In this paper I propose a new approach to the foundation of mathematics: non-monotonic set theory. I present two completely different methods to develop set theories based on adaptive logics. For both theories there is a finitistic non-triviality proof and both theories contain (a subtle version of) the comprehension axiom schema. The first theory contains only a maximal selection of instances of the comprehension schema that do not lead to inconsistencies. The second allows for all the instances, also the inconsistent (...) ones, but restricts the conclusions one can draw from them in order to avoid triviality. The theories have enough expressive power to form a justification/explication for most of the established results of classical mathematics. They are therefore not limited by Gödel’s incompleteness theorems. This remarkable result is possible because of the non-recursive character of the final proofs of theorems of non-monotonic theories. I shall argue that, precisely because of the computational complexity of these final proofs, we cannot claim that non-monotonic theories are ideal foundations for mathematics. Nevertheless, thanks to their strength, first order language and the recursive dynamic (defeasible) proofs of theorems of the theory, the non-monotonic theories form (what I call) interesting pragmatic foundations. (shrink)

In this article complexity results for adaptive logics using the minimal abnormality strategy are presented. It is proven here that the consequence set of some recursive premise sets is $\Pi _1^1 - complete$ . So, the complexity results in (Horsten and Welch, Synthese 158:41–60,2007) are mistaken for adaptive logics using the minimal abnormality strategy.

This paper presents eight (previously unpublished) adaptive logics for belief revision, each of which define a belief revision operation in the sense of the AGM framework. All these revision operations are shown to satisfy the six basic AGM postulates for belief revision, and Parikh's axiom of Relevance. Using one of these logics as an example, we show how their proof theory gives a more dynamic flavor to belief revision than existing approaches. It is argued that this turns belief revision (that (...) obeys Relevance) into a more natural undertaking, where analytic steps are performed only as soon as they turn out to be necessary in order to uphold certain beliefs. (shrink)

We present two defeasible logics of norm-propositions (statements about norms) that (i) consistently allow for the possibility of normative gaps and normative conflicts, and (ii) map each premise set to a sufficiently rich consequence set. In order to meet (i), we define the logic LNP, a conflict- and gap-tolerant logic of norm-propositions capable of formalizing both normative conflicts and normative gaps within the object language. Next, we strengthen LNP within the adaptive logic framework for non-monotonic reasoning in order to meet (...) (ii). This results in the adaptive logics LNPr and LNPm, which interpret a given set of premises in such a way that normative conflicts and normative gaps are avoided ‘whenever possible’. LNPr and LNPm are equipped with a preferential semantics and a dynamic proof theory. (shrink)

This paper defines provably non-trivial theories that characterize Frege’s notion of a set, taking into account that the notion is inconsistent. By choosing an adaptive underlying logic, consistent sets behave classically notwithstanding the presence of inconsistent sets. Some of the theories have a full-blown presumably consistent set theory T as a subtheory, provided T is indeed consistent. An unexpected feature is the presence of classical negation within the language.

Pluralism has many meanings. An assessment of the need for logical pluralism with respect to scientific knowledge requires insights in its domain of application. So first a specific form of epistemic pluralism will be defended. Knowledge turns out a patchwork of knowledge chunks. These serve descriptive as well as evaluative functions, may have competitors within the knowledge system, interact with each other, and display a characteristic dynamics caused by new information as well as by mutual readjustment. Logics play a role (...) in the organization of the chunks, in their applications and in the exchange of information between them. Epistemic pluralism causes a specific form of logical pluralism. Against this background, the occurrence of inconsistencies will be discussed together with required reactions and systematic ways to explicate them. Finally, the place of inconsistencies in the sciences will be considered. Seven theses will be proposed and argued for. The implications of each of these for pluralism will be considered. The general tenet is that paraconsistency plays an important role, bound to become more explicit in the future, but that the occurrence of inconsistencies does not basically affect the need for pluralism. (shrink)