1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Propositional Logic - There Exists dual

  1. Dec 3, 2007 #1
    Propositional Logic - "There Exists" dual

    1. The problem statement, all variables and given/known data
    "State and prove the [itex]\exists[/itex]-dual of 6.1.8"

    Section 6.1.8 shows how to change [itex]\vdash (\forall x)(\forall y)A \equiv (\forall y)(\forall x)A[/itex] via this method:

    (Line 1) [itex](\forall x)(\forall y)A[/itex] (Hypothesis)
    (Line 2) [itex](\forall y)A[/itex] (Line 1 + spec)
    (Line 3) [itex]A[/itex] (Line 2 + spec)
    (Line 4) [itex](\forall x)A[/itex] (Line 3 + gen; Line 1 has no free x)
    (Line 5) [itex](\forall y)(\forall x)A[/itex] (Line 4 + gen; Line 1 has no free y)

    The [itex]\exists[/itex]-dual of this would be to show [itex]\vdash (\exists x)(\exists y)A \equiv (\exists y)(\exists x)A[/itex].

    2. Relevant equations
    A crucial formula here is that [itex](\forall x)A \equiv \neg (\exists x) \neg A[/itex]


    3. The attempt at a solution
    I potentially solved it, but something seems off.

    This is what I did:

    [itex]\vdash (\exists x)(\exists y)A \equiv (\exists y)(\exists x)A[/itex]

    (Line 1) [itex](\exists x)(\exists y)A[/itex]
    (Line 2) [itex]\neg (\forall x)\neg (\exists y)A[/itex] Definition of [itex]\exists[/itex]
    (Line 3) [itex]\neg (\forall x)\neg \neg (\forall y)\neg A[/itex] Definition of [itex]\exists[/itex]
    (Line 4) [itex]\neg (\forall x)(\forall y)\neg A[/itex] Double negative
    (Line 5) [itex]\neg (\forall y)\neg A[/itex] Spec
    (Line 6) [itex]\neg \neg A[/itex] Spec
    (Line 7) [itex]\neg (\forall x)\neg A[/itex] Gen
    (Line 8) [itex]\neg (\forall y)(\forall x)\neg A[/itex] Gen
    (Line 9) [itex]\neg (\forall y)\neg \neg (\forall x)\neg A[/itex] Double negative
    (Line 10) [itex]\neg (\forall y)\neg (\exists x)A[/itex] Definition of [itex]\forall[/itex]
    (Line 11) [itex](\exists y)(\exists x)A[/itex] Definition of [itex]\forall[/itex]

    What worries me is going from Line 5 to 6, what happens to the first [itex]\neg[/itex] symbol? Can I just leave it there, and then insert my [itex](\forall x)[/itex] between the two [itex]\neg[/itex]s in Line 7 like that?
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?