ROBERT MARTY
Université
de Perpignan
EXTENDING CONCEPT LATTICES BEYOND ATTRIBUTES TO RELATIONS
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EXTENDING CONCEPT LATTICES BEYOND ATTRIBUTES TO RELATIONS |
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To that end we are going to generalize the notions of context and of lattice concepts such as they are defined by R. Wille [4] as follows: A context is a triple (G,M,1) where G is a set of objects,
M is a set of attributes, and I is a binary relation between G and
M indicating by DEFINITION 13. Let J and K be two posets and |
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The
morphisms
DEFINITION
14:. The triple If The notion of representation
context seems interesting as a theoretical framework for various
formalizations. For instance, if In the case of Touretzky's
semantic net (see Figure 5),
DEFINITION 15. If Now, for all sets A of
objects of DEFINITION 16. All pairs
Thus, a concept in Wille's sense is a semiotic in which every attribute is a sign of a set of objects and the intention of a concept is the set of attributes related to the concept. Similarly every object is signified by a set of attributes, and the extension is the set of objects related to the concept. By analogy we say that A is the extension of the semiotic (A, B) and B its intention. DEFINITION 17. Let
The set The notion of the representation
context can be adapted in order to respond to particular situations.
The most interesting cases seem to be when the ordered set Semantic networks can also
be interpreted with the help of representation contexts, The definition of our representation contexts raises the same questions as are raised by Wille's lattice- concepts: How do we determine the semiotics of empirical representation contexts? How do we describe the order graph on the set of the semiotics of a representation context in order to optimize the choice of the semiotics in particular examples (for instance, choice of notations or graphical conventions, conception of a signalling system, the representation of a problem with a view to its computerisation, etc.)? However, taking in account the possibility of the foliation of the set of the phenomenological morphisms we can reduce the general problems of the representation to a few underlying problems which are their corresponding expression in the different "sheets" which are associated with them. Thus it will be possible to categorize notational differences which depend on phenomenological differences in each problem. In this respect we should remind ourselves of D. J. Israel's remark: the kind of things and the qualities of things belong to different phenomenological categories. Therefore we can associate
with every category of phanerons the categories of phanerons obtained
by decomposing it and then assembling according to their category
the decomposition of their diagrams and their morphisms as previously
indicated. This being done for the Thus, the questions of the determination,categorization and the description of these semiotics come down to the study of the same questions in the cases of elementary representation contexts which are of one of the six following types:
where the index numbers
of the letters a and 0 indicate the type (arity) of the relational
structures and those of the letter These elementary representation contexts are naturally ordered so as to form a lattice. This order corresponds to the hierarchy of the "cenopythagorean categories." We obtain the lattice represented in Figure 6. Moreover, we have similar lattices with the semiotics of every elementary representation context,. Going back to semantic networks regarded as particular representation contexts we see that the introduction of the phenomenological aspect leads to a foliated conception (meaning that we can distinguish between different hierarchized levels which can be compared with geological strata) through the resulting distribution of the nodes and relations possibly reduced to their elementary forms between the different levels corresponding to the classes in Figure 6. |
Figure 6. The lattice of the different possible
species of "representation context" depending on the fundamental
categories (1, 2 or 3) of the entities on either side of our (Galois)
"representation relations." R. Wille's lattice concepts
are of type [2, 1] because they consist of sets of dyads objects
described using monad attributes. There are five other representation
possibilities, depending on the "arity" of the entities.
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As an example we will deal with Winograd's network,
(Figure 1) based on our analytical method and we would suggest a
few additions in order to ensure phenomenological harmony and consistency.
Our proposal is the diagram, Figure 7, in which the levels of the
elementary representation contexts are the planes and the relations
are dotted lines. |
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We consider that the representation contexts are determined and connected by means of social processes and agreements which govern all inter individual communications. This latter only actualizes the social consent about representation of things by other things. However, this is a process which changes at the same time as the real world changes. Many discussions and conflicts of a semantic order come from disagreements and changes which occur during this process. In other words, we communicate always within a framework imposed by society, but we can create new connections and/or new categories of phanerons which will or will not be incorporated in our culture. Nevertheless, most of the time, the social consent is sound and we communicate without difficulty. We find in the writings of Peirce the notion of "Commens" or "Cominterpretant," an entity which is required so that communication can take place, which is very close to this conception. In order to take these remarks into account, we can undertake our formal approach, which generalises the notion of the "fusion" of individual contexts proposed by Wille [22], to the fusion of representation contexts. In these formalisms, the phenomenological reduction can effectively contribute to the simplification of the description of the phenomena. |
that the object 
has the attribute 
. There is a natural Galois connection between G and
M (which we indicate with "prime" marks) defined by 
for 
,and 
for 
. A concept of the context (G, M,1) is a pair (A, B) with 
, 
A' = B, and B' = A. The set L(G, M, I) of the concepts of 
is partially ordered by the definition 
if and only if 
. 
is a complete lattice called the lattice-concept of (G,
M, I). Wille's concept lattices are described in his article in
this volume. From our standpoint that will mean considering each
attribute as representing all objects to which it is related.
and 
the corresponding categories of phanerons. 
and 
are said to be connected if, for at least 
there exists at least 
such that, if 
' is a phenomenological morphism between two objects 
and 
of D then there exists a phenomenological morphism 
between two objects 
of D' and two phenomenological morphisms 
and 
such that we have the commutative diagram: 
and 
are called connections between 
and 
and the set 
of these connections is called the "connecting set"
between 
and 
where 
and 
are categories of phanerons and 
a connecting set between them is called a "representation-
context." We call it a representation context because this
notion captures the substitution of a part of a phaneron of any
type by a part of a phaneron of another type. 
and 
are categories of phanerons of which all the objects are diagrams
reduced to one element, then the connecting set is a binary relation
between two sets which are named respectively objects and attributes
by Wille and our representation context is in accordance with his
definition of a context. 
is a part of a language in which objects are names and morphisms are
predicates we can to talk of "linguistic context of representation";
if 
is a set of dots and lines with rules of combination we can to talk
of "graphic context"; likewise for "pictorial context,"
"musical context," etc. 
is the set of all particular elephants, particular cars and particular
motorcycles provided with the complex of real relations existing
in the world between them (and present to Touretzky's mind). These
relations are monadic, dyadic and also triadic since Touretzky accepts
that the elephants are thinking animals. J is the underlying partial
order on this set (for instance Clyde < Royal Elephant because
there exists a phenomenological morphism of which the source is
Clyde into any one of the Royal Elephants). Between Elephants and
Motorcycles there is no phenomenological morphism. 
is Touretzky's net and the connecting set is constituted by the set
of phenomenological morphisms which, for instance, maps every elephant
onto the pair constituted by a dot and the word "Elephant,"
and maps also every phenomenological morphism between Royal Elephants
and Elephants onto the arrow linking the corresponding dots. It
maps every individual preference of every elephant relatively to
cars and motorcycles onto the triple of the words (elephant, car,
motorcycle) linked as in the network. This representation combines
graphical and linguistic representation contexts. 
is a representation context and if 
, every 
such that there exists a connection of 
connecting D and D' is a sign of D in this representation context. (This
definition agrees with Peirce's meaning of a "sign".)
, we define the subset A' of objects of 
which are signs of all objects of A and, for all subsets B of
we define the subsets B' of all objects of 
of which the objects of B are signs. The maps 
and 
form a Galois connection between the power sets of 
and 
. 
such that A = B' and B=A' are called semiotics of the representation
context. 
and 
be two semiotics of the same representation context, we define an order
of relationship for the set of semiotics of this representation
context by: 
if and only if 
and 
of the semiotics of the representation context 
is partially ordered by this relation 
, and we will say that the semiotic 
is better than the semiotic 
because it represents more objects using less signs. 
is a lattice or better, when it is a complete lattice with a
top, because in this latter case there exists a universal semiotic
which represents everything. 
being constituted by a unique object (a labelled graph
or hypergraph present to the mind of the reader) connected with
a category of phanerons
of which it must express some knowledge. For instance, in Winograd's
network (Figure 1), 
is a state of things present, any day, to Winograd's mind
concerning supposed physical entities (Kazuo, Fido) which are instances
of subconcepts (person, dog) of a superconcept (animal), facts concerning
these entities (to eat, to own, to be an ingredient) and laws governing
these facts. The connecting set 
contains two classes of elements: the first (graphical convention)
is a functor which maps the physical entities and their related
concepts onto the nodes of the graph on the one hand, and maps the
relations between entities, subconcepts and superconcepts onto the
arrows of the graph, on the other; the second (linguistic convention)
maps the same elements of 
onto the labels with the implicit rule according to which
the labels of the nodes occupy the corresponding place markers of
the 
predicates labels on the arrows which join them. 
and 
(since they are simply examples of diagrams) of a representation
context we now decompose the morphisms of 
: we will finally obtain in a natural way a "foliation" of
the original representation context into a poset of elementary representation-contexts.
These latter are
of six types (arities) according to the previous results. This property
is naturally transmitted to the "semiotics" of the elementary
representation contexts. 
the types of elementary morphisms. 
can be the designated and 
the designata, and 
is the representation relation itself. 
we have the concepts or monadic predicates (_ is a person, _ is a dog, ..
. ) and in this plane we have relations between subconcepts and
superconcepts; at the level [1,1] they are at the same time the
tokens of the monadic and dyadic concepts of the superior levels
and [3,2] so that the blanks of the latter are taken up by the
former. We have added at the level 
the general quality "carnivorous" which is implicit and at the
level [2,1] many tokens of the concepts of "cow" 
and of "meat of the cow" which take up the blanks of a dyadic
relation of ingredience because the existent Fido cannot feed himself
on the concept of meat. We notice that level [2] is missing because
its fundamental characteristic is actuality here and now. And the
same goes for level [3] of the triadic predicates because there
is no argument in this basic network. In the case of Touretzky's
network, we obtain the foliations of Figure 8. 
by which the structures in the net represent structures
in the world. As shown in Winograd's example they must be shared
by the utterer and the reader. Since phaneroscopy is the theory
of all that is present to an individual mind the first answer will
be necessarily of a sociological type considering these connections
as social institutions appealing to the dialecticized concept of
institution as described in [7].