A Game Semantics for System P (Extended Version) Johannes Marti and Riccardo Pinosio February 15, 2016 Abstract In this paper we introduce a game semantics for System P, one of the most studied axiomatic systems for non-monotonic reasoning, conditional logic and belief revision. We prove soundness and completeness of the game semantics with respect to the rules of System P, and show that an inference is valid with respect to the game semantics if and only if it is valid with respect to the standard order semantics of System P. Combining these two results leads to a new completeness proof for System P with respect to its order semantics. Our approach allows us to construct for every inference either a concrete proof of the inference from the rules in System P or a countermodel in the order semantics. Our results rely on the notion of a witnessing set for an inference, whose existence is a concise, necessary and sufficient condition for validity of an inferences in System P. We also introduce an infinitary variant of System P and use the game semantics to show its completeness for the restricted class of well-founded orders. 1 Introduction System P is an inference system which formalizes core principles of non-monotonic consequence relations as studied in artificial intelligence [8]. It is also the non- nested fragment of a conditional logic developed in philosophy and linguistics [10, 3, 15]. The standard semantics for System P is based on orders and evaluates a non-monotonic inference or conditional by minimization in the order. A similar order semantics is also used in the theory of AGM belief revision [5], which can be recast in the setting of conditional logic [1]. In this paper we introduce a game semantics for the validity of inferences in System P. The study of logical systems with game-theoretic methods was initiated independently by Lorenzen and Lorenz [11] and Hintikka [6]. Hintikka’s approach, known as game theoretic semantics, uses a game to establish the truth of a formula in a given model. Lorenzen and Lorenz developed what is known as dialogical logic. A dialogical game is a game in which two players debate the validity of an inference in a logical system. The main difference between dialogical games and game theoretic semantics is that Lorenzen and Lorenz 1