Obligation, Free Choice, and the Logic of Weakest Permissions Forthcoming in The Review of Symbolic Logic Albert J.J. Anglberger 1 , Norbert Gratzl 1 , and Olivier Roy 2 1 Munich Center for Mathematical Philosophy, LMU Munich 2 Universit¨ at Bayreuth June 18, 2015 Abstract We introduce a new understanding of deontic modals that we call obli- gations as weakest permissions. We argue for its philosophical plausibility, study its expressive power in neighborhood models, provide a complete Hilbert-style axiom system for it and show that it can be extended and applied to practical norms in decision and game theory. 1 Introduction In this paper we study the logic of what we call obligations as weakest permis- sions. The basic idea is this. An action type ϕ is obligatory only in cases where the following two conditions hold. • ϕ is permitted; • If ψ is also permitted then one cannot do ψ in the present situation while not doing ϕ. In that sense ϕ is the logically weakest permitted action type that the agent can achieve in the situation she is in. This notion is a very natural one: Suppose that ϕ is obligatory, but that there is some logically weaker and permitted ϕ ′ . This means that you may do ϕ ′ without ensuring the obligatory ϕ. But this seems odd. In Section 2 we study and motivate our understanding of obligations and permissions in more detail. This understanding of obligations and permissions results in an interesting non-normal deontic logic. In Section 3 we show some of its core properties, and provide a complete Hilbert-style axiom system for it. In Section 4 we show that the system can be extended to capture stronger 1