Re: [lojban] Re: A (rather long) discussion of {all}

[John E Clifford <clifford-j@sbcglobal.net>]

--- Jim Carter <jimc@math.ucla.edu> wrote:

> On Wed, 12 Jul 2006, Jorge Llamb�as wrote:
> > On 7/12/06, Maxim Katcharov <maxim.katcharov@gmail.com> wrote:
> > 
> > > No, I want to know how you explain why the singular is the only one
> > > that is not subject to collectivity.
> > 
> > You need at least two things before you can have a distinction between
> > distributing or not distributing something among them. Isn't that obvious?
> 
> 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, distributive
> > and non-distributive give identical results.
> 
> In a database query you often sort (order) the result, and it's important 
> to do so, even if you don't know in advance whether the result will have 
> zero, one or multiple members, and any of those outcomes happen often.  You 
> expect to be able to produce an ordered set with no irrelevant complaints 
> about the lack of plurality.
> 
> Another example: "An Army of One".  Usually battle involves teams of 
> soldiers, but it happens, often enough to mention and often enough to try 
> to give the soldiers some training, that the outcome hinges on the actions 
> of a team of one soldier.  The relation between the circumstances of battle 
> 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 
> this:  if a 1-1 relation exists between 2 sets they are said to have the 
> "same count" (or cardinality), and this is an equivalence relation, so that 
> each set is in exactly one of the equivalence classes of equal count sets.  
> The equivalence classes are the integers.  Bertrand Russell proved back in 
> the 1950's (or earlier?) that a particular list of examples had a unique 
> member in every equivalence class, and thus was a representation of the 
> integers.  The list member for 0 is the empty set (represented {}; all the 
> members of the set can be viewed between the brackets).  The member for 1 
> is {{}} (set containing the empty set).  The member for 2 is {{} {{}}} (set 
> containing the list member for each smaller integer (1 and 0 follow the 
> same definition)), and so on recursively. The point is, each of these is a 
> set, and it doesn't work if you elide the set nature of the non-plural 
> {{}}, which cannot be taken to be "the same as" its unique member {}.  And 
> similarly it's important that procedures work correctly when applied to all 
> the members of the empty set (look up St. Anselm's ontological proof of the 
> existence of God).
> 
> So distributing a relation over all the one or zero members of the smaller 
> 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  90095-1555
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