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Re: Quantifying over functions in first-order logic

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  • cedricstjl
    I don t see why one would ever want to quantify over call_function. I don t think that there is an equivalent to that in regular logic. I can quantify over
    Message 1 of 9 , Jun 14, 2006
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      I don't see why one would ever want to quantify over call_function. I
      don't think that there is an equivalent to that in "regular" logic.

      I can quantify over reflexive by adding another level of indirection.
      reflexive(f) becomes call_function_2(reflexive, f). As long as all
      functions have a clearly defined level, it should work. Furthermore,
      using levels is, from what I've read, a common way of avoiding
      Russell's paradox in SOL. Does the "SOL is strictly more expressive
      than FOL" claim apply only to SOL without levels?

      Cedric

      --- In aima-talk@yahoogroups.com, "Peter Norvig" <peter@...> wrote:
      >
      > I agree that for most of what you would reasonably want to do, it
      > wouldn't matter. But the point is that when you quantify over all
      > functions/relations (represented as objects), you wouldn't be
      > quantifying over "call_function" nor "reflexive".
      >
      > -Peter
      >
      > On 6/13/06, cedricstjl <cedricstjl@...> wrote:
      > > Why does it matter? I'm quantifying over "objects that behave exactly
      > > like FOL functions" instead of quantifying directly over FOL
      functions.
      > >
      > > I could write a program, which takes as input a second-order logic
      > > sentence, does inference in FOL with the conversion from my first
      > > post, and translates back into second-order logic. Or are there SOL
      > > sentences that wouldn't be translatable?
      > >
      > > Thank you for your time,
      > >
      > > Cedric
      > >
      > > --- In aima-talk@yahoogroups.com, "Peter Norvig" <peter@> wrote:
      > > >
      > > > You could do that, and that could be a way of expressing
      reflexivity,
      > > > but it wouldn't be quantifying over FOL functions -- it would be
      > > > quantifying over objects that you happen to call "functions" but are
      > > > regular objects as far as the logic is concerned.
      > > >
      > > > -Peter
      > > >
      > > > On 6/8/06, cedricstjl <cedricstjl@> wrote:
      > > > > I have some difficulty seeing the difference in expressiveness
      between
      > > > > first-order logic and higher-order logics. Wikipedia (as well
      as AIMA,
      > > > > AFAICT) says that F-O logic cannot quantify over functions. But
      > > > > couldn't it be achieved equivalently by reifying functions and
      > > > > predicates? I.e.:
      > > > >
      > > > > x = fun(var)
      > > > >
      > > > > becomes
      > > > >
      > > > > x = call_function(fun, var)
      > > > >
      > > > > and then one can express reflexivity:
      > > > >
      > > > > for all fun, a, b: reflexive(fun) <=> call_function(fun, a, b) =
      > > > > call_function(fun, b, a)
      > > > >
      > > > > Cedric
    • cedricstjl
      Actually, I think that I don t even need levels. for all fun, a, b: call_function(reflexive, fun) call_function(fun, a, b) = call_function(fun, b, a) seems
      Message 2 of 9 , Jun 14, 2006
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        Actually, I think that I don't even need levels.

        for all fun, a, b: call_function(reflexive, fun) <=>
        call_function(fun, a, b) = call_function(fun, b, a)

        seems to work fine to me.

        Cedric

        --- In aima-talk@yahoogroups.com, "cedricstjl" <cedricstjl@...> wrote:
        >
        > I don't see why one would ever want to quantify over call_function. I
        > don't think that there is an equivalent to that in "regular" logic.
        >
        > I can quantify over reflexive by adding another level of indirection.
        > reflexive(f) becomes call_function_2(reflexive, f). As long as all
        > functions have a clearly defined level, it should work. Furthermore,
        > using levels is, from what I've read, a common way of avoiding
        > Russell's paradox in SOL. Does the "SOL is strictly more expressive
        > than FOL" claim apply only to SOL without levels?
        >
        > Cedric
        >
        > --- In aima-talk@yahoogroups.com, "Peter Norvig" <peter@> wrote:
        > >
        > > I agree that for most of what you would reasonably want to do, it
        > > wouldn't matter. But the point is that when you quantify over all
        > > functions/relations (represented as objects), you wouldn't be
        > > quantifying over "call_function" nor "reflexive".
        > >
        > > -Peter
        > >
        > > On 6/13/06, cedricstjl <cedricstjl@> wrote:
        > > > Why does it matter? I'm quantifying over "objects that behave
        exactly
        > > > like FOL functions" instead of quantifying directly over FOL
        > functions.
        > > >
        > > > I could write a program, which takes as input a second-order logic
        > > > sentence, does inference in FOL with the conversion from my first
        > > > post, and translates back into second-order logic. Or are there SOL
        > > > sentences that wouldn't be translatable?
        > > >
        > > > Thank you for your time,
        > > >
        > > > Cedric
        > > >
        > > > --- In aima-talk@yahoogroups.com, "Peter Norvig" <peter@> wrote:
        > > > >
        > > > > You could do that, and that could be a way of expressing
        > reflexivity,
        > > > > but it wouldn't be quantifying over FOL functions -- it would be
        > > > > quantifying over objects that you happen to call "functions"
        but are
        > > > > regular objects as far as the logic is concerned.
        > > > >
        > > > > -Peter
        > > > >
        > > > > On 6/8/06, cedricstjl <cedricstjl@> wrote:
        > > > > > I have some difficulty seeing the difference in expressiveness
        > between
        > > > > > first-order logic and higher-order logics. Wikipedia (as well
        > as AIMA,
        > > > > > AFAICT) says that F-O logic cannot quantify over functions. But
        > > > > > couldn't it be achieved equivalently by reifying functions and
        > > > > > predicates? I.e.:
        > > > > >
        > > > > > x = fun(var)
        > > > > >
        > > > > > becomes
        > > > > >
        > > > > > x = call_function(fun, var)
        > > > > >
        > > > > > and then one can express reflexivity:
        > > > > >
        > > > > > for all fun, a, b: reflexive(fun) <=> call_function(fun, a, b) =
        > > > > > call_function(fun, b, a)
        > > > > >
        > > > > > Cedric
        >
      • Joe Hendrix
        ... 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,
        Message 3 of 9 , Jun 14, 2006
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          > 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
        • 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 4 of 9 , Jun 15, 2006
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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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