ONT Re: Apposite Purposes Of Logical Languages Objectified (APOLLO)
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Having acquired a bit more experience with the sorts
of relationships that are likely to come into play
between the objects of a logical language and the
varieties of syntactic forms that such languages
display, let us now return to the objectives
that I enumerated toward the beginning of
this investigation:
| 1. What is it that I expect a calculus to help
| me do in the case of a propositional domain?
|
| 2. What is it that I want to be able to do in
| regard to a given language of propositions?
|
| Given a propositional expression, I would like to be
| capable of quickly figuring out just about where it
| is in the lattice or partial order of propositions.
| This means finding the logical neighborhood of the
| expression in question, in the following senses:
|
| a. To recognize the logical equivalents of an expression.
| b. To know what expressions are above it in the lattice.
| c. To know what expressions are below it in the lattice.
| d. To recognize simpler expressions that approximate it.
A generic picture of the situation may be drawn like this:
| Object Domain Syntactic Domain
|
| o-----------o
| o~~~~~~~~~~~~~~~~~~/| s s s ... |\
| / \ / o-----------o \
| / \ / \
| / \ o-----------o \
| o~~~~~~~\~~~~~~~~~~~| s s s ... | \
| \ \ o-----------o \
| \ \ \ o-----------o
| \ o~~~~~~~~~~\~~~~~~~~| s s s ... |
| \ / \ o-----------o
| \ / \ /
| \ / \ o-----------o /
| o~~~~~~~~~~~~~~~~~~\| s s s ... |/
| o-----------o
|
| Figure. Objects Inducing A Sign Partition
The objects in the object domain, with all of its structure intact,
correspond to equivalence classes of signs in the syntactic domain.
Any language that is adequate at all to this object structure will
have a sufficient diversity of signs, generally speaking, far more
than enough, to fill out the parts of the requisite partition in a
way that echoes the structure of the object domain.
For example, in the case of the various languages that we have been
considering for ZOL(p, q), the languages Bin, Dec, Vec are tantamount
to sets of proper names, in so far as they have a single sign for the
object in question in each of their equivalence classes. In the case
of quasi-continuous sign systems like Cub, Ven, Ver, the question of
multiplicity depends on whether one reduces the continuum of tokens
according to their topological types before counting them or not.
However, a useful calculus will always afford a wide diversity
of ways to express the same object, and yet a variety that is
not so excessive as to obviate the recognition of the object.
The reason for this is not far to find, but I'm afraid that
it will have to keep until another time.
Jon Awbrey
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