A Game Semantics for System P

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.

A Discrete Duality Between Nonmonotonic Consequence Relations and Convex Geometries

“…We do not know of an analogous result in the literature that works also for infinite convex geometries. However, Theorem 4 in [9] and the completeness results for System P ∞ in [26] suggest that such a representation might also be obtainable for well-founded infinite convex geometries.…”

“…It follows from Example 7 together with Propositions 11 and 12 in [26] that (Or ∞ ) is a consequence of the axioms in Definition 1.…”

“…Let us see why this rule follows from another more common variant of the cautious cut rule, which in turn is shown in Lemma 5.3 of [24] and in Example 13 of [26] to follow from the axioms in Definition 1. This more common variant of the cut rule is the following rule (CCut): If A|∼ B and A ∩ B|∼ C then A|∼ C. Thus assume (CCut) and that the premises A|∼ B and B|∼ C of (CCut') hold.…”

“…The axioms for nonmonotonic consequence relations in Definition 1 are an infinitary version of the rules in the well-known System P [24]. They are precisely the rules of System P ∞ from [26] and differ from the rules of System P only in that in System P the infinitary (And ∞ ) is replaced with its finitary variant (And), which holds for all A, C, D ∈ PW :…”

Conditional Logic Is Complete for Convexity in the Plane

“…In [17, 32] it is observed that for developing proof systems for preferential conditional logic it is beneficial to lift the implicit assumption that the family of sets of worlds, relative to which the conditional is evaluated, is closed under intersections. To achieve this they use a simplified semantic clause from [29] that is sensitive to closure under intersections. When one uses the conditional with this semantic clause relative to a family of sets of worlds that is not closed under intersection different formulas turn out to be true than would be true relative to the same family of sets of worlds using the semantic clause from premise semantics.…”