Re: universal quantifiers, because g?(g?¹(x)) = g(y) [1/2] Re: how

On 05/07/2024 02:28 PM, FromTheRafters wrote:

Jim Burns used his keyboard to write :

There is a different post which
makes this thread more understandable.
>
Message-ID: <1e875f8d-d6f8-4a16-b7a3-68424dc89a89@att.net>

On 5/6/2024 4:15 PM, Jim Burns wrote:

On 5/6/2024 3:36 PM, Ross Finlayson wrote:

On 05/05/2024 03:02 PM, Jim Burns wrote:

>

For my wish,
I would like everyone to be clear on what
standard.issue quantifiers and variables
mean.
>
I think that,
way off in that glorious future,
both you and I will be able to be
satisfactorily understood.
>
And what more could there be
to wish for?

>
Well, one might aver that extra-ordinary
universal quantifiers are merely syntactic sugar,
yet there's that in the low- and high- orders,
or the first and final, that what they would
reflect of the _effects_ of quantification,
something like
>
for-any A?
for-each A+
for-every A*
for-all A$
>
that it is so that the sputniks or extras
of the quantification in the extra-ordinary,
have that a quantifier disambiguation:
is in the syntax.
>
Then for the rest of it,

>
Before you move on,
could you explain what your notation means?
Thanks in advance.

>
On 5/6/2024 9:00 PM, Ross Finlayson wrote:

On 05/06/2024 01:16 PM, Jim Burns wrote:

[...]

>
Well, first of all, it's after pondering that there
is quantifier comprehension artifacts of the extra sort,
as of a set of all sets, order type of ordinals, a universe,
set of sets that don't contain themself, sets that contain
themselves, and so on.
>
Then, English affords "any, "each, "every, "all".
>
The -any means for example that "it's always a fragment".
So in this sense the usual universal quantifier is for-each.
>
Then, for-each, means usual comprehension, as if an enumeration,
or a choice function, each.
>
Then, for-every, means as a sort of comprehension, where it
so establishes itself again, any differently than -each,
when -each and -every implies both none missing and all gained.
>
Then, "for-all", sort of is for that what is so "for-each"
and "for-every" is so, "for-all", as for the multitude as
for the individual.
>
Then, I sort of ran out of words, "any", "each", "every", "all",
then that seems their sort of ordering, about comprehension,
in quantification, in the universals, of each particular.
>
About sums it up, ....

>
Are there differences in syntax between
'for.any' 'for.each' 'for.every' 'for.all' ?
>
I take the following to be _standard.issue syntax_
I am cribbing it from
Elliott Mendelson's _Introduction to Mathematical Logic"
https://www.karlin.mff.cuni.cz/~krajicek/mendelson.pdf
|
| ∀x:B(x) ⇒ B(t)
|
| ∀x:(B⇒C(x)) ⇒ (B⇒∀x:C(x))
|
| B  |-  ∀x:B
|
| ∃x:B ⇔ ¬∀x:¬B
>
Mendelson's chapter "Quantification Theory"
discusses why _that syntax_ starting from
how Mendelson and a standard.issue mathematician
expect truth to behave, and deriving
_that syntax_ from those expectations.
>
It's not fair to compare you and Mendelson,
since he has behind him that horde of
All.The.Standard.Issue.Mathematicians
whose work he is passing on to a new generation.
>
But I was hoping for something closer to Mendelson
in nature from you.
>
I think the question you will have to answer
eventually is:
Are there differences in syntax between
'for.any' 'for.each' 'for.every' 'for.all' ?
or
you must be be reconciled to your distinctions
being pointless.

>
I was thinking the same thing, but my thinking is limited to a certain
few disciplines. I lump 'each and every' (which seems redundant when put
together like that) with 'any' because it seemed to me that they all say
essentially the same thing -- that it will be true of any choice you
make from the set. I reserve 'all' for 'the set of all' more like a
'group noun' for 'any, each and every'.

Well said.
The terms like A? A+ A* A$ evoke the syntax of regular expressions,
with the Kleene star and question as a usual option, with that
plus indicate one or more and star actually usually indicates
zero or more, while dollar matches the end.
So, it's sort of why it's A? A+ A* A$ instead of
A? A* A+ A$, is that it seems where the "match" as
it were, indicates along the lines of that the top
of the deck match, turning over each card matches,
turning over each card none don't match, then
that turning over all the cards now the other
sides aren't visible.
https://en.wikipedia.org/wiki/Kleene_star
It sort of carries in the negation, for example,
"not: not-a-heap, not: not-a-heap, ..., was-a-heap,
not-a-heap".
It makes a stronger statement about the transfer
principle, about when that which applies to the
members applies to the collection, and when it
does, then it's inseparable in syntax to remain
correct, because, it's part of its model.