for all and for every , how to avoid the subtle contradiction on using qualifiers?

[Philip Lee]

For these two statements,
(1). There exists a positive real number ε/2 smaller than every positive real number ε.
(2). There is no positive real number smaller than every positive real number ε.

Questions :
Both statements seems right , but both also seems contradicting with each other , what should have in mind in writing to avoid such subtle contradictory representations ? It would be much better to recommend some reading materials on this topic, thanks!

[Zelos Malum]

1 is obviously false if the statement isn't "for every positive real number ε, ε>ε/2

[John Gabriel]

Nonsense. It's obviously, _true_ except that it doesn't state *every* real number.

As usual your lack of common sense logic and ineptitude at math is shocking.

[John Gabriel]

On Thursday, 19 October 2017 05:17:02 UTC-4, Philip Lee wrote:
> For these two statements,
> (1). There exists a positive real number ε/2 smaller than every positive real number ε.
> (2). There is no positive real number smaller than every positive real number ε.

>
> Questions :
> Both statements seems right , but both also seems contradicting with each other,

They do NOT contradict each other.

[burs...@gmail.com]

Or (1) true and (2) is true.

(1) could be also simply:
forall ε>0 (ε/2 < ε)
(2) could be simply:
~exists δ>0 forall ε>0 (δ<ε)

[John Gabriel]

On Thursday, 19 October 2017 09:55:14 UTC-4, burs...@gmail.com wrote:
> Or (1) true and (2) is true.

Nope. They are both CORRECT. There is NO "or" and it has NOTHING to do with TRUE or FALSE stupid!!!!

>
> (1) could be also simply:
> forall ε>0 (ε/2 < ε)
> (2) could be simply:
> ~exists δ>0 forall ε>0 (δ<ε)

Natural language works just fine my little twerp.

[Archimedes Plutonium]

Well, when you have the correct understanding of finite versus infinite, coupled with the correct understanding of what numbers are-- Grid Numbers. Then all this nonsense stuff is kindergarten stuff.

So, let us walk through your kindergarten nonsense using just 10 Grid numbers.

(1). There exists a positive real number ε/2 smaller than every positive real number ε.

In 10 Grid take for example .1, is there a .05 in 10 Grid, no, so your first claim is false

Now examine your second kindergarten nonsense

(2). There is no positive real number smaller than every positive real number ε.

Again, taking the 10 Grid or perhaps any other Grid, take the 10^20 Grid. They all start out with 0. So, 0 is the smallest, and smaller than any other Grid number except for 0 itself.

You see, when you lack the proper and correct understanding of what finite and infinite border is, and you lack the understanding of what Numbers truly are, then everyday, you can come up with puzzling brain teasers that you are never able to answer.

But, if you have the true infinity concept nailed by a borderline and have the true Numbers that exist, every dolt with his "brain teaser for the day" is a vanquished dolt.

AP

[John Gabriel]

True crank you are.

[burs...@gmail.com]

Well here is a proof that they are true in the reals.
(only showing (1) to give you an idea)

In the reals we have :

a < b => a+c < b+c

So from:

0 < ε

We get:

ε < 2*ε

In the reals we have:

0 < c & a < b => a/c < b/c

So we get:

ε/2 < ε

We can discharge the assumption and variable so we get:

forall ε (0 < ε => ε/2 < ε)

Since the above is a CONSEQUENCE of the axioms and
theorems about reals like models, the above is
also TRUE in our intended reals model.

Discharge is a movement, for example found
in natural deduction, see also here:

Natural Deduction Rules, Calgary Remix
of forall x intro logic text by P. D. Magnus
http://openlogicproject.org/2017/10/18/natural-deduction-rules-in-forall-x-calgary/

[burs...@gmail.com]

Of course this truth crumbles, if the axioms
are not true. But lets not be such a pessimist.

LoL

Anyway here is a playable logic:
http://proofs.openlogicproject.org/

[Mike Terry]

The problem here is with the order of quantifiers. (Quantifiers being
"for every/all" and "there exists".)

Spelling it out, for (1) it would be clearest and I think most natural
to say:
(1) For every positive real number e, there exists a real number
smaller than e.

The fact that e/2 is such a real number provides a proof of (1), and I
would prefer to see this separately from the statement (1) itself.
After all, there are many other such numbers, and the choice of e/2 is
nothing special, and does not need to be highlighted as you have done.

