On the expressive power of quantifiers in continuous logic

TL;DR

This paper compares the expressive power of two variants of continuous logic, showing they are equivalent even when simplifying the quantifiers, thus clarifying their relative expressive capabilities.

Contribution

It demonstrates the equivalence in expressive power between real-valued and compact-valued continuous logic, even with a simplified quantifier structure.

Findings

Both logics have the same expressive power.

Replacing multiple quantifiers with a single one does not reduce expressiveness.

The result clarifies the foundational understanding of continuous logic.

Abstract

In this short note we compare the expressive power of real-valued continuous logic (or just continuous logic, in recent literature) with that of compact-valued continuous logic, proposed by Chang and Keisler. We conclude that the two logics have the same expressive power, and moreover, that this remains true if we replace the plethora of potential quantifiers of compact-valued logic with a single ''primordial'' one.