This paper answers the philosophical contentions defended in Horsten and Welch (2007, Synthese, 158, 41-60). It contains a description of the standard format of adaptive logics, analyses the notion of dynamic proof required by those logics, discusses the means to turn such proofs into demonstrations, and argues that, notwithstanding their formal complexity, adaptive logics are important because they explicate an abundance of reasoning forms that occur frequently, both in scientific contexts and in common sense contexts.
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 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.
In this paper I present a non-trivial but inconsistent set theory based on the axioms of naive set theory. The theory is provably non-trivial and strong enough for most of the applications of regular mathematics. This is realized by distinguishing between strong and weak set membership and allowing for the derivation of strong membership from weak membership whenever this is not problematic (it does not lead to paradoxes). This idea of applying rules whenever unproblematic is formalized by means of an adaptive logic.
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 nonmonotonic 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. * I am very indebted to Laszlo Kosolosky, Diderik Batens and two anonymous referees who suggested many useful corrections and improvements on a former version of this paper.† Peter Verdée is a post-doctoral fellow of the Fund for Scientific Research -Flanders.
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