However, the main point here is that in (1) the intention is we FIRST
choose the number e, and THEN (once e is fixed) get to choose another
number which is smaller than e, e.g. e/2. This could also be argued out
rather verbosely as:
"Let e be any positive real number. Then we see that e/2 is also a
positive real number, and e/2 < e. Hence for every positive real
number, there exists another positive real number smaller than that number."

OK, that's a bit long winded, but the point is at least it makes clear
the order of choices to be made: first choose e, THEN (based on this
choice) we get to choose a smaller number.

The problem with your wording for (1) is that it does not respect this
ordering, and requires the reader to jump forwards and backwards within
the sentence to (hopefully) work out what's going on. Reading in the
natural order, the reader first encounters "there exists e/2", but this
is not yet in a context where e has been introduced or explained, so
it's encouraging the reader to think this is the FIRST choice being
made, not a second (dependent) choice. In fact, I would just say NEVER
word it this way! (I don't think any maths book author would do so.)

The wording of your (2) is OK, because the order of quantification is
clear, it being in the order naturally encountered during reading of the
sentence.

Moving away from english wording, another approach could be to write (1)
and (2) in more formal symbolic language, e.g. perhaps

(1) Ax: (x in R & x>0 --> Ey: (y in R & y>0 & y<x))
(2) Not (Ey: y in R & y>0 & Ax: (x in R & x>0 --> y<x))

Or more simply if it is understood we are only considering positive reals:

(1) Ax: (Ey: (y<x))
(2) Not (Ey: (Ax: (y<x))

Ax: is my ASCII notation for "for all x", and is properly written with
an upside down A. Similarly Ey is "there exists y" and should be a
backwards E (reflected left/right).

The symbolic formulas make absolutely clear the order of
quantifications, due to the bracketing etc. but often this would be
considered overkill, and English is fine as long as the quantification
order is made clear. Often a combination of symbols like Ax: is used in
conjunction with less formal English wording, to get something
reasonably concise/clear, while still remaining reasonably natural to read.

For reading material, I think any introductory book on e.g. Real
Analysis has to deal with this issue over and over again, so see how
they do it, and you'll soon get used to the conventional wordings employed.

Mike.

[John Gabriel]

Good points.

>
> Moving away from english wording, another approach could be to write (1)
> and (2) in more formal symbolic language, e.g. perhaps
>
> (1) Ax: (x in R & x>0 --> Ey: (y in R & y>0 & y<x))
> (2) Not (Ey: y in R & y>0 & Ax: (x in R & x>0 --> y<x))

No need for long winded symbolic statements when natural language is very clear. You mentioned "ordering", but also your definition (2) requires one to jump back and forth. One has to evaluate everything to the right of Not and then negate. Not good.

[burs...@gmail.com]

LoL, laughing my ass off.

Well, extra for JG, we introduce a postfix negation,
written as A Bozo. So its not anymore a prefix negation.

You can then writen (2) as:

2) (Ey: y in R & y>0 & Ax: (x in R & x>0 --> y<x)) Bozo

I guess is not anymore "Not good",
but now is "Good". Right?

Hint: For formal systems it doesn't matter whether you
use infix, postfix, reverse polish notation or whatever.

The ordering that we or Mike Terry or the OP might
refer to is another ordering. Its not a cosmetic ordering.

Its an ordering which has an impact to the semantics,
and not an ordering which doesn't have any impact at all.

For exampke:

Ax Ey P(x,y)

Is not the same as:

Ey Ax P(x,y)

So the ordering of quantifiers Ey and Ax is very important,
and some formulations errors stem from using the wrong ordering.

See also:
The Diversity of Quantifier Prefixes
H. Jerome Keisler and Wilbur Walkoe, Jr.
The Journal of Symbolic Logic
Vol. 38, No. 1 (Mar., 1973), pp. 79-85
https://www.jstor.org/stable/2271729

[burs...@gmail.com]

Anyway bird brain John Gabriel, you make
it too easy, Dans Christensens banner:

*** THE IDIOCIES JUST KEEP ON COMING! ***

Is your only archivement so far.

[William Elliot]

On Thu, 19 Oct 2017, Philip Lee wrote:

> For these two statements,

> (1). There exists a positive real number ε/2 smaller than every
> positive real number ε.

You stated that there is a positive real number e/2 such that for
every positive real e (meaning 2*the given e/2), e/2 < e. That is
trivial.

