Re: universe of discourse

[bob@MEGALITH.RATTLESNAKE.COM]

[My apologies; I have been too busy with family estate matters to
follow closely the last two weeks.  I will try to fill in soon.]

   >> The number 3 does not exist in the universe of discourse which
   >> is restricted to the set of even numbers.

 > What is a universe of discourse?

That topic within which you and your interlocutor are talking.

The phrase is based on a container metaphor.  The idea is that a
discussion may stay, like two people, within a garden, house or
particular cosmos.

A second metaphor might refer to traditional Western music and say
that a specific discussion remains within a particular key.
(Hmmm... you will note here that the music metaphor is expressed using
a territorial or container metaphor, too... )

A third metaphor might use the concept of saliency: that those
utterances that are meaningful within a discussion are those that leap
out or poke at you, `are salient'.

Utterances or knowledge that is not salient to the discussion have no
effect on the truth value of assertions in that discussion.  This is
why, in at least one universe of discourse, you can truthfully say
that `unicorns leap like gazelles'.


As for even numbers:

Some years ago, one of my teachers pointed out that there is a kind of
mathematical container or universe that relates to addition and
multiplication among even numbers that does not occur with odd
numbers.

  * When you add two even numbers, the sum is an even number; and
    when you multiply two even numbers, the product is an even number.

    The result of the two operations is a number that is also and
    always even.

    But

  * when you add two odd numbers, the product is an even number; and
    when you multiply two odd numbers, the product is an odd number.

    The result of the two operations is a number that may be odd and
    may be even.

In the jargon I was taught many years ago, the even integers
{..., -4, -2, 0, 2, 4, ...}  with the usual operations of addition
and multiplication form a "commutative ring".  The odd integers do not.

    <even number> + <even number> ==> <even number>
                2 + 2              => 4
    <even number> * <even number> ==> <even number>
                2 * 6              => 12

    <odd number> + <odd number>   ==> <even number>
               3 + 5               => 8
    <odd number> * <odd number>   ==> <odd number>
               3 * 7               => 21

The universe of even numbers and the universe of odd numbers are very
different.

--

    Robert J. Chassell               bob@rattlesnake.com
    P. O. Box 693                    bob@ai.mit.edu
    Stockbridge, MA 01262-0693 USA   (413) 298-4725