Theory of Multitudes as an Alternative to the Set Theory

Introduction

Since the times of Cantor the classical set theory has evolved a lot and now, in tight connection with the classical predicate logic, is the ground for formalization of many theories of mathematics, as well as other fields of knowledge. Modern tools of knowledge engineering and representation essentially based on these two formalisms allow building ontologies, conceptual graphs and other formalized conceptualizations widely used in the science, education, industry and business. Various extensions of the set theory allow to account for fuzzy membership of an element in a set (Zadeh, 1965), multiple occurrence of an element in a set as “copies” (multiset theory of Blizard, 1989), ordering of elements in a set (posets in order theory – see, e.g., Davey and Priestley, 2002, or trees in descriptive set theory – see Kechris, 1995) and other important peculiarities of sets.

Nonetheless, there remains an issue that is of vital importance for knowledge representation but, in our view, may not be correctly addressed by the classical set theory in its present form, nor by any existing extension. This is the issue of meaning of concepts that underlies ontology design and related techniques.

Along with personality in psychology or field in physics, meaning is an entity as widely used as poorly understood. The abundance of theories of meaning, from competing semantic views of Frege, Russel and Tarsky to linking it to syntax by Chomsky, skepticism of Wittgenstein and, in addition, intentional-behaviorist views of Grice, Donaldson and many others (see, e.g., a review of Speaks, 2016, and references therein) fairly well illustrate this fact. This urged us, considering the meaning, develop our own vision of the subject, mainly based on actual use of this term in the knowledge engineering and representation.

As might be concluded from the cornerstone publications and numerous application reports, concepts in ontologies (Gruber, 1993), conceptual graphs (Sowa, 2006), entity-relationship diagrams (Chen, 1976) and, presumably, other knowledge representation constructs are commonly treated as potentially non-empty sets. Still, the adequacy of this approach looks debatable. For instance, compare two cases: (i) only green apples were delivered to the supermarket X and (ii) apples can be green, yellow or red. In the former case, one surely may treat the concept “green apples” as a set, while in the latter there is no confidence if apples ever exist. If to substitute apples with flying hills, and colors, with “disintegrating”, “aggregating” and “stable”, we get empty sets, but the logic of relations remains the same, and meanings, however unexpected, are still there. Furthermore, “atomic” has been considered in the science as a synonym to “further indivisible”. However, whatever atom to take, in the real world it appears not only splittable into pieces (e.g., apple can be sliced) but, yet more painful, often consisting of notably different parts (e.g., apple consists of skin, flesh, and seeds). Perhaps only quarks in the physics are considered absolutely atomic, but these, by the same theory, may not occur alone, i.e., may not be treated separately. In addition, “atoms” recognized in the real world are not warranted to avoid coalescence, disintegration or complete disappearance.

Moreover, we have serious problems identifying “elements” (i.e., smallest indivisible parts) of water, air, or any continuous medium, as well as energy, light, and the like. Are these molecules/atomic particles or “elementary units”? If the former, their properties may have nothing to do with the properties of the whole thing (as different are the micro- and macroworld), and if the latter, how big these units are? Unlike apples, here the choice of an element is absolutely provisional. Obviously, taking “apples”, “water”, “air”, “energy”, “light” for sets, we have to admit that they are filled with some other “type of existence” than elements.