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Positive logics.

Saharon Shelah^1,2, Jouko Väänänen^3,4

¹Institute of Mathematics, Hebrew University, Jerusalem, Israel.

Archive for Mathematical Logic

|January 23, 2023

Summary

This summary is machine-generated.

Lindström's Theorem defines first-order logic. This study explores extensions of first-order logic, finding no single strongest positive logic that retains key model-theoretic properties.

Keywords:

Abstract model theory Existential second order Lindström’s Theorem

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Area of Science:

Mathematical Logic
Model Theory

Background:

Lindström's Theorem identifies first-order logic as maximal under specific properties.
These properties are the Compactness Theorem and the Downward Löwenheim-Skolem Theorem.

Purpose of the Study:

To investigate extensions of first-order logic that preserve the Compactness and Downward Löwenheim-Skolem Theorems.
To determine if a strongest logic exists within negation-less (positive) logics.

Main Methods:

Examining existential second-order logic as a baseline extension.
Analyzing a family of proper extensions of existential second-order logic.
Focusing on logics not closed under negation (positive logics).

Main Results:

Existential second-order logic has multiple proper extensions satisfying the specified theorems.
No strongest extension exists within the class of positive logics.

Conclusions:

The search for a maximal logic under these properties is more nuanced when negation is excluded.
This suggests a rich landscape of logics beyond first-order logic with unique model-theoretic characteristics.