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Math Help - This logic is nonsense

  1. #1
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    This logic is nonsense

    I got this from an MIT lecture note:

    "If pigs fly, then you can understand the Chernoff bound," which is P \Rightarrow Q

    If P is false, and Q is true, the logical implication must be true.

    In my case: Whether pigs can fly or not, I cannot understand the Chernoff bound, but

    in Mr. Chernoff's case: Whether pigs can fly or not, he of course can understand his own bound.

    There must be a better example than this. Does anyone have one?
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  2. #2
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    Quote Originally Posted by novice View Post
    I got this from an MIT lecture note:

    "If pigs fly, then you can understand the Chernoff bound," which is P \Rightarrow Q

    If P is false, and Q is true, the logical implication must be true.

    In my case: Whether pigs can fly or not, I cannot understand the Chernoff bound, but

    in Mr. Chernoff's case: Whether pigs can fly or not, he of course can understand his own bound.

    There must be a better example than this. Does anyone have one?

    I think you're confusing here logical validity with true, or actual, facts: nobody knows and nobody cares whether you can understand Chernoff's proof or whether pigs can fly, but if you can understand that proof (i.e., if Q is T), then it doesn't matter whether pigs can or can't fly (i.e., whether P or ~P is T), the implications P --> Q , ~P --> Q are both valid (i.e., they get both a T value).
    The first question here could perhaps be: what have to do pigs, either flying or not, with understanding some proof? In actual life the answer seems to be clear: nothing. That though wasn't the point, but what truth values are you assigning to the different premises, and whether the logical outcome is valid or not.

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    I think you're confusing here logical validity with true, or actual, facts: nobody knows and nobody cares whether you can understand Chernoff's proof or whether pigs can fly, but if you can understand that proof (i.e., if Q is T), then it doesn't matter whether pigs can or can't fly (i.e., whether P or ~P is T), the implications P --> Q , ~P --> Q are both valid (i.e., they get both a T value).
    The first question here could perhaps be: what have to do pigs, either flying or not, with understanding some proof? In actual life the answer seems to be clear: nothing. That though wasn't the point, but what truth values are you assigning to the different premises, and whether the logical outcome is valid or not.

    Tonio
    Thanks Tonio,
    I see now. You have answered my question better than anyone else have.
    Yah, you are right: When we are in the ivory tower, what do we care about reality. Swim with the flow of current is the way to go.

    At my school the pigs fly. No kidding. Yes, honest.
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  4. #4
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    Tonio,

    I just had a good chat with a graduate student in computer engineering at my chow hall at dinner. I told him what you told me. He laughed at me as if I was an idiot, then kindly he explained it to me.

    Since you are a nice fellow, I thought I ought to share this with you.

    Here is what he said:

    The reason that truth table in formal logic being nonsensical is that it is a computer logic. That's how the logic gates in the integrated circuit work. It's not a reasoning logic, but useful for controlling on-off witches in the IC.

    Cheers.
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  5. #5
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    Quote Originally Posted by novice View Post
    Tonio,

    I just had a good chat with a graduate student in computer engineering at my chow hall at dinner. I told him what you told me. He laughed at me as if I was an idiot, then kindly he explained it to me.

    Since you are a nice fellow, I thought I ought to share this with you.

    Here is what he said:

    The reason that truth table in formal logic being nonsensical is that it is a computer logic. That's how the logic gates in the integrated circuit work. It's not a reasoning logic, but useful for controlling on-off witches in the IC.

    Cheers.

    Computer logic my buttox: these rules are on looooong before anything ressembling an electronic computer (leaver machines and abacos aside) were even a dream.
    And truth tables aren't nonsensical: they are what they are, namely formal human inventions which, in this case, are based on informal, previous ways of thinking and deducing stuff, and they work pretty fine even if some people find them a little/some/a lot anti-intuitive.

    But no wonder you got that laugh from an engineer: engineers are sore because they know bananas about mathematics and, of course, about mathematical logic, and they cannot offer anything better that empty laughs.

    Tonio
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  6. #6
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    I agree with Tonio. Implication defined by the truth table in the discrete mathematics course, which is often called "material implication", is the same implication that is used to formulate mathematical theorems of the form "Suppose that ... Then ..." for the last couple of thousands years. For example, it is natural to think that if a natural number n is divisible by four, then it is divisible by two. However, what if n = 3 or n = 2? We don't want the general fact to become false because of this.

