You wrote:
''The ordering relationship and the definition of U is the critical issue,
in
terms of ontology construction and entailments. But we need to look closely
at what a Boolean lattice is and what is proposed as a lattice of theories.
What is the binary relationship in this lattice of theories?'' (01)
This is John's jurisdiction, not mine. If he insists on the idea, then
something must be behind. As a matter of fact, there are several levels of
the conceptual representations: a framework, or generic theory modeling the
properties and relationships of all things of a given domain (as Maxwell
equations); a specific theory or theoretical model representing some
features of some things of a given domain (like the accounts of specific
physical effects); a diagram or graph sketching the composition and
structure of a thing of a given domain; and a schema just listing the
specific properties of a thing in a given domain. I only can guess that he
is meaning a latticeordered structure of generic and specific theories.
Such a lattice of theories may constitute a collection of bodies of
knowledge (as a partially ordered sets of statements) with their reference
classes, primitives, and axioms, all hierarchically ordered by the set
inclusion relations.
To be on the safe side, I would prefer to speak of the lattices of
realities, properties, or relationships formally modelled as the lattices of
concepts, predicates or the lattices of meanings. Then you can put up a
Boolean lattice for the whole class of the sets of realities, where the
least element is the null individual 0, Nonbeing or Nothing, the last
element U is the World as a whole, Being, Thing, Entity, or the Universe.
And where the standard equalities for unary and binary operations can be
held such as associativity, commutativity, absorption, distributivity,
inversion and complementation. Depending on the sort of realities
[substances, states, changes, or relationships], the entities will be
ordered either with respect to the partwhole relationships or wrt causal
connections, or both. So the lattice theory as a generalization of set
theory and the core of mathematical logic having received a real world
interpretation can constitute the unified framework ontology or universal
formal ontology (UFO) where Causal Mathematics (see the introduction to
''Standard Ontology for Machines and People) will play a basic role.
Your persistence on the ontological interpretation of the lattice theory has
a good reason; for the ontology of the lattice theory is what the modern
science and engineering are in need at the first place, since it gives a
universal mathematical theory concerning the major properties of the world;
while the reference class of the lattice of theories is just a set of
conceptual bodies. (02)
Azamat (03)
<<I changed the name of this thread from (04)
Theories, Models, Reasoning, Language, and Truth (05)
because the key concern/excitment I have is that the pure mathematical
concept of a Boolean lattice might be revealed in a simple fashion. (06)
Your note opens onto such a simple explanation. (07)
http://colab.cim3.net/forum/ontacforum/200512/msg00127.html (08)
Lattice theory in physics, particularly in thermodynamics, does not have the
same properties as does the Boolean lattice (abstracted from the ( U,
subset, smallest element, largest element ) ) . Lattice theory (as in
thermodynamics) is something I know a little about since I published in this
area as early as 1987. (09)
The ordered quadruple designation, you use, for the Boolean algebra is
incomplete, but this is not important technical issue. (010)
The ordering relationship and the definition of U is the critical issue, in
terms of ontology construction and entailments. But we need to look closely
at what a Boolean lattice is and what is proposed as a lattice of theories.
What is the binary relationship in this lattice of theories? (011)
It seems to some of us that one needs to be able the talk separately about
the definition of the elements of an ontology, and that these elements
should be in all cases "concepts". (012)
The entailment of a set of concepts is then in two parts: (013)
inferential entailment: how the concepts are used by some type of machine
or formal inference (014)
structural entailment: how the concept referent sits inside of specific
situations. (015)
So if one starts with a "universal set" as a set of concept representations,
or taxa in structural bioinformatics, we can then impose an ordering
relationship to organize this universal set into a tree like structure
(taxonomy). This does not produce a Boolean lattice, it just produces a
tree (a specific type of graph). (016)
So how are lattice of theories developed?>> (017)
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