Decidability of Admissibility: On a Problem by Friedman and Its Solution by Rybakov

Abstract

Rybakov (1984a) proved that the admissible rules of IPC are decidable. We give a proof of the same theorem, using the same core idea, but couched in the many notions that have been developed in the mean time. In particular, we illustrate how the argument can be interpreted as using refinements of the notions of exactness and extendibility. * Support by the Netherlands Organisation for Scientific Research under grant 639.032.918 is gratefully acknowledged.

Uniform Interpolation and Admissible Rules

“…For some parts of this overview we rely on the detailed historical overview with precise references to primary sources by Goudsmit (2015) and add recent literature. This section is also meant as an informal introduction on admissible rules, where we intuitively define different concepts.…”

“…Other explicit bases in the realm of classical modal logic can be found in (Rybakov et al, 2000). Explicit bases are also investigated in for example intermediate logics (Iemhoff, 2005(Iemhoff, , 2006Goudsmit and Iemhoff, 2014;Goudsmit, 2018), Łukasiewics logic (Jeřábek, 2010), and {¬, →}-fragment of IPC (Cintula and Metcalfe, 2010).…”

“…Modern approaches have replaced these interpretations of 'extensions.' In particular, in an analysis of Rybakov's work, Goudsmit (2021) provides a new semantics for admissible rules in IPC in terms of adequately exact models, which in IPC correspond to so-called adequately extendible models (see also Goudsmit, 2015).…”

“…One can avoid the meta-language by representing the rules by propositional variables, such as (MP) p, p → q q, and using substitutions to present any instance of rule (which is in fact a rule itself). This approach is taken in for instance (Goudsmit, 2015;Jeřábek, 2008). This approach seems technically more justified, and will therefore be the approach that we will take throughout this part of the thesis.…”

“…So-called finitary, structural consequence relations are becoming standard practice in the study of admissible rules and are used by, among others, Jeřábek (2005), Cintula andMetcalfe (2010), andGoudsmit (2015). We follow this line of research.…”