SUO: Re: Lattices, Objects, Signs
[Reposting after 3 hours]
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LOS. Note 3
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I have always shared the Icarian tendencies of our age,
or that just passing age, to begin at the top and try
to work upward, but many hard knocks have taught me
that my place is among the rude mechanicals, who
must start a bit nearer the bottom of the heap,
and shore up the dreams of higher realities.
So I always get edgy when we do not have concrete examples
of the abstract things we vaunt to talk about, as I cannot
forget the many times in my experience when the very first
concrete example that I ran across reduced to rubbish most
of what I had speculated in absentia of it.
With all that rueful experience in mind, I think that it would be smart
to begin examining some very simple examples of lattices, together with
a broader family of related orders, of the sort that might arise in our
present enterprise.
I have often regaled you with the complications that unfold from a universe
of discourse where there are but two predicates to discourse with, so let me
now attempt to underwhelm you with the topography of a landscape that has but
a single feature to note.
Figure 1 depicts a 1-dimensional universe of discourse.
o-------------------------------------------------o
| X |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | | |
| | Q | |
| | | |
| o o |
| \ / |
| \ / |
| \ / |
| \ / |
| o-------------o |
| |
| |
o-------------------------------------------------o
Figure 1. One-Dimensional Universe Of Discourse
In this picture, there is supposed to be pictured a universe X,
a predicate or proposition q : X -> B, and the set Q c X that
some people call the "fiber of truth in X under q", defined
as Q = (q^(-1))(1) and conveniently symbolized as [|q|].
It needs to be understood in using such pictures that only the
predicates themselves are being formally treated -- all of the
rest of the interpretive furniture, for instance, all concepts
of elements, functions, individuals, and sets, are still being
used, for the sake of exposition, in a casual or informal way,
and could be supplanted by many other interpretive apparatus.
In order to conduct some manner of formal discourse about this
universe of discourse it is necessary to introduce some manner
of formal language, often called a "propositional calculus".
As it happens, there are many such languages that arise to
the purpose, and so we have the task of translating among
these calculi in a way that respects their common object.
The nature of the situation is illustrated in Figure 2.
o-----------------------------o-------------------o-----------------------------o
| Language 1 | Object Domain | Language 2 |
o-----------------------------o-------------------o-----------------------------o
| |
| o----------o o----------o |
| /| "T" |\ 1 /| " " |\ |
| / | "q => q" |~\~~~~~~~~~~~~~~~~o~~~~~~~~~~~~~~~~/~| "(q(q))" | \ |
| / | ... | \ / \ / | ... | \ |
| / o----------o \ / \ / o----------o \ |
| / \ / \ / \ |
| o----------o \ / \ o----------o \ |
| | "q" | \ q / \ | "q" | \ |
| | "T => q" |~~~~~~~~~~~~~\~~~~o~~~~~~~~~~~\~~~~~~| "((q))" | \ |
| | ... | \ \ \ | ... | \ |
| o----------o o----------o \ \ o----------o o----------o |
| \ | "~q" | \ \ \ | "(q)" | |
| \ | "q => F" |~~~~~~\~~~~~~~~~~~o~~~~\~~~~~~~~~~~~~| "(q(()))"| |
| \ | ... | \ /(q) \ | ... | |
| \ o----------o \ / \ o----------o |
| \ / \ / \ / |
| \ o----------o / \ / \ o----------o / |
| \ | "F" | / \ / \ | "()" | / |
| \ | "q & ~q" |~/~~~~~~~~~~~~~~~~o~~~~~~~~~~~~~~~~\~| "q(q)" | / |
| \| ... |/ 0 \| ... |/ |
| o----------o o----------o |
| |
o-------------------------------------------------------------------------------o
Figure 2. Lattice of Objects Inducing a Diversity of Sign Partitions
For a 1-dim universe, the object lattice has 4 elements.
These are, that is to say, these may be interptreted as,
the 4 functions of the type p : X -> B that are derived
from a distinguished predicate q : X -> B. They may be
recognized as the constant predicate 0 : X -> B and the
constant predicate 1 : X -> B, the predicate q : X -> B,
and its negation, the predicate ~q : X -> B.
Here, I am conforming to a convention, customary in mathematics,
of regarding symbols like "0" and "1" as incomplete notations,
which must be completed by type indicators, often relegated
to context, in order to tell whether they are intended to
denote elements of B, functions of type X -> B, or some
other type of object.
The object lattice is ordered by logical implication,
with 0 at the bottom, 1 at the top, and the pair of
incomparable elements q and ~q in the middle.
Thus we have the following order relations:
| 0 => 0, q, ~q, 1
|
| q => q, 1
|
| ~q => ~q, 1
|
| 1 => 1
The so-called "Hasse diagram" shown in the middle of Figure 2
represents these order relations, omitting the self-loops and
the arcs that can be obtained by transitivity from the others.
If our enterprise were one of "pure" mathematics, we would promptly
settle on any convenient language that is up to the task of talking
about the objective structures of interest, and proceed to put away
as much as possible, not only the business of coordinating distinct
languages for the same objects, but most of the fuss about semantic
and semiotic issues, like how one uses signs to denote objects, and
how one can do this in a computationally meaningful and viable way.
Life will just not be that easy here.
Jon Awbrey
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