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Anomaly set theoretic equivalent to material implication

  1. May 16, 2015 #1
    OK, this is embarrassing, but I never looked carefully at this elementary point. We say that if
    p implies q
    P is the set of all things for which p is true
    Q is the set of all things for which q is true
    then Q ⊆ P.
    Also that the set of all things for which p&q is true equals P∩Q
    But p & q implies p, so (from the above) P ⊆ P ∩ Q, which is in general false.
    What is wrong?
    Thanks
     
  2. jcsd
  3. May 16, 2015 #2

    micromass

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    Are you sure that shouldn't be ##P\subseteq Q##?
     
  4. May 16, 2015 #3
    that was also my reaction, micromass, but I read a number of sources which stated it as I have put it, including HallsofIvy on this forum: https://www.physicsforums.com/threads/set-theory-representation-of-material-implication.206811/ :
    /If P is the set of all things for which statement p is true and Q the set of all things for which statement q is true, then "If p then q" can be represented as "Q is a subset of P"./
    This is apparently the reason that the symbol for material implication is often the (backward) subset sign: "p implies q" is often written as p⊃q, as you can see in http://en.wikipedia.org/wiki/Material_conditional and other sources. (In fact, this usage is what got me started on this question.)
    So .....????
     
  5. May 16, 2015 #4

    micromass

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    Let p = divisible by 4, let q = divisible by 2. Then ##P = 4\mathbb{Z}## and ##Q = 2\mathbb{2}##. We have ##p\rightarrow q## and we have ##P\subseteq Q##. So I don't see how it could be differently.

    The answer of Halls in that thread is wrong on another level to. It is right that the set theoretic version of ##p \wedge q## is ##A\cap B## and so on. But the set theoretic version of a operation should be a set. So it is false that the set theoretic analog of ##p\rightarrow q## is ##A\subseteq B## (or ##B\subseteq A##), since that is not a set. The set theoretic analog is rather equal to ##A^c \cup B## (in classical logic, the theory becomes much more complicated and more beautiful in other logics).

    The reason why we sometimes use ##p\subset q## for ##q\rightarrow p## is an unfortunate historical coincidence, see http://math.stackexchange.com/quest...the-symbol-supset-when-it-means-implication-a
     
  6. May 16, 2015 #5
    Thanks very much, micromass. Whew, that's a relief; my initial doubt was like suddenly thinking that maybe 1+1 isn't really 10. This historical accident sounds a bit like the unfortunate convention in physics that switched positive and negative in current. Also thanks for pointing the very good and easily overlooked point about sets versus proper classes.
     
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