In this paper, I will present a Fitch–style natural deduction proof theory for modal paralogics (modal logics with gaps and/or gluts for negation). Besides the standard classical subproofs, the presented proof theory also contains modal subproofs, which express what would follow from a hypothesis, in case it would be true in some arbitrary world.

In this paper, I will characterize a new class of inconsistency-adaptive logics, namely inconsistency-adaptive modal logics. These logics cope with inconsistencies in a modal context. More specifically, when faced with inconsistencies, inconsistency-adaptive modal logics avoid explosion, but still allow the derivation of sufficient consequences to adequately explicate the part of human reasoning they are intended for.

Most logic–based approaches characterize abduction as a kind of backwards deduction plus additional conditions, which means that a number of conditions is specified that enable one to decide whether or not a particular abductive inference is sound . Despite the fact that these approaches succeed in specifying which formulas count as valid consequences of abductive inference steps, they do not explicate the way people actually reason by means of abductive inferences. This is most clearly shown by the absence of a (...) decent proof theory. Instead, search procedures are provided that enable one to determine the right abductive consequences. However, these do not by far resemble human reasoning. In order to explicate abductive reasoning more realistically, an alternative approach will be provided in this article, namely, one that is based on the adaptive logics programme. Proof theoretically, this approach interprets the argumentation schema affirming the consequent as a defeasible rule of inference. This comes down to the fact that the abductive consequences obtained by means of AC are accepted only for as long as certain conditions are satis.ed—e.g. as long as their negation has not been derived from the background theory. In the end, only the unproblematic applications of AC are retained, while the problematic ones are rejected. In this way, the adaptive logics approach to abduction succeeds to provide a more realistic explication of the way people reason by means of abductive inferences. Moreover, as multiple abduction processes will be characterized, this article may be considered as the first step in the direction of a general formal approach to abduction based on the adaptive logics programme. (shrink)