ONT Re: Zeroth Order Theories (ZOT's)
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Our next Example illustrates the use of the Cactus Language
for representing "absolute" and "relative" partitions, also
known as "complete" and "contingent" classifications of the
universe of discourse, all of which amounts to divvying it
up into mutually exclusive regions, exhaustive or not, as
one frequently needs in situations involving a genus and
its sundry species, and frequently pictures in the form
of a venn diagram that looks just like a "pie chart".
Example. Partition, Genus & Species
The idea that one needs for expressing partitions
in cactus expressions can be summed up like this:
| If the propositional expression
|
| "( p , q , r , ... )"
|
| means that just one of
|
| p, q, r, ... is false,
|
| then the propositional expression
|
| "((p),(q),(r), ... )"
|
| must mean that just one of
|
| (p), (q), (r), ... is false,
|
| in other words, that just one of
|
| p, q, r, ... is true.
Thus we have an efficient means to express and to enforce
a partition of the space of models, in effect, to maintain
the condition that a number of features or propositions are
to be held in mutually exclusive and exhaustive disjunction.
This supplies a much needed bridge between the binary domain
of two values and any other domain with a finite number of
feature values.
Another variation on this theme allows one to maintain the
subsumption of many separate species under an explicit genus.
To see this, let us examine the following form of expression:
( q , ( q_1 ) , ( q_2 ) , ( q_3 ) ).
Now consider what it would mean for this to be true. We see two cases:
1. If the proposition q is true, then exactly one of the
propositions (q_1), (q_2), (q_3) must be false, and so
just one of the propositions q_1, q_2, q_3 must be true.
2. If the proposition q is false, then every one of the
propositions (q_1), (q_2), (q_2) must be true, and so
each one of the propositions q_1, q_2, q_3 must be false.
In short, if q is false then all of the other q's are also.
Figures 1 and 2 illustrate this type of situation.
Figure 1 is the venn diagram of a 4-dimensional universe of discourse
X = [q, q_1, q_2, q_3], conventionally named after the gang of four
logical features that generate it. Strictly speaking, X is made up
of two layers, the position space X of abstract type %B%^4, and the
proposition space X^ = (X -> %B%) of abstract type %B%^4 -> %B%,
but it is commonly lawful enough to sign the signature of both
spaces with the same X, and thus to give the power of attorney
for the propositions to the so-indicted position space thereof.
Figure 1 also makes use of the convention whereby the regions
or the subsets of the universe of discourse that correspond
to the basic features q, q_1, q_2, q_3 are labelled with
the parallel set of upper case letters Q, Q_1, Q_2, Q_3.
| o
| / \
| / \
| / \
| / \
| o o
| /%\ /%\
| /%%%\ /%%%\
| /%%%%%\ /%%%%%\
| /%%%%%%%\ /%%%%%%%\
| o%%%%%%%%%o%%%%%%%%%o
| / \%%%%%%%/ \%%%%%%%/ \
| / \%%%%%/ \%%%%%/ \
| / \%%%/ \%%%/ \
| / \%/ \%/ \
| o o o o
| / \ /%\ / \ / \
| / \ /%%%\ / \ / \
| / \ /%%%%%\ / \ / \
| / \ /%%%%%%%\ / \ / \
| o o%%%%%%%%%o o o
| ·\ / \%%%%%%%/ \ / \ /·
| · \ / \%%%%%/ \ / \ / ·
| · \ / \%%%/ \ / \ / ·
| · \ / \%/ \ / \ / ·
| · o o o o ·
| · ·\ / \ / \ /· ·
| · · \ / \ / \ / · ·
| · · \ / \ / \ / · ·
| · Q · \ / \ / \ / ·Q_3 ·
| ··········o o o··········
| · \ /%\ / ·
| · \ /%%%\ / ·
| · \ /%%%%%\ / ·
| · Q_1 \ /%%%%%%%\ / Q_2 ·
| ··········o%%%%%%%%%o··········
| \%%%%%%%/
| \%%%%%/
| \%%%/
| \%/
| o
|
| Figure 1. Genus Q and Species Q_1, Q_2, Q_3
Figure 2 is another form of venn diagram that one often uses,
where one collapses the unindited cells and leaves only the
models of the proposition in question. Some people would
call the transformation that changes from the first form
to next form an operation of "taking the quotient", but
I tend to think of it as the "soap bubble picture" or
more exactly the "wire & thread & soap film" model
of the universe of discourse, where one pops the
parts of the soap film that stretch across the
anti-model regions of the positional space.
o-------------------------------------------------o
| |
| X |
| |
| o |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| o Q_1 o |
| / \ / \ |
| / \ / \ |
| / \ / \ |
| / \ / \ |
| / \ / \ |
| / Q \ |
| / | \ |
| / | \ |
| / Q_2 | Q_3 \ |
| / | \ |
| / | \ |
| o-----------------o-----------------o |
| |
| |
| |
o-------------------------------------------------o
Figure 2. Genus Q and Species Q_1, Q_2, Q_3
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