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Phrasing mathematical statements

  1. Nov 7, 2013 #1
    I'm working on a project on Logic and as part of it I want to write a selection of sentences in mathematical/logical syntax. I've consulted a mathemagician for the the first sentence but need confirmation on the other five, would someone be able to tell me if my maths lingo is correct?

    What I have so far:

    1. For Frege, there should be nothing that could not be described in logical terms.
    (∄ x): ∄ DL(x)

    For example…

    2. Everybody loves Frege
    (∀ x) F(x, Gottlob)

    3. Everybody loves somebody
    (∀ x) (Ǝ y) F(x,y)

    4. There is somebody whom everybody loves
    (Ǝ y) (∀ x) F(x,y)

    5. There is somebody whom no one loves
    (Ǝ y) (∀ x) ¬F(x,y)

    6. And there is somebody whom Frege does not love
    (Ǝ x) ¬F(Gottlob,x)

    With 4 and 5 I’m not sure about the order of Ǝ and ∀ – the way it is now implies that the “somebody” is the same for each person. We're going to illustrate this with an animation in which we’ll have a group of people all loving their mums. Now that’s fine for the concept “their mum”, but the sentence implies that there is a single somebody and we don’t all have the same mum.

    Also, I’d be tempted to put in colons after the Ǝ signs, if only for grammatical reasons. Any comments much appreciated!

    Thanks,
    Hari
     
  2. jcsd
  3. Nov 7, 2013 #2

    mfb

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    Staff: Mentor

    Please use our homework section for homework-like problems (even if they are not homework), I moved your thread.

    No, that existing somebody is loved by all persons // is [not loved] by all persons.

    That is a completely different problem.

    That's exactly what you have to (and did) express.

    The formulas are right if F(x,y) means "x loves y", I just don't understand the difference between "Frege" and "Gottlob".

    In (1), I would expect a "not" symbol instead of the second "does not exist".
     
  4. Nov 7, 2013 #3
    Sorry for posting in the wrong section and thank you for your comments!

    I should've clarified my function, yes F(x,y) means "x loves y" and Gottlob Frege is one and the same person, I should also use the same name in the sentence and the expression.

    If I wanted to write "there is somebody whom everybody loves" and that somebody is different for each x then is it essentially the same statement as 3: "everybody loves somebody" and so there's no progression in thought?

    I'll have to adjust the animation...

    So instead of "there does not exist an x for which there does not exist a logical description of x" it's "there does not exist x for which a description of x is not logical".
     
  5. Nov 7, 2013 #4

    mfb

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    There are multiple statements that look similar, but are not:

    Everybody loves someone = Everbody loves at least one other person, this does not have to be the same for all = There is no person that loves nobody
    There is somebody whom everybody loves = There is at least one single person that is loved by all
    All love their mom = For every person, there is a specific other person they love. (In general, the moms of different persons will be different.)

    I interpreted DL(x) as "describable in logic terms".
     
  6. Nov 7, 2013 #5
    Could I make the y in number 4. x-dependent?

    There is somebody whom everybody loves, where yx∈{x's mum}

    (Ǝ yx) (∀ x) F(x,yx)
     
  7. Nov 7, 2013 #6

    mfb

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    Staff: Mentor

    Then you get (3).

    The index x of y is meaningless before you introduce x. Swap the order of those, and it is fine.

    A set of moms looks strange. I would write yx = x's mum. Then you can just write (∀ x) F(x,yx)
     
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