Subject: Re: [lojban] More about quantifiers

From: [email-masked]
Date: Wed, 13 Mar 2002 17:30:44 EST

In a message dated 3/13/2002 1:17:30 PM Central Standard Time, 
[email-masked] writes:

> What do you mean by "actual quantification"?
> 
Quantity and quality (universal-particular, affirmative-negative) as well as 
import.

<Right, except I'm not sure what you mean with those {..}.
You cannot insert another term in there for the equality to hold.>

In case broda has arguments attached.

<Which one is the traditional system?>

All + with the assumption that all classes mentioned as subject are non-null 
(and maybe a few less certain things as well). Using {ro broda} and {ro da 
poi} as the usual quantifiers of course already guarantees the assumption -- 
in true sentences anyhow.

<If you mean (A-,E-,I+,O+),>

I never would, since it is unlojbanic in its first two members and lacks 
subalternation.

<
>If you check other pages, you will find that Helman is not dealing with
>restricted quantification as such but using the notation as a stage in the
>process of translating English into symbols in ordinatry first-order logic.

On the contrary, he specifically defines restricted universal
quantification and unrestricted universal quantification, and
then gives the following "principle of equivalence":

(Ax: Sx) Px =||= Ax (Sx -> Px).

(where should be inverted). What's more, in the chapter
about existential quantification he also has the equivalence
between the restricted and unrestricted forms:

(Ex: Sx) Px =||= Ex (Sx & Px)

and also the principle of obversion:

¬ (Ax: Sx) Px =||= (Ex: Sx) ~Px

which works only if the restricted universal is A-.

>Once the block attached to the quantifier is correctly filled in, the whole
>can then be correctly moved into the formula in the usual way. But the
>"restricted quantifier" (as the regular use of "thing" suggests) is just a
>passing phase of translation, not a part of the logic.>

Well, I didn't read the whole book, just a few sections that talked about 
restricted quantification. I never saw any evidence that it was developed as 
a separate system. In fact all the cases I saw were parts of translation 
exercises, like the one you sent me to originally: "(AxFx)Gx" as a more or 
less Englishy sentence that could then be converted into "Ax(Fx => Gx)", but 
not used in proofs or derivations. Obversion is just a device for making 
"not every" a bit more readable, as I read him. But I will look at some more 
(and of course it works for an all positive set as well -- under the standard 
condition -- no empty subjects).
Of course, the restricted quantifier is -, since it just is the ultimate form 
in a minorly gussied up way. Part of the gussying is, alas, to hide the real 
subject of the of the final quantifier, namely the universal class.

<Saying "you can work it out" is unconvincing. Maybe if you actually
did work it out it would be more convincing, but since in the end
it is just a matter of definitions...>

Every universal quantifier (in a non-empty universe) entails every instance 
of its matrix, every matrix with a free term entails its particular closure 
on that term:
AxFx therefore Fa therefore ExFx. That is about as thorough a working out as 
I can think of.