Paul wrote: ''But
... I do not understand this notion of a lattice of
theories''.
Paul,
In your several messages you have been asking the
same but good questions. Below I paste a passage from the book ''Standard
Ontology for Machines and People'' regarding the nature and ontological
relevancy of lattice theory. Hope it will a bit clarify the mistery surrounding
mathematical lattices being most fundamental to UFO.
''The lattice
theory with set theory involve such effective abstractions as the universe, class, set, property, object, relation, partial orderings, order preserving or isotonicity,
universal bounds, duality, partially ordered sets, and lattice. The Boolean
algebra introduces the notions of isomorphism or bijection, Cartesian
(direct) product, Boolean polynomial;
the relation algebras - binary
relation and composition; the category theory - the notions of objects, morphisms (or maps or
transformations), composition of morphisms and the duality principle.
The main
concern of the lattice theory is an abstract structure L of families of subsets
X, Y, Z, ...of a given set, class, or aggregate (the universe) U. Ordered by an
inclusion relation £, it is marked by having a
largest element, denoted by the intersection of subsets X Ù Y, and a smallest element,
denoted by the set-union operation X Ú Y: L = < U,
Ù, Ú, £ >. The inclusion
relation of holding £ is characterized by the
laws of reflexivity X£ X, transitivity, if X
£ Y and Y£ Z, then X
£ Z, and antisymmetricity,
if X£ Y and Y£ X, then X =Y. All lattices
share the properties of idempotency, associativity, commutativity, and
absorption. What is more important, these algebraic structures are subject to
dualization: if the union Ú is replaced by the
intersection Ù and vice versa and the
inclusion relation £ is interchanged with the
opposite relation ³, the lattice will keep its
original structure, although turning upside-down.
In a general form, the lattice structure L is an ordered
quadruple of the universal aggregate U, a binary operation of relation
°, the least or bottom
element 0, the null thing as well as the last, universal, top element, which is
the world W: L=< U, °, 0, W > .
Accordingly, the models
of the abstract structure may be as diverse as the types of
relation:
|
the partial
orderings;
the implication
relation;
the subsumption
relation;
the part-whole
relation;
the cause and effect relation.
Basing
on a selected type of relations, one can realize the models as diverse
as
partially ordered
sets;
deductive logical
systems;
classificatory
taxonomies;
mereology;
natural, mental or social
systems. |
|
By bringing
meaning, semantics, and content into the formal structure through specific
interpretations, the lattice structure may serve for whatever model of reality
one would like to prefer: the totality of things, facts, objects and events, the
totality of properties, etc. For instance, the universal class U may be
represented as a collection of set variables, where its variables X, Y, . . . ,
range both over sets and individuals. The structure of reality is to be modeled
as a lattice produced by substantial individuals ordered by the part-whole
relation. This structure will culminate in a world individual and bottom with a
least ontic individual. As a result, a
least unit or null element in such the lattice of individual entities is
supposed to form the bottom level or
the ultimate reality of the ontological structure. The smallest element in such
a model of the world ontology of facts is supposed to be the individual, the indivisible, the simple,
the particular, the atom, the point, the element, the fact, the item, the
single, or the particle. By the agency of the binary operations as
juxtaposition or superposition, the multitude of individuals forms the second grade or scale of reality. It
includes all kinds of wholes or complex unities: the composition, the aggregation, the
aggregate, the assemblage, the group, the association, the set, the number, the
collection, the mass, the body, the substance, the matter, the quantity, the
class, the compound, the mixture, or the system. The entities are here
ordered by the part-whole relation corresponding in set theory to the class membership relation Î of an individual and its
collection. The third grade of the
physical world is made up of different bodies, masses, complexities,
compositions, aggregates, aggregations, collections, systems, and wholes
composed of individual parts. This level is composed of a number of large
collections of collections ordered by the class
inclusion relationship Í. At the apex, the
universal element is the totality of all individuals presented as the class of
all classes of individuals. It is the world presented as the many rather than
the one, as the total sum of particulars, as a multitude of objects, facts,
events, and processes constrained by particular causation''.
Regards,
Azamat Abdoullaev
EIS Ltd
Pafos, CYPRUS
http://www.eis.com.cy-----
Original Message ----- From: "Paul S Prueitt"
<psp@xxxxxxxxxxxxxxxxxx> To: "ONTAC-WG General Discussion"
<ontac-forum@xxxxxxxxxxxxxx> Sent: Tuesday, December 20, 2005 10:39
PM Subject: RE: [ontac-forum] Theories, Models, Reasoning, Language, and
Truth > > > The lattice that you are refering to is
a construction, or potential > construction. In this way it is
similar to the set of counting numbers. > The set of positive integers can
be regarded as existing only as a > potential... given that there is no
way to get the entire construction put > somewhere. But this is a
mute point, as you point out. > > > In the counting numbers
one has a total, sometimes called linear, order. > This order extends to
the real numbers but not the complex numbers. > > The lattice can
be a structure with a partial order having a minimal element > and a
maximal element. > > An example of this is the set of subsets can
be organized in such a fashion. > John, is your , and Tarski's, notion of
a lattice of theories such a > structure? This is what I just do not
see. How do you or Tarski compare to > theories? Suppose that
all theories are the special ones that are > formalized to a degree
necessary. > > Lattice geometries are discussed at > >
http://www.hermetic.ch/compsci/lattgeom.htm > > > and I may
say that in 1987 I published a paper on spin glass lattices in the >
journal "Complex Systems". I recognize how really interesting it is to
say > that there is a relationship .... > > > In my
recent work I develop the notion of a non specific relationship > (without
order) as a means of mapping data structure into computer memory. >
> But ... I do not understand this notion of a lattice of
theories. > > > > >
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