Full Idea
When we ask 'is there a number?' in its inferential role (or internalist) reading, then we ask whether or not there is a true instance of 't is a number'. When we ask in its domain conditions (externalist) reading, we ask if the world contains a number.
Gist of Idea
The inferential quantifier focuses on truth; the domain quantifier focuses on reality
Source
Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 03.6)
Book Reference
Hofweber,Thomas: 'Ontology and the Ambitions of Metaphysics' [OUP 2018], p.94
A Reaction
Hofweber's key distinction. The distinction between making truth prior and making reference prior is intriguing and important. The internalist version is close to substitutional quantification. Only the externalist view needs robust reference.