ONT Re: Zeroth Order Theories (ZOT's)
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We are contemplating the sequence of initial and normal forms
for the Consat problem and we have noted the following system
of logical relations, taking the enchained expressions of the
objective situation o in a pairwise associated way, of course:
Logue(o) <=> Model(o) <=> Tenor(o) => Sense(o).
The specifics of the propositional expressions are cited here:
http://suo.ieee.org/ontology/msg03722.html
If we continue to pursue the analogy that we made with the form
of mathematical activity commonly known as "solving equations",
then there are many salient features of this type of logical
problem solving endeavor that suddenly leap into the light.
First of all, we notice the importance of "equational reasoning"
in mathematics, by which I mean, not just the quantitative type
of equation that forms the matter of the process, but also the
qualitative type of equation, or the "logical equivalence",
that connects each expression along the way, right up to
the penultimate stage, when we are satisfied in a given
context to take a projective implication of the total
knowledge of the situation that we have been taking
some pains to preserve at every intermediate stage
of the game.
This general pattern or strategy of inference, working its way through
phases of "equational" or "total information preserving" inference and
phases of "implicational" or "selective information losing" inference,
is actually very common throughout mathematics, and I have in mind to
examine its character in greater detail and in a more general setting.
Just as the barest hint of things to come along these lines, you might
consider the question of what would constitute the equational analogue
of modus ponens, in other words the scheme of inference that goes from
x and x=>y to y. Well the answer is a scheme of inference that passes
from x and x=>y to x&y, and then being reversible, back again. I will
explore the rationale and the utility of this gambit in future reports.
One observation that we can make already at this point,
however, is that these schemes of equational reasoning,
or reversible inference, remain poorly developed among
our currently prevailing styles of inference in logic,
their potentials for applied logical software hardly
being broached in our presently available systems.
Jon Awbrey
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