Metainferential Levels and Inferential Networks

Metainferentialism is the view that a logic is determined by its valid metainferences. It comes in two variants. The finitist version holds that logical consequence is determined by derivability relations between generalised sequents. This is always a Tarskian consequence relation. According to the transfinitist version, one cannot stop here. Instead, all the possible metainferential levels should be taken into consideration when specifying a logic. We defend the finitist version against tranfinitist criticism, clarifying, on the way, the central tenets of finitist metainferentialism. By way of concluding we offer a novel generic interpretation of sequents which breaks with their usual construal as reified inferences.