Université de Perpignan




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 with the 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 as a category; we call "category of phanerons of type" 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.

     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 priori by 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 is substituted for 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 theory is consequently 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!