Translating Metainferences Into Formulae: Satisfaction Operators and Sequent Calculi

ROFFÉ, ARIEL JONATHAN; PAILOS, FEDERICO

Australasian Journal of Logic

Australasian Association of Logic

Año: 2021

1448-5052

In this paper, we present a way to translate the metainferences of a mixed metainferential system into formulae of an extended-language system, called its associated sigma-system. To do this, the sigma-system will contain new operators (one for each satisfaction standard), called the sigma operators, which represent the notions of "belonging to a (given) satisfaction standard". We first prove, in a model-theoretic way, that these translations preserve (in)validity. That is, that a metainference is valid in the base system if and only if its translation is a tautology of its corresponding sigma-system. We then use these results to obtain other key advantages. Most interestingly, we provide a recipe for building unlabeled sequent calculi for sigma-systems. We then exemplify this with a sigma-system useful for logics of the ST family, and prove soundness and completeness for it, which indirectly gives us a calculus for the metainferences of all those mixed systems. Finally, we respond to some possible objections and show how our sigma-framework can shed light on the "obeying" discussion within mixed metainferential contexts.