SUO: Re: Examples! Examples! Examples!
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EEE. Note 31
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Let us pause for a moment and look at what we've been doing from
a semiotic point of view. A semiotic process is a transformation
of signs. One very often views it by picking a representative sign
and following its "curve", "orbit", or "trajectory", as it is often
called, through a sequence of changes under the continuation or the
iteration of the transformation in question. A very important type
of semiotic process is one that takes place in a partitioned space
of signs and where every orbit remains within a single part of the
partition. In this case, we say that the semiosis "preserves" or
"respects" the partition or its associated equivalence relation.
Figure 19 shows how these ideas apply to the present Example.
o-------------------------------o------------------------------------------o
| Object Domain TLC^ ~=~ 2^TLC | Language Domain L(!TLC!) |
o-------------------------------o------------------------------------------o
| |
| o---------------o |
| /| (()) |\ |
| 1 / | (!a! (!c!)) | \ |
| o~~~~~~~~~~~~~~~~~~~~~~/~~| (!c! (!a!)) | \ |
| / \ / | ((!a!),(!c!)) | \ |
| / \ / | ... | \ |
| / \ / o---------------o \ |
| / \ / \ |
| / \ / o---------------o |
| / \ / | !a! | |
| / \ / | !c! | |
| / !a! o~~~~~~/~~~~~~~~~~~~~~~~~~| BE<!a!> | |
| / / / | BE<!c!> | |
| / / / | ... | |
| / TLC^ / o---------------o o---------------o |
| / / | (!a!) | / |
| / / | (!c!) | / |
| (!a!) o~~~~~~~~~~~~~~~/~~~~~~~~~| BE<(!a!)> | / |
| \ / | BE<(!c!)> | / |
| \ / | ... | / |
| \ / o---------------o / |
| \ / \ / |
| \ / \ o---------------o / |
| \ / \ | () | / |
| \ / \ | !a! (!c!) | / |
| o~~~~~~~~~~~~~~~~~~~~~~\~~| !c! (!a!) | / |
| 0 \ | (!a! , !c!) | / |
| \| ... |/ |
| o---------------o |
| |
o--------------------------------------------------------------------------o
Figure 19. Lattice of Propositions Inducing a Partition of Sentences
The Figure shows a sample of four object elements in the lattice TLC^
and four logical equivalence classes of propositional expressions in
the cactus language L(!TLC!). () and (()) are the usual appearances
of the constant false and the constant true proposition, respectively.
BE<!q!> is a special kind of "boolean expansion" of the expression !q!.
This is the type of normal form, akin to the disjunctive normal form,
that we take as the canonical equivalent of the given expression !q!,
and it constitutes the output of the Model function in Theme One.
As usual in these types of situations, there are an infinite number
of sentences in L that belong to each logical equivalence class and
that correspond to each element of the object lattice.
The process by which we passed from the TLC axiom !a! to its normal
form BE<!a!> was a graphical transformation that took the parse graph
of !a! through a sequence of logical equivalents, finally arriving at
BE<!a!>, a traversal and a projection of which gave the corresponding
outline of models. Likewise for !c! and BE<!c!>. Finally, we tested
the equivalence of !a! and !c! by computing the BE of !a! (!c!) and
!c! (!a!), which were found to be in the logical equivalence class
of (), that is, the proposition with no models. This means that
their negations (!a! (!c!)) and (!c! (!a!)), respectively, are
in the logical equivalence class of (()), that is, the valid
proposition. But these latter expressions are the cactus
forms for !a! => !c! and !c! => !a!, respectively, and
so we have come to the conclusion that !a! <=> !c!.
Jon Awbrey
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