Full Idea
The 'logica magna' [of the Fregean tradition] has quantifiers ranging over a fixed domain, namely everything there is. In the Boolean tradition the domains differ from interpretation to interpretation.
Gist of Idea
Big logic has one fixed domain, but standard logic has a domain for each interpretation
Source
John Mayberry (What Required for Foundation for Maths? [1994], p.410-2)
Book Reference
'Philosophy of Mathematics: anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.410
A Reaction
Modal logic displays both approaches, with different systems for global and local domains.
Related Idea
Idea 17786 The mainstream of modern logic sees it as a branch of mathematics [Mayberry]