When we study logic we define various metatheoretic properties for logical systems and first-order theories, and then ask whether particular systems or theories have these properties. "Consistent" and "complete" are two examples. We check first-order logic and say it's both consistent and complete, that's good. We check Peano Arithmetic and find it's not complete if it's consistent, too bad. 

My first question is what to call properties like consistency and completeness. "Goodness criteria" or "adequacy criteria" come to mind. Is there a term in general use that's better than these?

My second is whether anyone can provide a reference to any discussion of why we use these "goodness criteria" rather than others. 

(Motivation: I think I've found one or two new criteria for modal systems, and need to explain what they are.) 

Thanks.