a much more specific meaning. Hence, your assertion that paradoxes do not exist in the real world is meaningless mumbo-jumbo without a clear definition of what you mean by paradox.
     
  13. May 29, 2004 #12

    Tom Mattson

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    There is no inconsistency in the statements, "paradoxes exist in logic", "logic describes the real world", and "paradoxes exist in the real world". That is because the paradoxes of logic only arise when statements are self-referential or vague, not when they describe the observable universe.
     
  14. May 29, 2004 #13
    "
    This dog is mine.
    This dog had puppies.
    So, this dog is a mother.
    This dog is mine and this dog is a mother.
    Therefore, this dog is my mother.
    "
    :smile:
     
  15. May 30, 2004 #14

    Tom Mattson

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    :rofl: :rofl: :rofl:

    Very funny, but that's a fallacy, not a paradox.
     
  16. May 30, 2004 #15
    There is a possiblity that paradoxes could exist in our physical world. Many of these are predicted by physics theorems and models. For example, a branch of superstring theory suggests that if a spaceship were traveling at just the right velocity very close to a position where two superstrings(very dense 'strings' that cruise around the universe at high speeds) crossed each other, the intense gravity would pull the path of the spaceship into a tight enough loop that the acceleration would cause it to arrive before it left. Now here's the real question:

    If the spaceship arrives before it leaves, does it hit itself?

    If so, it would produce a physical paradox, as indeed many time-travel situations do. Scientists are divided on what such a paradox would 'do'. Some believe it would destroy the universe, but that's rather out of a melodramatic constitution than out of any real science in my opinion. A popular opinion which possibly finds its roots in sci-fi is that it would 'rip the fabric of time-space' but this positin is also scientifically unfounded. We simply don't know what would happen.

    Hope that helps,
    jeffceth
     
  17. May 30, 2004 #16

    Tom Mattson

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    But models are only that: models. They have no efficacy in the real world. And if a theoretical physicist were to come up with something that predicted a paradoxical effect, then the theory would most likely be modified or rejected on that basis. For instance noninteger values of the quantum number l are rejected as they lead to two different wavefunctions corresponding to the same quantum state (which would be paradoxical). Similarly, periodic boundary conditions are imposed on angular wavefunctions specifically to avoid multiply-valued wavefunctions (which would also be paradoxical).

    When we encounter a paradox in theoretical physics, I think it should be viewed as a giant neon sign that says, "You are making a mistake!"
     
  18. May 30, 2004 #17
    Or your context is too vague.

    For example, I could assert that life, the universe, and everything is ultimately paradoxical. However, such is the realm of philosophy rather than physics. The context is simply too broad and vague to have any demonstrable meaning in the context of physics.
     
  19. May 31, 2004 #18

    Les Sleeth

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    Yes, and when we come up against a paradox through reasoning, the same neon sign should be flashing. Or if one is reasoning correctly maybe the sign could say something like, "You are reasoning without all the information you need to resolve what merely appears to be a paradox."
     
  20. Jun 1, 2004 #19

    loseyourname

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    I don't see any way you could seriously assert that everything is paradoxical. I can name plenty of things that are not.
     
  21. Jun 1, 2004 #20
    Exactly my point, you can name plenty of things that are apparently not paradoxical and I could name plenty that are. For all either of us can prove life, the universe, and everything is ultimately ineffable. Without providing a specific context there is no way to meaningfully discuss the issue. Merely asserting that plenty of things are or are not apparently paradoxical proves nothing about the big picture.
     
    Last edited: Jun 1, 2004
  22. Jun 1, 2004 #21

    Tom Mattson

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    But the specific context was provided in the opening post. The comparison is being drawn between formal logic, which has inherent paradoxes, and the observable universe, in which the law of excluded middle seems to never be violated. The question is, given that, how can we say that logic corresponds in any way to the universe.

    I don't see what your point is.
     
  23. Jun 1, 2004 #22
    There are (at least) two types of paradox:

    1. consequences that seem to violate accepted presuppositions

    2. consequences that are antinomous (direct contradictions)

    .

    The answer to the 1 type might be to change those presuppositions (e.g. nature of infinite sets);
    the answer to the 2 type might be to acquire some kind of insurance (e.g. set of all sets);
    .

    (Logic, as a science, must watch its own back. Some limits and restrictions might be in order. Just exactly what needs to be done for safety may remain a longterm problem.)
     
  24. Jun 1, 2004 #23
    Ahhh, I see what you mean. Here is the original post again.

    This is a nonsensical statement by all the standards of logic. Paradoxes have no truth value whatsoever according to traditional formal logic. However, some types of formal logics, especially extentions of fuzzy logic, give them a truth value of "indeterminate" (note: much like QM.)

    In addition, you cannot prove or disprove a negative. You cannot prove paradoxes do not exist any more than you can prove angels do not dance on the heads of pins. Thus, by the standards of formal logic it is impossible to prove or disprove the existence of a paradox in the real world. All we can do is note what is apparently paradoxical, and look for resolutions.

    This is similar to the EPR thought experiment. Einstein, Podolsky, and Rosen could not prove paradoxes were impossible, but they could point out that QM is apparently inherently paradoxical and just how ridiculous that sounded. Likewise, I would add, it is equally ridiculous (paradoxical?) to insist everything must have some sort of rational explanation.

    For all these reasons, the statement has no meaningful context. Given a specific context, then something meaningful can be said on the subject.
     
    Last edited: Jun 1, 2004
  25. Jun 5, 2004 #24

    Tom Mattson

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    It's not nonsensical at all. Sikz made an observation about a property of logic which is well understood, and then made another observation that that same property is not evinced by any observational data. The question is then, "Is logic appropriate for describing the universe?"

    I think he's misunderstanding things, for the reason I stated, but the question is perfectly intelligible.

    But those fuzzy logics only describe quantum mechanical systems inbetween measurements. Make a measurement, and QM predicts no violation of the law of excluded middle. You will find that electron at precisely one location, not two.

    This was acknowledged in the first post.

    "Sounding ridiculous" is not the same as "being inherently paradoxical".

    What's this got to do with the question that Sikz asked?

    Well, the first post has a perfectly meaningful context, whether you recognize it or not. The idea that logical paradoxes exist, while physical paradoxes have not been observed, is well-defined enough for anyone to be able to understand it.
     
  26. Jun 6, 2004 #25
    Sorry, but this boy can not understand the statement. I will try to make myself as clear as possible.

    As far as I can tell, it is either too vague to be intelligable or downright paradoxical. Logic without paradox is like up without down, patently impossible. To then say that logic describes everything we observe, but we have never observed a paradox is a contradiction.

    There are two rudamentary types of paradoxes, the Heap paradox and the Liars paradox. The liars paradox is more substantiated by the rules of logic, while the heap paradox is more applicable to linguistics and concerns the vagueness of statements. Because we are talking about the "real" world, it is the heap paradox with it's emphasis on vagueness and linguistics that takes precident.

    Again, words only have demonstrable meaning according to their function in a given context. Natural language is repleat with vague terms while logistics are not. Mixing the two then requires a great deal of care in order to maintain coherency. :tongue2:
     
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