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Math Help - Logical Implication

  1. #1
    Junior Member
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    Logical Implication

    Let \theta be \forall{x}\forall{y}\exists{z}((R(x,z)\wedge {R(y,z)})\wedge \forall{w}((R(x,w)\wedge {R(y,w)}) \rightarrow R(z,w))) and let \phi be \forall{x}\forall{y}\exists{z}((R(z,x)\wedge {R(z,y)})\wedge \forall{w}((R(w,x)\wedge {R(w,y)}) \rightarrow R(w,z)))

    Where R is a binary predicate symbol.

    I have been asked to find an L structure consisting of 3 elements such that \theta \not \models \phi

    My Professor hasn't really given us a method to do this, any help would be appreciated.
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  2. #2
    Senior Member
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    Hi

    If I understand, you want a 3 elements structure where \theta is true but \phi isn't. In these sentences, precising a binary relation symbol properties, you can try to see R as an order relation (perhaps the most common binary predicates are order or equivalence relations).
    In such context, \theta says that for any x,y, there is a \sup(x,y) (for R), and \phi that for any x,y, there is a \inf(x,y).

    Then I guess you can quite easily define a partial order on three elements such that any two of them have a sup, but there are two with no inf (and in fact more: with no lower bound).
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