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From: FromTheRafters <FTR@nomail.afraid.org>
Newsgroups: sci.math
Subject: Re: =?ISO-8859-15?Q?universal_quantifiers,_because_g=3F(g=3F=B9(x?=
	=?ISO-8859-15?Q?))_=3D_g(y)_[1/2]_Re:_how?=
Date: Tue, 07 May 2024 17:28:59 -0400
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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'.