Full Idea
In the standard semantics of second-order logic, by fixing a domain one thereby fixes the range of both the first-order variables and the second-order variables. There is no further 'interpreting' to be done.
Gist of Idea
In standard semantics for second-order logic, a single domain fixes the ranges for the variables
Source
Stewart Shapiro (Foundations without Foundationalism [1991], 3.3)
Book Reference
Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.73
A Reaction
This contrasts with 'Henkin' semantics (Idea 13650), or first-order semantics, which involve more than one domain of quantification.
Related Idea
Idea 13650 Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro]