Our fundamental hypothesis consists precisely in interpreting
the eidetic structure obtained in the previous section in a phenomenological
sense.
Figure2. An eidetic structure for reality lies behind different
perception of it.
Figure3. The letters "i" represent relations of inherence
(of the color to matter) and the letters "a" relation
of "adjacence" between two rectangles which have a common
side
To this end it will suffice to consider that an object is "present
to the mind" if and only if this mind "forms" its eidetic structure.
Indeed, since this structure is included (that is to say present)
in each perceptual judgement (mental semantic network) of every
member of the community, we automatically obtain its "presence
to the mind." For every perceiving entity there is a one to
one correspondence between perceived objects of the world and a
set of connected relational structures of . This point of view agrees perfectly with the principle of the "identity
theory" enunciated by Jackendoff : "the distinction between mental and physical events is purely
one of mode of description."
The morphisms between eidetic structures are naturally interpreted
as relations between world objects. Finally this modelization consists
in substituting for the exterior world, considered as
a "vague" category with objects and relations between
these objects, tile precise categorywith the aid of a "metaphorical functor" supported by the mechanism
of the perception of the exterior world itself. This functor is
created, so to speak, from the very fact of the ierception of objects
and of their relations.
The objects of the
world are, most of the time, present to the mind simultaneously
with ether objects, if only through their own constituent parts
with which they are related. In other vords the objects participate
in what Peirce calls a "phaneron" (synonymous for him
with "phenomenon"). The phaneron is "the collective
total of all that is in any way or in any sense present o the mind,
quite regardless of whether it corresponds to any real thing or
not" (8, CP 1 2841. recording to the previous hypothesis, it
follows that a phaneron can be produced by the perception of a set
of related objects of the exterior world, or internally constituted
by the mind which associates their eidetic structures already memorized,
or by a combination of both. These eidetic structures are combined
by means of their possible morphisms. (Changeux speaks of "associative
properties of mental objects.") Again using Category Theory,
we will build the formal framework 'or dealing with the phaneron
(phenomenology), that is to say the structure of all that can be
present to the mind.
Before that, we
will establish a reduction theorem and give a few definitions (in
keeping with ;he philosophical speculations and the formal intuitions
of C. S. Peirce) which permit us to describe more easily the phaneron
and the relations between phanerons.
DEFINITION 7. Let be a relational structure of type n on base set X and a relational structure of type m on the same base set; we define
the relative product of (X, ) by , and we call the relational structure of type n + m- 2 defined by the following subset
of the Cartesian product set of :
such that there exist
such
that and .
DEFINITION 8. A relational structure is relatively decomposable on a set of relational structures if is the relative product of a set of relational structures of
K
DEFINITION 9. A relational structure of type n is relatively reductible on k if and only if is relatively decomposable on a set of relational structures on K of type smaller than n. If a relational
structure is not relatively reductible on K it is relatively irreductible
on K.
It is easy to establish that
every relational structure of a type equal to 3 or smaller than
3 is relatively irreductible within any domain.
THEOREM 1. Let be a relational structure of type and a set of the same cardinality as ; we consider and the relational structure to be of type n defined on by the n tuples of a. Then is relatively reductible on a set of relational structures of
type 3.
PROOF (n = 4): Let a bijection of onto . We call the corresponding element of . We define two triadic relations on by:
.
We form the relative product and let .
There exists such that and and there are and .
Clearly, since is a bijection and , we get , , , . Hence .We conclude that ; is obvious and we have .
There is no difficulty in extending this
proof to considering the triadic relations defined as follows:
where . The method used above is in accordance with the principles enunciated
by C. S. Peirce in [15].
This reduction thesis6 proved differently
in terms of extensional logic by H. G. Herzberger and in terms of intentional logic by R. W. Burch seems to be in opposition to a well known result of W. V. O.
Quine (18) according to which every n adic predicate can be reduced
to a combination of dyadic predicates. This discussion has no place
in this work, but we note simply that it is, in our opinion, a difference
of viewpoint as to the importance accorded to the notion of "teridentity"
(see [17,19] and the article by R. W. Burch in this volume).
Now, the reduction theorem allows us
to describe every relational structure on X of type n by means of
the relative product of relational structures of type 1, 2 or 3,
called elementary relational structures. We call "monads"
the elements of relational structures of type 1 (1- tuples); we
call "dyads" the ones of type 2 (2- tuples) and "triads"
the ones of type 3 (3- tuples). Each monad corresponds to a simple
"quality of feeling," each dyad to an existent individual
or a fact, and each triad to a "concept, law or something expressible
by universal proposition." This is the empirical decomposition
of the phaneron into indecomposable elements many a time described
by Peirce.
If is an eidetic structure of an object of type by introducing enough new individuals like for every such that , we can consider that every object of the world is represented
in our model by a relational structure of that is to say a relational structure of type {1, 2, 3}. But, considering
that the perceptual judgement which assembles n qualities
of feeling of an n- tuple assembles a fortiori all the k-
tuples (n < k) obtained erasing n- k of
these qualities of feeling, we define a special relational structure
of type { 1, 2, 3} as follows:
DEFINITION 10. A relational structure of type {1, 2, 3} on the set X with , , is a phenomenological structure of type if the following conditions are fulfilled:
(i) if, then ,, ,,
(ii) if , then and
By combination of
(i) and (ii) we have:
(iii) if , then , and
PROPOSITION 1. The relations R defined by the above condition (i) between and , R defined by between and , defined by (iii) between and , define a category
called of which objects are the , and and the morphisms and the identities on , and
Indeed we have . Henceforth, we will represent the phenomenological structure
associated to an object of the world by where X is a set and the category defined in Proposition 1. We recall that, if X and Y are two
sets, a functional correspondence between X and Y is a relation
such that if , and then
DEFINITION 11. Let
and two phenomenological structures and a functional correspondence between X and Y; we consider the following sets:
and
the corresponding relations (called elementary morphisms)
is called a phenomenological
morphism if determines a functor of the category into the category by means of the relations .
For
instance, the functional correspondences r are such that we have
the following commutative diagrams which are phenomenological morphisms:
PROPOSITION 2.
There are exactly 10 phenomenological morphisms between two phenomenological
structures; these phenomenological morphisms are ordered in a structure
of lattices by the natural transformations of functors.
Indeed
it will suffice to write the 10 functors and to note the pairs of
these functors for or which a natural transformation is defined.
PROPOSITION 3.
The set of the phenomenological structures provided withthe
phenomenological morphisms is a category called the Phenomenological Category of
the Objects PHo.
Indeed the composition of phenomenological
morphisms is defined by the composition of the functional correspondences
combined with the composition of the functors.
The interpretation of these definitions
and results are as follows: the phenomenological structures represent
the objects of the world (present to the mind), whereas the phenomenological
morphisms represent the relations between these objects, that is
to say the modes
of being. There are six fundamental classes
of modes of being: and corresponding respectively to Authentic Thirdness, Degenerate
Thirdness at the first degree, Degenerate Thirdness at the second
degree; and corresponding respectively to Authentic Secondness and degenerate Secondness;
corresponding to Firstness.
These modes
of being correspond with the so called "cenopythagorean categories"
of Peirce and their degenerated forms (see
[20]). Moreover these results agree perfectly with the results of
Figure 4. Some possible triadic relations between an elephant,
a car and a motorcycle.The one at left, involving thought, is the
only one requiring true Thirdness.
Figure 5. Elephants prefer cars to motorcycles. But elephants
do not prefer compact cars to Fred's Harley
R. W. Burch ( and in this volume). Moreover, if is not empty, then with and is not empty; therefore the presence in a phenomenological structure
of a mode of being corresponding to implies the presence of a mode of being corresponding to . In fact the modes of being are organized in a lattice of six
elements and this framework permits us to develop a complete formalization
and various extensions of the Peircean semiotic [7].
For example, referring to a net
proposed by Touretzky [21, p. 199 which will be considered below,
we have for the triadic relation "1 prefers 2 to 3" (see
the left side of Figure 4).
Now we note in Figures 4 and 5
that the feelings of the elephants for the cars are an independent
reality of their feelings for the motorcycles. It is realized in
acts (in the physical world including the mind's acts, i.e., Secondness)
and represented in Figure 4 in an iconic linguistic manner (i.e.,
physical relation of proximity with a graphical link between the
common nouns "Elephant" and "Car"). However
we must note, with respect to this example, that we are obliged
to use a linguistic representation, that is to say that we are speaking
of the phenomenology within a special (linguistic) phenomenology.
The notion of representation context (see below) will allow us to
take into account this difficulty.
Now we have something corresponding
to a phaneron considered as a collective totality of objects present
to the mind with relations between these objects. In our model,
the phaneron will be represented by a `diagram'in the phenomenological
category of objects PHo.
DEFINITION 12. Let J be a poset considered asa category; wecall "category of phanerons
of type" the category
of which objects arediagrams of scheme J on PHo,
that is to say covariant functors , and the morphisms are natural transformations of functors.
Remember that a natural transformation
of functors is a family of morphisms such that, for each morphism of , we have the commutative diagram:
We have moved up another level.
For a suitable poset J, we can obtain all possible forms of configurations
of objects which can be present to any mind, that is to say all
abstract forms of phanerons. These forms are determined a prioriby
the forms of the perception, present perceptions or previously memorized
perceptions.
Now, we are able to approach the
phenomenology of representation
in formal terms.
In a representation one collective totality stands for another,
that is to say that one phenomenon issubstitutedfor another. Therefore, in our model a diagram is substituted
for another. The complex relations that these two totalities can
have in the exterior world are clearly represented by the corresponding
connections of diagrams. A representation
theoryisconsequently
possible using the categories of phanerons . In this way the problem of the representation of knowledge
by means of semantic networks can be reduced, because we have the
possibility of "foliation" by means of our elementary
sub networks.
5This hypothesis is also considerably substantiated
by neurophysical considerations. Indeed the "mental object"
is identified by J. P. Changeux [13] with the physical state created
by the electrical and chemical activity of large "assemblies" of neurons, mathematically described by a graph, discrete,
close and autonomous. Now, each neuron is formally equivalent to n-adic relations between the other neurons belonging to
the assembly with which it is linked. Therefore, it seems right
to consider that this assembly is, formally a connnected relational
structure.
6The interest of a reduction thesis lies not only
in the result and its consequences but also in the mechanism of
proof based on the addition to every n tuple of of a new element which corresponds to the "quality of feeling"
of the n-tuple itself. This element can be interpreted as the partial
determination of the mind which precisely constitutes this n- tuple
in a greater entity in the perceptual judgement. This establishes
a correspondence between the "external" relation structure
(built by the community) and the "internal" structure
in which all adicities (arities) have been increased by 1. This
correspondence is defined as follows:
Moreover, this interpretation allows us to conceive the
external structure as a cause of the internal structure in a dialectic
process (see). Then, eidetic structures can be interpreted as causal structures
assigned to external objects the forms of which are imposed by the
specific mechanisms of the perception. This remark also justifies
the removal of the division between an external category and an
internal category, the former becoming a subcategory of the latter.
Therefore, we can consider that the external category contains objects
without physical materiality.
7Pytagoreans, followers of Pythagoras, assigned deep
mystical significance to certain low integers. Peirce realised that
he was doing the same!
if and only if this mind "forms" its eidetic structure.
Indeed, since this structure is included (that is to say present)
in each perceptual judgement (mental semantic network) of every
member of the community, we automatically obtain its "presence
to the mind." For every perceiving entity there is a one to
one correspondence between perceived objects of the world and a
set of connected relational structures of 
. This point of view agrees perfectly with the principle of the "identity
theory" enunciated by Jackendoff 
: "the distinction between mental and physical events is purely
one of mode of description." 
with the aid of a "metaphorical functor" supported by the mechanism
of the perception of the exterior world itself. This functor is
created, so to speak, from the very fact of the ierception of objects
and of their relations. 
be a relational structure of type n on base set X and 
a relational structure of type m on the same base set; we define
the relative product of (X,
) by 
, and we call 
the relational structure of type n + m- 2 defined by the following subset
of the Cartesian product set of :
such that there exist 
and 
. 
is relatively decomposable on a set 
of relational structures if 
is the relative product of a set of relational structures of
K
of type n is relatively reductible on k if and only if 
is relatively decomposable on a set 
of relational structures on K of type smaller than n. If a relational
structure is not relatively reductible on K it is relatively irreductible
on K.
