From nobody@digitalkingdom.org Wed Jul 12 16:19:45 2006 Received: with ECARTIS (v1.0.0; list lojban-list); Wed, 12 Jul 2006 16:19:46 -0700 (PDT) Received: from nobody by chain.digitalkingdom.org with local (Exim 4.62) (envelope-from ) id 1G0nzJ-00018v-Dn for lojban-list-real@lojban.org; Wed, 12 Jul 2006 16:19:25 -0700 Received: from simba.math.ucla.edu ([128.97.4.125]) by chain.digitalkingdom.org with esmtps (TLS-1.0:DHE_RSA_AES_256_CBC_SHA1:32) (Exim 4.62) (envelope-from ) id 1G0nzF-00018o-MY for lojban-list@lojban.org; Wed, 12 Jul 2006 16:19:25 -0700 Received: by simba.math.ucla.edu (Postfix, from userid 500) id DA67E3BADA; Wed, 12 Jul 2006 16:19:19 -0700 (PDT) Received: from localhost (localhost [127.0.0.1]) by simba.math.ucla.edu (Postfix) with ESMTP id B9BE83BAD4 for ; Wed, 12 Jul 2006 16:19:19 -0700 (PDT) Date: Wed, 12 Jul 2006 16:19:19 -0700 (PDT) From: Jim Carter To: lojban-list@lojban.org Subject: [lojban] Re: A (rather long) discussion of {all} In-Reply-To: <925d17560607121113y4d0be37y1fe4757a46c030f4@mail.gmail.com> Message-ID: References: <20060711233003.36140.qmail@web81310.mail.mud.yahoo.com> <925d17560607120531v7de5bfefwa96db493b274fbdf@mail.gmail.com> <925d17560607121113y4d0be37y1fe4757a46c030f4@mail.gmail.com> MIME-Version: 1.0 Content-Type: MULTIPART/MIXED; BOUNDARY="1635810564-1773356151-1152746359=:20555" X-Spam-Score: -2.6 (--) This message is in MIME format. The first part should be readable text, while the remaining parts are likely unreadable without MIME-aware tools. X-archive-position: 12159 X-ecartis-version: Ecartis v1.0.0 Sender: lojban-list-bounce@lojban.org Errors-to: lojban-list-bounce@lojban.org X-original-sender: jimc@math.ucla.edu Precedence: bulk Reply-to: lojban-list@lojban.org X-list: lojban-list --1635810564-1773356151-1152746359=:20555 Content-Type: TEXT/PLAIN; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable On Wed, 12 Jul 2006, Jorge Llamb=EDas wrote: > On 7/12/06, Maxim Katcharov wrote: >=20 > > No, I want to know how you explain why the singular is the only one > > that is not subject to collectivity. >=20 > You need at least two things before you can have a distinction between > distributing or not distributing something among them. Isn't that obvio= us? No. I can't help jumping in here... > Because there is no distinction to be made. Why does it not make > any difference to order a set of numbers from smallest to largest or > from largest to smallest when the set contains a single number? > Same thing with distributivity, if there is only one thing, distributiv= e > and non-distributive give identical results. In a database query you often sort (order) the result, and it's important= =20 to do so, even if you don't know in advance whether the result will have=20 zero, one or multiple members, and any of those outcomes happen often. Y= ou=20 expect to be able to produce an ordered set with no irrelevant complaints= =20 about the lack of plurality. Another example: "An Army of One". Usually battle involves teams of=20 soldiers, but it happens, often enough to mention and often enough to try= =20 to give the soldiers some training, that the outcome hinges on the action= s=20 of a team of one soldier. The relation between the circumstances of batt= le=20 and the teams are the same, regardless of how many people are in them. Yet another example: One formalism for defining the integers goes like=20 this: if a 1-1 relation exists between 2 sets they are said to have the=20 "same count" (or cardinality), and this is an equivalence relation, so th= at=20 each set is in exactly one of the equivalence classes of equal count sets= . =20 The equivalence classes are the integers. Bertrand Russell proved back i= n=20 the 1950's (or earlier?) that a particular list of examples had a unique=20 member in every equivalence class, and thus was a representation of the=20 integers. The list member for 0 is the empty set (represented {}; all th= e=20 members of the set can be viewed between the brackets). The member for 1= =20 is {{}} (set containing the empty set). The member for 2 is {{} {{}}} (s= et=20 containing the list member for each smaller integer (1 and 0 follow the=20 same definition)), and so on recursively. The point is, each of these is = a=20 set, and it doesn't work if you elide the set nature of the non-plural=20 {{}}, which cannot be taken to be "the same as" its unique member {}. An= d=20 similarly it's important that procedures work correctly when applied to a= ll=20 the members of the empty set (look up St. Anselm's ontological proof of t= he=20 existence of God). So distributing a relation over all the one or zero members of the smalle= r=20 sized sets is important and needs to be supported in the language. James F. Carter Voice 310 825 2897 FAX 310 206 6673 UCLA-Mathnet; 6115 MSA; 405 Hilgard Ave.; Los Angeles, CA, USA 90095-15= 55 Email: jimc@math.ucla.edu http://www.math.ucla.edu/~jimc (q.v. for PGP= key) --1635810564-1773356151-1152746359=:20555-- To unsubscribe from this list, send mail to lojban-list-request@lojban.org with the subject unsubscribe, or go to http://www.lojban.org/lsg2/, or if you're really stuck, send mail to secretary@lojban.org for help.