Topological Models of Columnar Vagueness

This paper intends to further the understanding of the formal properties of (higher-order) vagueness by connecting theories of (higher-order) vagueness with more recent work in topology. First, we provide a “translation” of Bobzien's account of columnar higher-order vagueness into the logic of topological spaces. Since columnar vagueness is an essential ingredient of her solution to the Sorites paradox, a central problem of any theory of vagueness comes into contact with the modern mathematical theory of topology. Second, Rumfitt’s recent topological reconstruction of Sainsbury’s theory of prototypically defined concepts is shown to lead to the same class of spaces that characterize Bobzien’s account of columnar vagueness, namely, weakly scattered spaces. Rumfitt calls these spaces polar spaces. They turn out to be closely related to Gärdenfors’ conceptual spaces, which have come to play an ever more important role in cognitive science and related disciplines. Finally, Williamson’s “logic of clarity” is explicated in terms of a generalized topology (“locology”) that can be considered an alternative to standard topology. Arguably, locology has some conceptual advantages over topology with respect to the conceptualization of a boundary and a borderline. Moreover, in Williamson’s logic of clarity, vague concepts with respect to a notion of a locologically inspired notion of a “slim boundary” are (stably) columnar. Thus, Williamson’s logic of clarity also exhibits a certain affinity for columnar vagueness. In sum, a topological perspective is useful for a conceptual elucidation and unification of central aspects of a variety of contemporary accounts of vagueness

A reconstruction of the multipreference closure

The paper describes a preferential approach for dealing with exceptions in KLM preferential logics, based on the rational closure. It is well known that the rational closure does not allow an independent handling of the inheritance of different defeasible properties of concepts. Several solutions have been proposed to face this problem and the lexicographic closure is the most notable one. In this work, we consider an alternative closure construction, called the Multi Preference closure (MP-closure), that has been first considered for reasoning with exceptions in DLs. Here, we reconstruct the notion of MP-closure in the propositional case and we show that it is a natural variant of Lehmann's lexicographic closure. Abandoning Maximal Entropy (an alternative route already considered but not explored by Lehmann) leads to a construction which exploits a different lexicographic ordering w.r.t. the lexicographic closure, and determines a preferential consequence relation rather than a rational consequence relation. We show that, building on the MP-closure semantics, rationality can be recovered, at least from the semantic point of view, resulting in a rational consequence relation which is stronger than the rational closure, but incomparable with the lexicographic closure. We also show that the MP-closure is stronger than the Relevant Closure.Comment: 57 page

Yet More Modal Logics of Preference Change and Belief Revision

We contrast Bonanno's `Belief Revision in a Temporal Framework' \cite{Bonanno07:briatfTV} with preference change and belief revision from the perspective of dynamic epistemic logic (DEL). For that, we extend the logic of communic

Kleene algebra with domain

We propose Kleene algebra with domain (KAD), an extension of Kleene algebra with two equational axioms for a domain and a codomain operation, respectively. KAD considerably augments the expressiveness of Kleene algebra, in particular for the specification and analysis of state transition systems. We develop the basic calculus, discuss some related theories and present the most important models of KAD. We demonstrate applicability by two examples: First, an algebraic reconstruction of Noethericity and well-foundedness; second, an algebraic reconstruction of propositional Hoare logic.Comment: 40 page

Knowledge Grounded on Pure Reasoning

In this paper I deal with epistemological issues that stem from the hypothesis that reasoning is not only a means of transmitting knowledge from premise-beliefs to conclusion-beliefs, but also a primary source of knowledge in its own right. The idea is that one can gain new knowledge on the basis of suppositional reasoning. After making some preliminary distinctions, I argue that there are no good reasons to think that purported examples of knowledge grounded on pure reasoning are just examples of premise-based inferences in disguise. Next, I establish what kinds of true propositions can to a first approximation be known on the basis of pure reasoning. Finally, I argue that beliefs that are competently formed on the basis of suppositional reasoning satisfy both externalist and internalist criteria of justification