Translating Metainferences Into Formulae: Satisfaction Operators and Sequent Calculi

PAILOS, FEDERICO MATÍAS; ROFFÉ, ARIEL

The Australasian Journal of Logic

Australasian Association of Logic and the Centre for Logic, Language and Computation at Victoria University

Año: 2022

1448-5052

In this paper, we present a way to translate the metainferences of a mixed metainferentialsystem into formulae of an extended-language system, called its associated σ-system. To dothis, the σ-system will contain new operators (one for each satisfaction standard), called theσ 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. Thatis, that a metainference is valid in the base system if and only if its translation is a tautologyof its corresponding σ-system. We then use these results to obtain other key advantages.Most interestingly, we provide a recipe for building unlabeled sequent calculi for σ-systems.We then exemplify this with a σ-system useful for logics of the ST family, and provesoundness and completeness for it, which indirectly gives us a calculus for the metainferencesof all those mixed systems. Finally, we respond to some possible objections and show howour σ-framework can shed light on the ?obeying? discussion within mixed metainferentialcontexts.