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A question on quantifiers

  1. May 26, 2015 #1
    1. The problem statement, all variables and given/known data

    This question may seem as an axiom to some. I also feel the same.
    Prove:
    ##\forall x\forall y\ p(x,y)\Leftrightarrow\forall y\forall x\ p(x,y)##


    3. The attempt at a solution

    ##Assume\ \forall x\forall y\ p(x,y)##
    ##Let\ x_0\in \mathbb{R}##
    ##\ \ \therefore \ \forall y \ p(x_0,y)##
    ##\ \ Let\ y_0\in \mathbb{R}##
    ##\ \ \ \ \therefore\ p(x_0,y_0)##

    From here onwards I'm stuck. Someone please help me to prove this (using only algebraic methods).
     
  2. jcsd
  3. May 26, 2015 #2

    WWGD

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    Do you have a rule for existential generalization to conclude first ## \forall x p(x,y_0) ## and then do the same for ##y_0##?
     
  4. May 26, 2015 #3
    I didn't get you....
    Do you mean this.....?

    ##Assume\ \forall x\forall y\ p(x,y)##
    ##Let\ y_0\in \mathbb{R}##
    ##\ \ \therefore \ \forall x \ p(x,y_{0})##
    ##\ \ Let\ x_0\in \mathbb{R}##
    ##\ \ \ \ \therefore\ p(x_0,y_0)##
    ##\ \ \therefore \forall x\ p(x,y_0)##
    ##\therefore \forall y\ \forall x\ p(x,y)##

    BTW: I don't know what existential generalization could do here. Universal specification and universal generalization are the only rules of inference I can think of.
    Thank You.
     
  5. May 31, 2015 #4
    Somebody tell me whether I'm right or wrong....
     
  6. Jun 1, 2015 #5

    WWGD

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    Could you please tell us the details of the rule of universal generalization that you use?
     
  7. Jun 1, 2015 #6
    universal generalization:if P(a) is true for all a in universe of discourse then we can say $$\forall x P(x)$$
     
  8. Jun 1, 2015 #7

    WWGD

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    Then it seems like from ##p(x_0,y_0)##, since each of ##x_0,y_0 ## is arbitrary, you could generalize to either ## \forall x p(x,y_0)## or ##\forall y p(x_0,y)## and then generalize again. I think that does it.
     
  9. Jun 1, 2015 #8
    Me too...
     
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