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Philosophy of Vagueness: A Topological Perspective

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Universitat de Barcelona. Facultat de Filosofia

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[eng] This dissertation is a contribution to the application of topology to the philosophical problem of vagueness. We pursue two main goals: The first goal is to give an account of the main features of vague concepts. In our proposed account, vague concepts that are structured around typical cases (called poles), are boundaryless and have borderline cases. We show that this account of vague concepts can be used to deal with the Sorites paradox and higher-order vagueness. The second goal is to provide a topological model for vague concepts in a conceptual space based on previous works by Ian Rumfitt and Thomas Mormann. We use this topological model to show how one can keep the two truth-valued semantics of classical logic while still reject the principle of Bivalence. While our main concern is to give an account of vagueness, Rumfitt cares about classical logic that has been threatened by vagueness, because it shakens the firm wall between the extensions of concepts. We share the idea with him that the principle of Bivalence does not hold, yet disagree with him in accepting the third truth value.
After the introduction, in Part II, through a literature review of some existing theo- ries of vagueness, we settle what is expected from a theory of vagueness. In Part III, we review the fundamental notions of topology to show how they can be fruitfully applied to better understand the structure of vague concepts. Part IV consists of three sections. Sections 5 and 6 are dedicated to a critical analysis of two other topological proposals, namely the Kantian model by Boniolo and Valentini and the topological approach of Weber and Colyvan which presents a continuous version of the Sorites paradox. Section 7 is a critical review of a prominent geometrical framework in cognitive science, namely Gärdenfors’ conceptual spaces, in which concepts are represented at a conceptual level. Conceptual spaces will be the base of our account. We discuss its pros and cons and following the recent works by Mormann on polar spaces, we show that conceptual spaces need a topological structure to be optimized. None of the previous views can answer or even aimed at answering all the questions relating to vagueness and finding a solution to the mentioned problems. Part V introduces a model for vagueness based on weakly scattered T0 Alexandroff spaces. Alexandroff spaces have a tight relation to modal logic and applications in computer science and image processing, among other fields. The model is a refinement and expansion of Rumfitt’s topological model and Mormann’s generalization of it. In order to make it apt to deal with the phenomenon of vagueness, we improve the model to a 3-layer-model by taking a closer look at the previous topological models to reveal their hidden properties and deficiencies. This new model reveals three layers in a concept: the first layer contains the typical cases of the concept, the second layer contains the almost typical cases of the concept and the third layer the borderline cases of the concept. The extension of a concept contains the typical and almost typical cases, i.e., it consists of the first two layers. These layers were hidden in the previous models. Then, we define the notions of borderline case and similarity relations in this model and we use them to explain in detail Rumfitt’s solution to the Sorites and sharp boundary paradoxes. The solution is based on the rejection of the tolerance principle in its strict sense. We propose a weak version of tolerance that holds in our model. We accept truth-value gaps but, pace Rumfitt, we do not accept a third alethic truth value. After that, we deal with the problem of higher-order vagueness and compare the proposed model to some of the dominant theories of vagueness. We end up with some suggestions to improve the model to overcome its limitations and to be able to answer further questions on vagueness.

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Philosophy of Vagueness: A Topological Perspective