Full Idea
The aim in setting up a model theory is that the technical notion of truth in all models should agree with the intuitive notion of truth in all instances. A model is supposed to represent everything about an instance that matters for its truth.
Gist of Idea
We aim to get the technical notion of truth in all models matching intuitive truth in all instances
Source
John P. Burgess (Philosophical Logic [2009], 3.2)
Book Reference
Burgess,John P.: 'Philosophical Logic' [Princeton 2009], p.43