I’m trying to understand how to justify Universal Generalization of a variable
introduced in an assumed premise. My book seems to indicate that a variable
introduced in an assumed premise stands for a particular but as yet unknown
member of a domain. At this point it is error to universally generalize on this
type of variable. But after the premise is discharged, we are free to consider
the variable as general for any member of the domain and can therefore
universally generalize it. What frees us to consider the variable type
differently? What restricts us to considering it an unknown particular in the
assumed premise?

Thank you for any help you can give.