Abstract: Non-monotonic reasoning clusters a number of logic-related formalisms that only share the fact of lacking the monotonicity property of classical logic. This very unspecific feature covers a number of almost unrelated calculi having various motivations, one of which being exception-tolerant reasoning. Knowledge representation and reasoning systems to this end have been pioneered by Dov Gabbay and Yoav Shoham,  then extensively developed by Daniel Lehmann, Judea Pearl and their colleagues from the mid 1980 till the mid 95. There exists natural connections between these formalisms and theories of uncertainty like  probability theory and possibility theory. The key-connection between uncertainty theories and exception-tolerant non-monotonic inference is the notion of conditioning, central to uncertain reasoning, and absent from classical logic. The aim of the talk is to show that conditional probability, once stripped from its numerical clothes, exactly corresponds to a non-monotonic conditional assertion captured by preferential inference of Lehmann and colleagues (without resorting to infinitesimals). Adding rational monotony leads to a form of conditional modelled by conditional possibility and an inference relation captured by possibilistic logic. This approach also accounts for a notion of accepted defeasible beliefs that  is closed under logical consequences, and closely related to theory revision,  contrary to the usual notion of probabilistic acceptance (studied especially by the late Henry Kyburg). Connections with imprecise probability and modal logics of belief can thus be laid bare as well. These results support the idea that the symbolic approach to exception tolerant reasoning stands as a backbone to probabilistic reasoning. The latter still obeys its basic principles (like cumulativity), but in a degenerated (numerical) form. In this sense this talk is a plea for a unified view of symbolic and numerical approaches toexception-tolerant reasoning.

Abstract: This talk reports on joint work with Olivier Roy and Norbert Gratzl. We explore a non-normal logic of beliefs for boundedly rational agents. The logic we study is the result of dropping positive introspection for knowledge in the system developed by Stalnaker . In that system beliefs are not closed under conjunction, but they are required to be pairwise consistent, a requirement that has been called agglomerativity elsewhere. While bounded agglomerativity requirements, i.e., joint consistency for every n-tuple of beliefs up to a fixed n, are expressible in that logic, unbounded agglomerativity is not.  We study an extension of this logic of beliefs with such an unbounded agglomerativity operator. We provide sound and complete axiomatizations for both logics, show that they have the finite model property and explore dynamical properties.

Abstract: We generalize the language of substructural logics interpreted over the ternary relational semantics. We introduce three symetric triples of connectives which are interconnected by means of cyclic permutations. The usual fusion, implication and co-implication connectives form one of these triples. This defines a logic that we call update logic. We define a cut-free display calculus for update logic which generalizes the display calculus for modal logic and a sequent calculus for update logic which generalizes the non-associative Lambek calculus. Then, we provide a display calculus as well as a sequent calculus for dynamic epistemic logic based on our display and sequent calculifor update logic.

Abstract: We present a framework based on a knowledge modality defined as a diamond operator over distributive non-associative full Lambek calculus with a negation. We deal with the relational semantics for distributive substructural logics, interpreting the elements of a relational frame as information states consisting of collections of data which may be incomplete or even inconsistent. We explicate the notion of knowledge as information confirmed by a reliable source. The system is modular in the sense that the axiomatization of the epistemic operator is sound and complete with respect to various background propositional logics, which makes the system potentially applicable to a wide class of epistemic contexts. Our system admits a weak form of logical omniscience (the monotonicity rule), but avoids stronger ones (the necessitation rule and the K-axiom) as well as some closure properties discussed in normal epistemic logics (like the positive and negative introspection). For these properties we provided characteristic frame conditions, so that they can be present in the system if they are considered to be appropriate for some specific epistemic context.