Open AccessTranslating Metainferences Into Formulae
Author(s) -
Ariel Jonathan Roffé,
Federico Pailos
Publication year - 2021
Publication title -
australasian journal of logic
Language(s) - English
Resource type - Journals
ISSN - 1448-5052
DOI - 10.26686/ajl.v18i7.6801
Subject(s) - soundness , tautology (logic) , sequent , completeness (order theory) , base (topology) , sequent calculus , calculus (dental) , mathematics , algebra over a field , computer science , key (lock) , theoretical computer science , pure mathematics , programming language , discrete mathematics , description logic , autoepistemic logic , medicine , mathematical analysis , geometry , dentistry , multimodal logic , mathematical proof , computer security
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 σ-system. To do this, the σ-system will contain new operators (one for each standard), called the σ operators, which represent the notions of "belonging to a (given) 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 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 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 σ-framework can shed light on the “obeying” discussion within mixed metainferential contexts