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Philosophical hypergraphics: some applications to philosophy of the theory of hypergraphs

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This thesis demonstrates the applicability of hypergraphs to philosophical problems. I employ and enrich the theory of transverse hypergraphs, the colouring theory of hypergraphs, and the novel harmonic theory of hypergraphs. I also demonstrate that the relationship between the latter two theories is one of logical duality. Because this thesis consists of a number of distinct articles, each representing a thematically diverse application of hypergraph theory to philosophy, it is difficult to speak of a unifying thread, except insofar as I may explain the general modus operandi in highly schematic terms. To that end, common to all of the articles is the exemplification of the following: A problem is given whereby there is a collection of objects and a question has arisen as to whether these objects stand in a particular relationship to one another. I use hypergraphs to represent the objects. A key feature of the objects is then modelled using either chromatic or harmonic number, or the notion of a transverse hypergraph. Lastly, properties of, or relations between chromatic number, harmonic number, or the notion of a transverse hypergraph, are shown to entail a solution to the problem. The main results in this thesis are summarized as follows: (1) It is possible to design a non-statistical polling technique which forms the basis of a representative political system. (2) The conditions under which a malfunction of a technical system is identical with its diagnosis can be characterized using equivalent maximality and minimality conditions on harmonic and chromatic number. (3) An axiomatization exists of extent of Wittgenstein's notion of family resemblance. (4) Taxonomic properties of identity can be discerned by exploring the mathematical relationship between diachronicity and synchronicity. (5) A new axiomatization of a class of weakly aggregative modal logics can be found by dualizing chromatic number, and exploiting harmonic number. (6) Completeness for classes of weakly aggregative and non-normal modal logics can be simplified by dualizing neighborhood semantics. (7) There is a relevant inference relation which is dual to the paraconsistent n-forcing relation, and which can be represented as a restriction of the classical provability relation.
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