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Re: Quantifying over functions in first-order logic

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  • cedricstjl
    Hmmm, I ve got a hard time translating your example, or those from the Wikipedia article into logical sentences (I have no idea how to express that a model is
    Message 1 of 9 , Jun 15 8:53 AM
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      Hmmm, I've got a hard time translating your example, or those from the
      Wikipedia article into logical sentences (I have no idea how to
      express that a model is finite, for instance). I guess I should pick
      up a book on the subject; my only background in logic is from AIMA and
      Hofstadter's book.

      Thanks for the info!

      Cedric

      --- In aima-talk@yahoogroups.com, Joe Hendrix <jhendrix@...> wrote:
      >
      > > On 6/8/06, cedricstjl <cedricstjl@> wrote:
      > > couldn't it be achieved equivalently by reifying functions and
      > > predicates?
      >
      > This is a useful technique. I've also seen it called currifying in
      > other papers, however it doesn't let you achieve the full power of
      > second-order logic, much less higher-order logic.
      >
      > A basic first-order logic result is that finiteness is not
      > expressible. That is there is no sentence P such that M is a model of
      > P iff. M is finite. However, in second order logic one can express
      > finiteness. Specifically, if P is the statement that all injective
      > functions are surjective, then M is a model of P iff. M is finite.
      >
      > The wikipedia article on second-order logic discusses some of the
      > differences between first-order and second-order logic at more length
      > than this email.
      >
      > Joe
      >
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