Difference between cosmology and ontology

Cosmology is the study of the origins of the universe, it's tied to astronomy.
Ontology, a branch of metaphysics, is the field of philosophy that discusses what kinds of being exist in the world.
Metaphysics is the field of philosophy that seeks to elucidate most general features of reality._{Apr 30, 2021}
Following the lead of Quine, ontology is traditionally said to deal with the question of what exists, while metaphysics, perhaps following Aristotle, works on a more shady area, dealing with some disparate issues such as being qua being, causality, theology, and in more recent times, the nature of modality, personal

Does a metaphysics have an ontology?

An ontology posits which entities exist in the world.
So, while a metaphysics may include:

an implicit ontology (which means

how your theory describes the world may imply specific things in the world)

they are not necessary the same field of study.
Let's consider an example that might clarify the distinction a little more:gravity.

What is the difference between cosmology and ontology?

Cosmology vs Ontology - What's the difference? is that cosmology is the study of the physical universe, its structure, dynamics, origin and evolution, and fate while ontology is the branch of metaphysics that addresses the nature or essential characteristics of being and of things that exist; the study of being qua being.

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