General normal forms for any additive logic

TL;DR

This paper introduces a unified framework for normal forms applicable to any logic with propositional components and distributive non-propositional connectives, including those with partial connectives, generalizing existing normal forms.

Contribution

It defines a broad class of normal forms for additive logics, encompassing partial connectives and unifying various known normal forms under a single framework.

Findings

Most known normal forms are special cases of the proposed general forms.

The normal forms extend to logics with partial connectives.

The framework improves upon previous distributive normal forms.

Abstract

In this article, we define general normal forms for any logic that has propositional part and whose non-propositional connectives distribute over the finite disjunctions. We do not require the non-propositional connectives to be closed on the set of formulas, so our normal forms cover logics with partial connectives too. We also show that most of the known normal forms in the literature are in fact particular cases of our general forms. These general normal forms are natural improvement of the distributive normal forms of J. Hintikka and their modal analogues, e.g. [Anderson] and [Fine].