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Math Help - Formalizing Predicate Logic (English to PL)

  1. #1
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    Formalizing Predicate Logic (English to PL)

    I'm stumped on how to formalize these two sentences properly into predicate logic.

    1: There's a city to which all roads lead.
    2: All roads lead to a city.

    The question says nothing about defining a Universe of Discourse.

    I tried using

    R = is a road
    C = is a city
    L = leads to

    1: If there exists a city and then every road leads to this city.

    (∀y)(∃x)[(Cx & Ry) -> Lyx)]

    I'm unsure about my use of the existential quantifier, it seems to me that what I wrote translates more into "If there exists a city and each road then each road leads to this city." I'm very confused at this point because this doesn't seem to make sense to me at all.

    2: Everything that is a road leads to a city.

    (∀y)(Ry -> Lyx)

    With this one I started writing it and I know that x is unqualified and unquantified because I realized that if I were to add something like (∃x)(Cx) then this one would be identical to the first question. Is that right? I'm sure it is not since the first one says that "All roads lead to a specific city" whereas the second question says that "All roads lead to at least one city".
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  2. #2
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    Quote Originally Posted by Knark View Post
    1: There's a city to which all roads lead.
    2: All roads lead to a city.
    This is the way I see them.
    1) \left( {\exists x} \right)\left( {\forall y} \right)\left[ {C_x  \wedge \left( {R_y  \to L_{yx} } \right)} \right]

    2) \left( {\forall x} \right)\left( {\exists y} \right)\left[ {R_x  \to \left( {C_y  \wedge L_{xy} } \right)} \right]
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  3. #3
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    That makes a lot more sense to me than my own answer, thank you very much. Goes to show how big of a difference the placement of brackets makes.
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