ONT Re: Zeroth Order Theories (ZOT's)
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Example. Molly's World (cont.)
In preparation for a contingently possible future discussion,
I need to attach a few parting thoughts to the case workup
of Molly's World that may not seem terribly relevant to
the present setting, but whose pertinence I hope will
become clearer in time.
The logical paradigm from which this Example was derived is that
of "Zeroth Order Horn Clause Theories". The clauses at issue
in these theories are allowed to be of just three kinds:
| 1. p & q & r & ... => z
|
| 2. z
|
| 3. ~[p & q & r & ...]
Here, the proposition letters "p", "q", "r", ..., "z"
are restricted to being single positive features, not
themselves negated or otherwise complex expressions.
In the Cactus Language or Existential Graph syntax
these forms would take on the following appearances:
| 1. ( p q r ... ( z ))
|
| 2. z
|
| 3. ( p q r ... )
The style of deduction in Horn clause logics is essentially
proof-theoretic in character, with the main burden of proof
falling on implication relations ("=>") and on "projective"
forms of inference, that is, information-losing inferences
like modus ponens and resolution. Cf. [Llo], [MaW].
In contrast, the method used here is substantially model-theoretic,
the stress being to start from more general forms of expression for
laying out facts (for example, distinctions, equations, partitions)
and to work toward results that maintain logical equivalence with
their origins.
What all of this has to do with the output above is this:
From the perspective that is adopted in the present work,
almost any theory, for example, the one that is founded
on the postulates of Molly's World, will have far more
models than the implicational and inferential mode of
reasoning is designed to discover. We will be forced
to confront them, however, if we try to run Model on
a large set of implications.
The typical Horn clause interpreter gets around this
difficulty only by a stratagem that takes clauses to
mean something other than what they say, that is, by
distorting the principles of semantics in practice.
Our Model, on the other hand, has no such finesse.
This explains why it was necessary to impose the
prerequisite "object" constraint on the Log file
for Molly's World. It supplied no more than what
we usually take for granted, in order to obtain
a set of models that we would normally think of
as being the intended import of the definitions.
Jon Awbrey
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