Rybakov (1984a) proved that the admissible rules of $\mathsf {IPC}$ 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.

Satisfaction systems and reductions between them are presented as an appropriate context for analyzing the satisfiability and the validity problems. The notion of reduction is generalized in order to cope with the meet-combination of logics. Reductions between satisfaction systems induce reductions between the respective satisfiability problems and (under mild conditions) also between their validity problems. Sufficient conditions are provided for relating satisfiability problems to validity problems. Reflection results for decidability in the presence of reductions are established. The validity problem in the meet-combination is proved to be decidable whenever the validity problem for the components are decidable. Several examples are discussed, namely, involving modal and intuitionistic logics, as well as the meet-combination of $\textrm {K}$ modal logic and intuitionistic logic.

A strong coloring on a cardinal $\kappa $ is a function $f:[\kappa ]^2\to \kappa $ such that for every $A\subseteq \kappa $ of full size $\kappa $ , every color $\unicode{x3b3} <\kappa $ is attained by $f\restriction [A]^2$ . The symbol

$$ \begin{align*} \kappa\nrightarrow[\kappa]^2_{\kappa} \end{align*} $$

$\kappa $

We introduce the symbol

$$ \begin{align*} \kappa\nrightarrow_p[\kappa]^2_{\kappa} \end{align*} $$

$f:[\kappa ]^2\to \kappa $

$p:[\kappa ]^2\to \theta $

$A\in [\kappa ]^{\kappa }$

$i<\theta $

$\unicode{x3b3} <\kappa $

$f\restriction ([A]^2\cap p^{-1}(i))$

We prove that whenever $\kappa \nrightarrow [\kappa ]^2_{\kappa }$ holds, also $\kappa \nrightarrow _p[\kappa ]^2_{\kappa }$ holds for an arbitrary finite partition p. Similarly, arbitrary finite p-s can be added to stronger symbols which hold in any model of ZFC. If $\kappa ^{\theta }=\kappa $ , then $\kappa \nrightarrow _p[\kappa ]^2_{\kappa }$ and stronger symbols, like $\operatorname {Pr}_1(\kappa ,\kappa ,\kappa ,\chi )_p$ or $\operatorname {Pr}_0(\kappa ,\kappa ,\kappa ,\aleph _0)_p$ , also hold for an arbitrary partition p to $\theta $ parts.

The symbols

$$ \begin{gather*} \aleph_1\nrightarrow_p[\aleph_1]^2_{\aleph_1},\;\;\; \aleph_1\nrightarrow_p[\aleph_1\circledast \aleph_1]^2_{\aleph_1},\;\;\; \aleph_0\circledast\aleph_1\nrightarrow_p[1\circledast\aleph_1]^2_{\aleph_1}, \\ \operatorname{Pr}_1(\aleph_1,\aleph_1,\aleph_1,\aleph_0)_p,\;\;\;\text{ and } \;\;\; \operatorname{Pr}_0(\aleph_1,\aleph_1,\aleph_1,\aleph_0)_p \end{gather*} $$

$+ \neg $