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

[Jim Carter <jimc@math.ucla.edu>]

On Wed, 12 Jul 2006, Jorge Llambías 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
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Email: jimc@math.ucla.edu    http://www.math.ucla.edu/~jimc (q.v. for PGP key)