be a relational structure of type 
and 
a set of the same cardinality as 
; we consider 
and the relational structure 
to be of type n defined on 
by the n tuples of a. Then 
is relatively reductible on a set of relational structures of
type 3.
a bijection of 
onto 
. We call 
the corresponding element of 
. We define two triadic relations on 
by: 
. 
and let 
. 
such that 
and 
and there are 
and 
. 
is a bijection and 
, we get 
, 
, 
, 
. Hence 
.We conclude that 
; 
is obvious and we have 
. 
considering the 
triadic relations defined as follows:
. The method used above is in accordance with the principles enunciated
by C. S. Peirce in [15].
and in terms of intentional logic by R. W. Burch 
seems to be in opposition to a well known result of W. V. O.
Quine (18) according to which every n adic predicate can be reduced
to a combination of dyadic predicates. This discussion has no place
in this work, but we note simply that it is, in our opinion, a difference
of viewpoint as to the importance accorded to the notion of "teridentity"
(see [17,19] and the article by R. W. Burch in this volume).
is an eidetic structure of an object of type 
by introducing enough new individuals like 
for every 
such that 
, we can consider that every object of the world is represented
in our model by a relational structure of 
that is to say a relational structure of type {1, 2, 3}. But, considering
that the perceptual judgement which assembles n qualities
of feeling of an n- tuple assembles a fortiori all the k-
tuples (n < k) obtained erasing n- k of
these qualities of feeling, we define a special relational structure
of type { 1, 2, 3} as follows: 
of type {1, 2, 3} on the set X with 
, 
, 
is a phenomenological structure of type 
if the following conditions are fulfilled:
, then 
,, 
,, 
, then 
and 
, then 
, 
and 
defined by the above condition (i) between 
and 
, R 
defined by 
between 
and 
, 
defined by (iii) between 
and 
, define a category 
of which objects are
the 
, and 
and the morphisms 
and the identities on 
, and 
. Henceforth, we will represent the phenomenological structure
associated to an object of the world by 
where X is a set and 
the category defined in Proposition 1. We recall that, if X and Y are two
sets, a functional correspondence between X and Y is a relation
such that if 
, and 
then 
and 
two phenomenological structures and 
a functional correspondence between X and Y; we consider the following sets: 
is called a phenomenological
morphism if 
determines a functor of the category 
into the category 
by means of the relations 
. 
and 
corresponding respectively to Authentic Thirdness, Degenerate
Thirdness at the first degree, Degenerate Thirdness at the second
degree; 
and
corresponding respectively to Authentic Secondness and degenerate Secondness;
corresponding to Firstness.
of Peirce and their degenerated forms (see
[20]). Moreover these results agree perfectly with the results of
and in this volume). Moreover, if 
is not empty, then 
with 
and 
is not empty; therefore the presence in a phenomenological structure
of a mode of being corresponding to 
implies the presence of a mode of being corresponding to 
. In fact the modes of being are organized in a lattice of six
elements and this framework permits us to develop a complete formalization
and various extensions of the Peircean semiotic [7].
" the category
of which objects are diagrams of scheme J on PHo,
that is to say covariant functors 
, and the morphisms are natural transformations of functors. 
is a family of morphisms 
such that, for each morphism 
of 
, we have the commutative diagram: 
. In this way the problem of the representation of knowledge
by means of semantic networks can be reduced, because we have the
possibility of "foliation" by means of our elementary
sub networks.
of a new element which corresponds to the "quality of feeling"
of the n-tuple itself. This element can be interpreted as the partial
determination of the mind which precisely constitutes this n- tuple
in a greater entity in the perceptual judgement. This establishes
a correspondence between the "external" relation structure
(built by the community) and the "internal" structure
in which all adicities (arities) have been increased by 1. This
correspondence is defined as follows:
). Then, eidetic structures can be interpreted as causal structures
assigned to external objects the forms of which are imposed by the
specific mechanisms of the perception. This remark also justifies
the removal of the division between an external category and an
internal category, the former becoming a subcategory of the latter.
Therefore, we can consider that the external category contains objects
without physical materiality.