Université de Perpignan




The previous sections set out the basis of a representation theory which is phenomenologically coherent. Sticking to the representation of individual objects we find a framework for the development of a theory of signs in the classical sense. We have proposed in L'Algèbre des Signes [7]such a theory in which we have emphasized the bonds between psychological and sociological phenomena, starting from the central Peircean notion of  "interpretant" and formalizing one of its definitions of the sign according to "a sign is a medium for the communication or extension of a Form (or feature)." Our method consists in describing the "travel" of the eidetic structure of the object represented, partially incorporated in the representing sign and rebuilt by the mind into this initial form by an inferential process (named "semiosis"). Then, the application of the phenomenological distinctions (see Section 4) allows us to show that the Peircean taxonomies of signs are organised by lattice structure and beyond this to create new notions and new lattices. We obtain [23] the lattice of the ten classes of the interpreted triadic signs, by duality, the lattice of the ten classes of the uttered triadic signs (named cosigns), the lattice of the fiftyclasses of elementary communications (defined as a pair: cosign/sign and classified according to the compatibility of phenomenological categories), and the lattice of the twenty-eight classes of  hexadical signs8. Every one of these lattices can be regarded as a guide for inferences in the meaning process.

Figure 8. Our "foliation" of Touretzky's network.


Figure 9. The marks in the columns [1], and [3,1] indicate the modes of being of the elements represented by the words of the first column. In the 4th column the dyadic predicates appear and the triadic in the 5th column [3]. These predicates are represented by vertical lines in their column. Every element of the first column is linked by a horizontal line with the blanks of the predicates which it fulfills and picked out by one of the numbers 1, 2 or 3. The predicates appearing by vertical dotted lines are implicitly presupposed by predicates of superior levels. For example, "elephants prefer cars to motorcycles" presupposes "elephants love cars" and "elephants love motorcycles".

Moreover, still using elementary Category-Theoretic notions of product and coproduct of a diagram, we have built a methodology for analysi sand synthesis of complex signs. The result is a diagram which shows the manner in which the elementary signs are algebraically combined with a view to producing global meanings.

The power of these formalizations, in so far as they become integrated in Artificial Intelligence, lies in the possibility of going beyond the IS-A or subsumption lattices or posets for inference which are classically used in semantic nets. Such lattices are complex and require considerable storage and processing. If we can make a foliation of the lattices into sub lattices we gain greatly because the program can concentrate on inference within the sub lattice without having to worry about cross links to other lattice levels. Presumably our method provides a suitable framework for introducing relational subsumption in semantic networks using the generalization hierarchy of relations which forms its own lattice/poset separate from but "governing'the concept lattice/poset by means of the families of morphisms Thus, we can obtain a new (hypergraph) knowledge net language allowing automatic treatment of entities which are undifferentiated by current A.I. programs. Programming methods generalizing the methods used for building the Galois lattice of a binary relation between two finite sets [24-27] would be the kernel of more practical future work.

We conclude emphasizing that the introduction of the phenomenological element to the theory of semantic networks produces problems which are not perhaps conducive to immediate computerisation. However, it brings to light an epistemological order which concerns nodes as well as relations which might prove to be incontrovertible. The difference between a foliated network and a non foliated network is the same as the difference between the doctrine of Peirce and the doctrine of Kempe discussed in [8, CP 3-423]. This difference is expressed by Peirce as follows (the diagrams of Kempe consist only "in spots and lines between pairs of spots"):

For in the first place, it is remarked that Mr. Kempe's conception depends on considering the diagram purely in its self contained relations, the idea of its representing anything being altogether left out of consideration; while my doctrine depends upon considering how the diagram is to be connected with nature."

Finally, our nets are foliated according to their mode of connection with the state of things that they represent.

8 The hexadic sign is obtained by means of new empirical distinctions within the triadic sign. Peirce distinguished the Dynamical Object (the reality at the origin of the sign), the Immediate Object (this reality such that represented in the sign), the Immediate Interpretant (the feeling produced by the sign), the Dynamical Interpretant (the physical or mental effect produced by the sign) and the Final Interpretant  (the habit, law or concept reactivated by the sign). The internal determinations are the following: determines which determines the sign, which determines which determines , which determines  (see [28], letter dated December 23, 1908).