From pycyn@aol.com Wed Mar 13 14:30:54 2002 Return-Path: X-Sender: Pycyn@aol.com X-Apparently-To: lojban@yahoogroups.com Received: (EGP: unknown); 13 Mar 2002 22:30:53 -0000 Received: (qmail 97607 invoked from network); 13 Mar 2002 22:30:52 -0000 Received: from unknown (216.115.97.171) by m10.grp.snv.yahoo.com with QMQP; 13 Mar 2002 22:30:52 -0000 Received: from unknown (HELO imo-m07.mx.aol.com) (64.12.136.162) by mta3.grp.snv.yahoo.com with SMTP; 13 Mar 2002 22:30:52 -0000 Received: from Pycyn@aol.com by imo-m07.mx.aol.com (mail_out_v32.5.) id r.73.1c499a40 (4587) for ; Wed, 13 Mar 2002 17:30:45 -0500 (EST) Message-ID: <73.1c499a40.29c12d14@aol.com> Date: Wed, 13 Mar 2002 17:30:44 EST Subject: Re: [lojban] More about quantifiers To: lojban@yahoogroups.com MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="part1_73.1c499a40.29c12d14_boundary" X-Mailer: AOL 7.0 for Windows US sub 118 From: pycyn@aol.com X-Yahoo-Group-Post: member; u=2455001 X-Yahoo-Profile: kaliputra X-Yahoo-Message-Num: 13691 --part1_73.1c499a40.29c12d14_boundary Content-Type: text/plain; charset="ISO-8859-1" Content-Transfer-Encoding: quoted-printable In a message dated 3/13/2002 1:17:30 PM Central Standard Time,=20 jjllambias@hotmail.com writes: > What do you mean by "actual quantification"? >=20 Quantity and quality (universal-particular, affirmative-negative) as well a= s=20 import. In case broda has arguments attached. All + with the assumption that all classes mentioned as subject are non-nu= ll=20 (and maybe a few less certain things as well). Using {ro broda} and {ro da= =20 poi} as the usual quantifiers of course already guarantees the assumption -= -=20 in true sentences anyhow. I never would, since it is unlojbanic in its first two members and lacks=20 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 =3D||=3D 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 =3D||=3D Ex (Sx & Px) and also the principle of obversion: =AC (Ax: Sx) Px =3D||=3D (Ex: Sx) ~Px which works only if the restricted universal is A-. >Once the block attached to the quantifier is correctly filled in, the whol= e >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=20 restricted quantification. I never saw any evidence that it was developed = as=20 a separate system. In fact all the cases I saw were parts of translation=20 exercises, like the one you sent me to originally: "(AxFx)Gx" as a more or= =20 less Englishy sentence that could then be converted into "Ax(Fx =3D> Gx)", = but=20 not used in proofs or derivations. Obversion is just a device for making=20 "not every" a bit more readable, as I read him. But I will look at some mo= re=20 (and of course it works for an all positive set as well -- under the standa= rd=20 condition -- no empty subjects). Of course, the restricted quantifier is -, since it just is the ultimate fo= rm=20 in a minorly gussied up way. Part of the gussying is, alas, to hide the re= al=20 subject of the of the final quantifier, namely the universal class. Every universal quantifier (in a non-empty universe) entails every instance= =20 of its matrix, every matrix with a free term entails its particular closur= e=20 on that term: AxFx therefore Fa therefore ExFx. That is about as thorough a working out = as=20 I can think of. --part1_73.1c499a40.29c12d14_boundary Content-Type: text/html; charset="ISO-8859-1" Content-Transfer-Encoding: quoted-printable In a message dated 3/13/2002 1:17:3= 0 PM Central Standard Time, jjllambias@hotmail.com writes:


What do you mean by "actual q= uantification"?

Quantity and quality (universal-particular, affirmative-negative) as well a= s 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 n= on-null (and maybe a few less certain things as well).  Using {ro brod= a} and {ro da poi} as the usual quantifiers of course already guarantees th= e 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 su= balternation.

<
>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 lo= gic.

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

             (A= x: Sx) Px =3D||=3D 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 =3D||=3D= Ex (Sx & Px)

and also the principle of obversion:

           =AC (Ax: Sx) P= x =3D||=3D (Ex: Sx) ~Px

which works only if the restricted universal is A-.

>Once the block attached to the quantifier is correctly filled in, the w= hole
>can then be correctly moved into the formula in the usual way.  Bu= t the
>"restricted quantifier" (as the regular use of "thing" suggests) is jus= t 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 r= estricted quantification.  I never saw any evidence that it was develo= ped as a separate system.  In fact all the cases I saw were parts of t= ranslation 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 = =3D> Gx)", but not used in proofs or derivations.  Obversion is jus= t 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 s= et as well -- under the standard condition -- no empty subjects).
Of course, the restricted quantifier is -, since it just is the ultimate fo= rm in a minorly gussied up way.  Part of the gussying is, alas, to hid= e the real subject of the of the final quantifier, namely the universal cla= ss.

<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.









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