ONT Re: Data Models, Ontologies, Logic

[Jon Awbrey <jawbrey@oakland.edu>]

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Note 23

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Sticking with the nitty-gritty details of the Family Interaction
data fragment, I am going to try and stay focused on another one
of those places where three roads meet, seeking the continuities
that exist, or may be induced to exist, among information theory,
logic, and statistics.  It's the sort of thing that Peirce spent
quite a bit of time thinking about, and had an amazing number of
rather remarkable and still revolutionary insights about.  But I
must plod along my concrete ways before I can get to any of that.

Before we go any further, let me show you one of the planks that
we shall use to build a bridge between boolean variables and any
discrete domain with a finite -- in practice, a relatively small --
number of values.

For the sake of concreteness and simplicity, let us imagine that the
variable u_1 for "family member initiating the action" is restricted
to the values 1, 2, 3 for child, father, mother, respectively.  Then
the linguistic strings "f1_child", "f2_father", "f3_mother" serve as
propositional expressions, that is, as functions from the data space
U of data points <u_0, u_1, u_2, u_3, u_4, u_5, u_6> into B = {0, 1}.
So "f1_child" is the name of an indicator function f1_child : U -> B.

In order to make the three functions f1_child, f2_father, f3_mother : U -> B
behave like three values of a discrete domain D_1 = {child, father, mother},
we impose the following propositional constraint, written in cactoid syntax
and translated into English:

    ((f1_child),(f2_father),(f3_mother))

    meaning:

    Exactly one of f1_child, f2_father, f3_mother is true.

This expression is a proposition g : U -> B like any other about U.
To "assert" g or to "impose" g as propositional constraint means to
ignore, to discount, or to rule out the regions of the data space U
where it fails to reign true, in one figure of speech that one often
hears, to take the "quotient" of U by g.

More generally, given the three propositions:

    u_j_a,  u_j_b,  u_j_c  :  U -> B

subject to the propositional constraint:

    ((u_j_a),(u_j_b),(u_j_c))

then the pattern of the corresponding fibers:

    U_j_a,  U_j_b,  U_j_c

is illustrated in the following venn diagram:

o-----------------------------------------------------------o
| U                                                         |
|                                                           |
|                      o-------------o                      |
|                     /```````````````\                     |
|                    /`````````````````\                    |
|                   /```````````````````\                   |
|                  /`````````````````````\                  |
|                 /```````````````````````\                 |
|                o````````` U_j_a `````````o                |
|                |`````````````````````````|                |
|                |`````````````````````````|                |
|                |`````````````````````````|                |
|                |`````````````````````````|                |
|                |`````````````````````````|                |
|             o--o----------o   o----------o--o             |
|            /````\          \ /          /````\            |
|           /``````\          o          /``````\           |
|          /````````\        / \        /````````\          |
|         /``````````\      /   \      /``````````\         |
|        /````````````\    /     \    /````````````\        |
|       o``````````````o--o-------o--o``````````````o       |
|       |`````````````````|       |`````````````````|       |
|       |`````````````````|       |`````````````````|       |
|       |`````````````````|       |`````````````````|       |
|       |`````````````````|       |`````````````````|       |
|       |`````````````````|       |`````````````````|       |
|       o````` U_j_b `````o       o````` U_j_c `````o       |
|        \`````````````````\     /`````````````````/        |
|         \`````````````````\   /`````````````````/         |
|          \`````````````````\ /`````````````````/          |
|           \`````````````````o`````````````````/           |
|            \```````````````/ \```````````````/            |
|             o-------------o   o-------------o             |
|                                                           |
|                                                           |
o-----------------------------------------------------------o

This discussion has demonstrated a perfectly generic construction
of a finite data domain, enabling the representation of a k-tomic
categorical variable u_j, ranging over a k-set of possible values,
in terms of k boolean variables, for instance, u_j_1, ..., u_j_k,
modulo the proposition ((u_j_1), ..., (u_j_k)).

Some folks call that "radio button logic".

Stay tuned ...

Jon Awbrey

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