    Properly speaking, in mathematics one usually does not ask why something is defined in a certain way, as long as the definition is useful in practice and/or gives rise to a rich and interesting theory. Mathematics starts when the definitions are given and one tries to find and prove their properties. The discussion of definitions is a part of philosophy, and there is plenty of discussions about implication: see, for example, Wikipedia and the Stanford Encyclopedia of Philosophy.

    Speaking about computer hardware: is implication really implemented as a primitive operations, or are negation, conjunction and disjunction (and possibly exclusive disjunction) primitive? If so, then what you call (not x) \/ y is not dictated by how computers work, is it?
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  7. #7
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    Well, emakarov,
    I cannot disagree with you or Tonio. I know if I tell the engineers what Tonio said, they will turn blue and not speak to me again. I just got my feet wet in math, and have yet to discover whether I swim or sink in math and logic.

    I enjoy listening to you and Tonio and many other good mathematicians here.
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  8. #8
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    Quote Originally Posted by tonio View Post
    Computer logic my buttox: these rules are on looooong before anything ressembling an electronic computer (leaver machines and abacos aside) were even a dream.
    And truth tables aren't nonsensical: they are what they are, namely formal human inventions which, in this case, are based on informal, previous ways of thinking and deducing stuff, and they work pretty fine even if some people find them a little/some/a lot anti-intuitive.

    But no wonder you got that laugh from an engineer: engineers are sore because they know bananas about mathematics and, of course, about mathematical logic, and they cannot offer anything better that empty laughs.

    Tonio
    Glad you have sense of humor. Thanks for the info. Next time the sob laugh at me, he will get strafed with the ammo you gave me, or at least he be told he knows banana about math.
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  9. #9
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by tonio View Post
    But no wonder you got that laugh from an engineer: engineers are sore because they know bananas about mathematics and, of course, about mathematical logic, and they cannot offer anything better that empty laughs.
    Haha!

    not to put engineers down, i'm sure there are some really smart ones out there, and yes, even ones that are good at math. but this is certainly true of any engineer i've ever met!

    Quote Originally Posted by novice View Post
    I got this from an MIT lecture note:

    "If pigs fly, then you can understand the Chernoff bound," which is P \Rightarrow Q

    If P is false, and Q is true, the logical implication must be true.

    In my case: Whether pigs can fly or not, I cannot understand the Chernoff bound, but

    in Mr. Chernoff's case: Whether pigs can fly or not, he of course can understand his own bound.

    There must be a better example than this. Does anyone have one?
    haha, when i was first learning this, this entry in the truth table bothered me too. now i agree with emakarov, when it comes to definitions, it is not the "why?" that is important, but the "is this useful?" that is. we do proofs to handle the "why"'s.

    anyway, i believe there is some reason to this odd definition, here is how i reconciled it to myself all those years ago.

    I make the statement, "If pigs fly, then you can understand Chernoff's bound."

    The statement "pigs fly" is obviously false (except at your school ). Now if you come to me and tell me, "I don't understand the bound". I can respond to you, "did I lie? I said if pigs fly, you would understand it, pigs don't fly, so of course you don't understand it!" if you come to me and say, "i do understand the bound!" i can respond, "did i lie? you simply understood it despite the fact pigs don't fly. i never said pigs flying was the only way to understand it. (that would be the bi-implication, which would definitely be false here)".

    So you see? if the first statement is false, then whether or not you understand the bound, i did NOT lie. That is, i did not make a false statement. Well, there are only two options here, true or false, if i didn't make a false one, i must have made a true one.

    P => Q means that if P happens, then Q will happen. The only way for this statement to be false, is if i lied and P happens but Q didn't. otherwise, it must be true.

    ...or do i know bananas about logic?
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    Quote Originally Posted by Jhevon View Post
    Haha!

    not to put engineers down, i'm sure there are some really smart ones out there, and yes, even ones that are good at math. but this is certainly true of any engineer i've ever met!

    haha, when i was first learning this, this entry in the truth table bothered me too. now i agree with emakarov, when it comes to definitions, it is not the "why?" that is important, but the "is this useful?" that is. we do proofs to handle the "why"'s.

    anyway, i believe there is some reason to this odd definition, here is how i reconciled it to myself all those years ago.

    I make the statement, "If pigs fly, then you can understand Chernoff's bound."

