# Rejecting inadmissible rules in reduced normal forms in S4

## TL;DR

This paper explores methods to identify and reject inadmissible rules in the modal logic S4 by analyzing reduced normal form rules and their associated rejection substitutions, extending previous approaches to rule sets.

## Contribution

It introduces a generalized method for rejecting inadmissible rules in S4, linking reduced normal form techniques with rejection substitutions and extending to rule sets.

## Key findings

- Established relations between reduced normal form rules and rejection substitutions.
- Generalized the method to handle sets of rules in S4.
- Provided a framework for rejecting inadmissible rules in modal logic S4.

## Abstract

Several methods for checking admissibility of rules in the modal logic $S4$ are presented in [1], [15]. These methods determine admissibility of rules in $S4$, but they don't determine or give substitutions rejecting inadmissible rules. In this paper, we investigate some relations between one of the above methods, based on the reduced normal form rules, and sets of substitutions which reject them. We also generalize the method in [1], [15] for one rule to admissibility of a set of rules.