ONT Re: Zeroth Order Theories (ZOT's)

[Jon Awbrey <jawbrey@oakland.edu>]

¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤

Here is a scaled-down version of one of my very first applications,
having to do with the demographic variables in a survey data base.

This Example illustrates the use of 2-variate logical forms
for expressing and reasoning about the logical constraints
that are involved in the following types of situations:

1.  Distinction:     A =/= B
    Also known as:   logical inequality, exclusive disjunction
    Represented as:  ( A , B )
    Graphed as:
    |
    |   A   B
    |   o---o
    |    \ /
    |     @

2.  Equality:        A = B
    Also known as:   logical equivalence, if and only if, A <=> B
    Represented as:  (( A , B ))
    Graphed as:
    |
    |   A   B
    |   o---o
    |    \ /
    |     o
    |     |
    |     @

3.  Implication:     A => B
    Also known as:   entailment, if-then
    Represented as:  ( A ( B ))
    Graphed as:
    |
    |   A   B
    |   o---o
    |   |
    |   @

Example of a propostition expressing a "zeroth order theory" (ZOT):

Consider the following text, written in what I am calling "Ref Log",
also known as the "Cactus Language" syntax for propositional logic:

|   ( male  , female )
|   (( boy  , male child ))
|   (( girl , female child ))
|   ( child ( human ))

Graphed as:

|                   boy   male     girl   female
|                     o---o child     o---o child
|  male   female       \ /             \ /          child   human
|     o---o             o               o               o---o
|      \ /              |               |               |
|       @               @               @               @

Nota Bene.  Due to graphic constraints -- no, the other
kind of graphic constraints -- of the immediate medium,
I am forced to string out the logical conjuncts of the
actual cactus graph for this situation, one that might
sufficiently be reasoned out from the exhibit supra by
fusing togteher the four roots of the severed cactus.

Either of these expressions, text or graph, is equivalent to
what would otherwise be written in a more ordinary syntax as:

|  male  =/=  female
|  boy   <=>  male child
|  girl  <=>  female child
|  child  =>  human

This is a actually a single proposition, a conjunction of four lines:
one distinction, two equations, and one implication.  Together, these
amount to a set of definitions conjointly constraining the logical
compatibility of the six feature names that appear.  They may be
thought of as sculpting out a space of models that is some subset
of the 2^6 = 64 possible interpretations, and thereby shaping some
universe of discourse.

Once this backdrop is defined, it is possible to "query" this universe,
simply by conjoining additional propositions in further constraint of
the underlying set of models.  This has many uses, as we shall see.

Jon Awbrey

¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