Positive logics

TL;DR

This paper explores the landscape of negationless logics, showing that positive logics, which extend first order logic without negation, have no maximal extension satisfying key model-theoretic properties.

Contribution

It demonstrates that in negationless logics, unlike classical logic, there is no strongest extension with Compactness and Löwenheim-Skolem properties.

Findings

Existential second order logic has multiple proper extensions with key properties.

No maximal positive logic extension of first order logic exists under the given properties.

Abstract

Lindstr\"om's Theorem characterizes first order logic as the maximal logic satisfying the Compactness Theorem and the Downward L\"owenheim-Skolem Theorem. If we do not assume that logics are closed under negation, there is an obvious extension of first order logic with the two model theoretic properties mentioned, namely existential second order logic. We show that existential second order logic has a whole family of proper extensions satisfying the Compactness Theorem and the Downward L\"owenheim-Skolem Theorem. Furthermore, we show that in the context of negationless logics, positive logics, as we call them, there is no strongest extension of first order logic with the Compactness Theorem and the Downward L\"owenheim-Skolem Theorem.