Perhaps you meant to state that
there is a positive real number e/2 such that for
for every positive real d, e/2 < d. Which is false.

> (2). There is no positive real number smaller than every positive
> real number ε.

Not some r > 0 with for all e > 0, r < e. That is correct.

In otherwords: for all r > 0, some e > 0 with not r < e.
Which is trivial.

>
> Questions : Both statements seems right , but both also seems
> contradicting with each other , what should have in mind in writing
> to avoid such subtle contradictory representations ? It would be
> much better to recommend some reading materials on this topic,

Let some symbolic logic including FOL logic with quantifers.

[Zelos Malum]

> Natural language works just fine my little twerp.

It doesn't you fag, that is exactly one of the main issues here, you are too stupid to understand the limitations of natural language. Then again that is where you thrive isn't it? In the vagueness, imprecision of natural language because as long as it is vague enough, if someone points out a flaw you can just hide behind a different part that they didn't adress, but is still within the meaning of the word/sentence.

>In 10 Grid take for example .1, is there a .05 in 10 Grid, no, so your first claim is false

Dipshit, he talks about real numbers, not your delusional crap.

>True crank you are.

You both are so why not buddy up?

>No need for long winded symbolic statements when natural language is very clear. You mentioned "ordering", but also your definition (2) requires one to jump back and forth. One has to evaluate everything to the right of Not and then negate. Not good.

Except natural language isn't necciserily clear, just the word "or" introduces natural ambiguity, does "or" include the choice of both options, or just either one? Some go for former, some for latter, etc.

The FOL reading is however entirely unambigious.

[Philip Lee]

Thanks, you have managed to get my meaning !
As you said :

> "Reading in the natural order, the reader first encounters "there
> exists e/2", but this is not yet in a context where e has been
> introduced or explained, so it's encouraging the reader to think this
> is the FIRST choice being made, not a second (dependent) choice. "

This is the problem on cognition caused by the irrespective of the order of quantifiers on English wording, thus statement (1) is very likely to convey the wrong meaning that "there exists a positive real number smaller than every positive real number", which contradicts with statement (2), this is the contradiction I called in the question.

I will also remember your practical tip on dealing with symbolic formulas and English wording, Thanks again , Mike !

> "The symbolic formulas make absolutely clear the order of
> quantifications, due to the bracketing etc. but often this would be
> considered overkill, and English is fine as long as the quantification
> order is made clear. Often a combination of symbols like Ax: is used
> in conjunction with less formal English wording, to get something
> reasonably concise/clear, while still remaining reasonably natural to
> read."

As for reading material, I mainly request it for dealing with the issue caused by the order of quantifiers with English wording, now you solved it !

[Philip Lee]

Yes!

> "Its an ordering which has an impact to the semantics, and not an
> ordering which doesn't have any impact at all. "

I think my question was caused by the irrespective of the order of quantifiers on English wording, I requests for reading materials to clear the fog in my mind caused by it. Does your recommendations cover the topic?
I think Mike Terry provided a good answer to my question, also have a look on my reply to him if you have interest .
Thanks for reply !

[Philip Lee]

Thanks for the 'or' example to show the vagueness, imprecision of natural language ! Is there any way to avoid it, especially in mathematics ?

[John Gabriel]

On Saturday, 21 October 2017 12:23:29 UTC-4, Philip Lee wrote:
> Yes!
>
> > "Its an ordering which has an impact to the semantics, and not an
> > ordering which doesn't have any impact at all. "
>
> I think my question was caused by the irrespective of the order of quantifiers on English wording, I requests for reading materials to clear the fog in my mind caused by it.

No. Your confusion resulted from your stupidity and your inability to understand natural language. All the same, when your own language (Chinese) is so hopelessly backward, it's a little cruel to be hard on you.

> Does your recommendations cover the topic?
> I think Mike Terry provided a good answer to my question, also have a look on my reply to him if you have interest .

Terry gave you some good advice, but not all of it is correct. You still have to think for yourself. That's something you ALWAYS need to do.

> Thanks for reply !

[FromTheRafters]

Philip Lee used his keyboard to write :

> Thanks for the 'or' example to show the vagueness, imprecision of natural
> language ! Is there any way to avoid it, especially in mathematics ?

XOR is used in mathematics as opposed to OR for the two meanings, in
plain language it is usually left to context.