Non-monotonic reasoning with normative conflicts in multi-agent deontic logic

We present proof-theoretical and semantical characterizations of two multi-agent deontic logics for dealing with normative conflicts. The resulting logics PMDL and PMDL are non-standard in at least two respects. First, they are nonclassical in the sense that they invalidate some inferences of the propositional fragment of Classical Logic (CL). Consequently, they also invalidate certain inferences of so-called Standard Deontic Logic (cfr. infra). The upshot of this non-classicality is that these logics consistently accommodate normative conflicts. Second, PMDL and PMDL are non-monotonic: previously derived conclusions may be withdrawn in the light of new premises. As such, these systems closely mirror actual normative and agentive reasoning. Next to the usual connectives ¬,∨,∧,⊃, and ≡, we make use of a set of modal operators for bringing about collective actions, and of two deontic operators for mandatory and permitted states of affairs. We work within a simplified a-temporal framework from which we exclude e.g. authorities and utilities of obligations, knowledge and beliefs of agents and groups, etc. The presentation of PMDL and PMDL proceeds in various steps. First, we define the monotonic, supraclassical multi-agent logic of action ML (Section 2). We illustrate how this logic deals with collective actions and discuss some further properties of our agentive modal operators. In Section 3 we extend ML with deontic modalities. The resulting logic is called MDL. We discuss some interesting properties of MDL related to collective obligations and (chains of) commands. Next, we weaken MDL in order to consistently model less idealized settings in which intraand interpersonal normative conflicts occur. In Section 4 we define the logic PMDL, a paraconsistent (yet monotonic) weakening of