A parametrised axiomatization for a large number of restricted second-order logics

TL;DR

This paper introduces a parametrized axiomatization for various restricted second-order logics, providing a strong completeness proof using Boolean algebra techniques, thereby advancing the formal understanding of these logical systems.

Contribution

It offers a unified infinitary axiomatization for a family of restricted second-order logics with a parameterized predicate range, along with a completeness proof.

Findings

Provides a strongly complete axiomatization for restricted second-order logics.

Uses Boolean algebra techniques for the completeness proof.

Applies to multiple systems with varying predicate ranges.

Abstract

By limiting the range of the predicate variables in a second-order language one may obtain restricted versions of second-order logic such as weak second-order logic or definable subset logic. In this note we provide an infinitary strongly complete axiomatization for several systems of this kind having the range of the predicate variables as a parameter. The completeness argument uses simple techniques from the theory of Boolean algebras.