SUO: Re: Comments on Whitten's Starter Ontology
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Chris, David,
I will not quote your dialogue since it will be some time before
I can work through it, but I do have some previously worked-out
thoughts on the relationship between assertions and expressions,
at least, in the sorts of languages that one intends to employ
for the purpose of describing some object domain. In addition,
this approach to the question helps to smooth out some of the
wrinkles between the descriptive and the functional properties
of the logical expressions being used. This is a little bit
reminiscent of the declarative/procedural dimension of choice,
since it is the functional properties of expressions that are
especially pertinent when you turn to the task of implementing
the language as a component of an active system -- especially
if you follow a functional programming paradigm as a surrogate
model for implementing procedures.
Disclaimer: I can only vouch for the servicability of this POV
at the level of propositional calculus (AKA sentential logic or
"zeroth order logic" (ZOL)). It's not much, but I always find
that a surprising fraction of the everyday problem complexity
is bound up at this level, so any easing of the bottleneck at
this level will usually help a lot with all the other tasks.
From this POV, propositional expressions are just notations
for boolean-valued functions f : X -> B, where B = {0, 1}.
Thus, the expression denotes a function.
As it is, such an f is just a function.
The distinction between assertions and expressions can now be treated
as not so much of a "logical" distinction as a "pragmatic" distinction,
in other words, not so much a question of what the expression denotes as
a question of how its denotation, namely, the function, is being used at
the particular moment in question.
When you assert an expression this is tantamount to asking your interpreter
to shift into a contemplation of the subset of X what is variously called:
1. The "fiber of truth" -- I did not make this up! --
in the function f, namely, f^(-1)(1) ç X,
2. The "antecedent", "level set", or "pre-image" of truth,
or of the functional value 1, under the function f,
3. The "indicated set" of the "indicator function" f.
In sum, assertion is a directive to "consider the models"
of the expression being asserted.
Anyway, this is how I more or less think about it.
Jon Awbrey
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