Phrasing mathematical statements

[hbails]

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

 

[mfb]

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

the way it is now implies that the “somebody” is the same for each person.

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

We're going to illustrate this with an animation in which we’ll have a group of people all loving their mums.

That is a completely different problem.

but the sentence implies that there is a single somebody

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".

 

[hbails]

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.

That is a completely different problem.

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...

In (1), I would expect a "not" symbol instead of the second "does not exist".

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".

 

[mfb]

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?

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.)

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".

I interpreted DL(x) as "describable in logic terms".

 

[hbails]

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.)

Could I make the y in number 4. x-dependent?

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

(Ǝ y_x) (∀ x) F(x,y_x)

 

[mfb]

Could I make the y in number 4. x-dependent?

Then you get (3).

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

(Ǝ y_x) (∀ x) F(x,y_x)

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 y_x = x's mum. Then you can just write (∀ x) F(x,y_x)