    The statement "pigs fly" is obviously false (except at your school ). Now if you come to me and tell me, "I don't understand the bound". I can respond to you, "did I lie? I said if pigs fly, you would understand it, pigs don't fly, so of course you don't understand it!" if you come to me and say, "i do understand the bound!" i can respond, "did i lie? you simply understood it despite the fact pigs don't fly. i never said pigs flying was the only way to understand it. (that would be the bi-implication, which would definitely be false here)".

    So you see? if the first statement is false, then whether or not you understand the bound, i did NOT lie. That is, i did not make a false statement. Well, there are only two options here, true or false, if i didn't make a false one, i must have made a true one.

    P => Q means that if P happens, then Q will happen. The only way for this statement to be false, is if i lied and P happens but Q didn't. otherwise, it must be true.

    ...or do i know bananas about logic?
    Oh, that was beautiful!

    Believe it or not, you do sound like you know all the bananas about logic.

    Oh, boy, aren't mathematicians a bunch stoic philosophers?
    Contrary! Contrary!
    They are frisky kittens.

    Finally, satisfied. If I may be one of you, I will meow on my way out.

    Lights off.
    Thread closed.
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  11. #11
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by novice View Post
    Oh, that was beautiful!

    Believe it or not, you do sound like you know all the bananas about logic.

    Oh, boy, aren't mathematicians a bunch stoic philosophers?
    Contrary! Contrary!
    They are frisky kittens.

    Finally, satisfied. If I may be one of you, I will meow on my way out.

    Lights off.
    Thread closed.
    haha, maybe you're a kitten. i'm a lion! like the lion king (mufasa, not simba)! roar on your way out please.

    i'll leave the thread open in case any one else wants to chime in. should be interesting to see the way people reconcile this to themselves
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  12. #12
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    Quote Originally Posted by Jhevon View Post
    haha, maybe you're a kitten. i'm a lion! like the lion king (mufasa, not simba)! roar on your way out please.

    i'll leave the thread open in case any one else wants to chime in. should be interesting to see the way people reconcile this to themselves
    Can hold myself. My side hurt from laughing.
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  13. #13
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    Quote Originally Posted by tonio View Post
    Computer logic my buttox: these rules are on looooong before anything ressembling an electronic computer (leaver machines and abacos aside) were even a dream.
    And truth tables aren't nonsensical: they are what they are, namely formal human inventions which, in this case, are based on informal, previous ways of thinking and deducing stuff, and they work pretty fine even if some people find them a little/some/a lot anti-intuitive.

    But no wonder you got that laugh from an engineer: engineers are sore because they know bananas about mathematics and, of course, about mathematical logic, and they cannot offer anything better that empty laughs.

    Tonio
    Whole heartedly agree. I'm an Engineer, but I've taken up studying Mathematics at night. I'm hoping to obtain an honours degree in it. Started 2 years ago, 4 more years to go : I'll be 39 by the time I've finished it. I think I missed my true calling in life. Ah well.

    Thanks for that explanation. That if/then true/false thing had me baffled. My first thought was "mathematicians are nuts." Makes sense now though. Cheers.
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  14. #14
    Super Member Matt Westwood's Avatar
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    "If pigs fly, then you can understand the Chernoff bound."

    Said in a sarcastic / ironic tone of voice, this can be considered the same construction as:

    "If England win a major sporting championship this year, then I'm a little German cake."

    The idea being: the probability of pigs flying is so remote (by implication zero), then it's a safe bet to assume that something less unlikely (i.e. you understanding something as abstruse as the Chernoff bound).

    Except the former happens all the time around here, they invented the "police helicopter".
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  15. #15
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    Hi guys,

    Just to continue this discussion about if/then:

    I've managed to confuse myself again, but I was thinking about the statement (1) " if it rains, then there are clouds."

    If the premise is true and the conclusion is false, then the implication is false - If it's raining, there must be clouds.
    If the premise is false and the conclusion is true, then the implication is true - it may be cloudy, but that does not mean that it is raining.
    If the premise is false and the conclusion is false, then the implication is true - No rain, no clouds.

    Makes sense.

    Then I thought about it another way.

    (2) "If there are clouds, then it is raining. "

    what I'm confused about is this: Lets say the premise in (2) is false, but the conclusion is true. According to the truth table, the implication is true. However, as far as I know, there can't be rain without clouds.

    I know I'm missing something. Anyone mind clarifying this for me ? Does it just mean that the assumption - or premise - is incorrect ? That is, if it's raining, then there must be clouds, and anyone who says there are no clouds is lying ?

    Cheers.
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