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Math Help - Predicate logic and validity

  1. #1
    Junior Member
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    Predicate logic and validity

    Hi,

    I have the following task:

    "Show that Pa V Pb -> Ex Px" where E stands for the existential quantifier.

    I have done the following:
    - Let M denote a model with domain D, and assume that M |= Pa V Pb
    - It suffices then to show that M |= Ex Px
    - Let s be an element in D, arbitrarily chosen
    - By assumption, we know that M |= Ps (since we consider disjunction, it suffices to only include one, from Pa V Pb)
    - Thus, it is the case that M |= Ex Ps
    - Since s was arbitrarily chosen, it will be so that M |= Ex Px, that Ex Px is true in M.

    Is this the right way to prove validity?
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  2. #2
    MHF Contributor
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    Re: Predicate logic and validity

    No, that's not right. For example, consider natural numbers with zero and let Px mean x = 0. Then P0 V P1, but it certainly does not mean that Ps is true for an arbitrary s.

    You just need to consider two cases depending on whether Pa or Pb is true in M.
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