Metatheoretical terms for logic 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.
 A: The part of logic you are referring to is strongly related to the relationship between syntax and semantics. A logical system is just a specification of how to construct well-formed-formulas and how to manipulate strings on a piece of paper which we then call proofs if they follow certain rules prescribed by the logical system. We then hope that there is some relationship between those things we can prove in that system and the truthfulness of the properties they refer to. 
So, there are two things going on here. Firstly, given a certain string $\phi$ of letters within the logical system that is a sentence, one can associate semantic meaning to this string of characters within a particular model. E.g., the string $\forall x \exists y s.t. y\cdot y=x$ is true when interpreted in $\mathbb R_+$ but false in $\mathbb R$. Secondly, there are the proofs. Again just symbols on a piece of paper constructed by following a set of rules. 
The adequacy of a logical system lies in a good interplay between syntax and semantics. A proof is supposed to affirm that something is true. Thus a syntactical object (characters on a piece of paper) are supposed to somehow imply something is true (semantics in a model). So, for the logical system to be any good it must be that the rules for creating proofs actually do imply semantic truth. This is called soundness. It means that if you prove something that it is actually semantically true in all models. 
Completeness is something different. It is a nice property to have but is not necessary for the system to be considered good. It simply says that if something is semantically true in all models, then it must have a proof. 
If a system is not sound then you really can't trust the proofs you produce to have any semantic value. If the system is sound and you manage to prove something, then you are certain that something is actually true in all models. But if you fail to prove something, then you are in the air, either a proof exists and you just didn't find it yet, or no proof exists. However, if the system is also complete, then if you can actually prove no proof exists, then it proves the statement is indeed false. 
I hope this clarifies matters as to the terminology.
A: A preliminary comment is needed : unfortunately, mathematical logic uses “completeness” in two different contexts : this does not help.
As Ittay says, we must start with the semantic notion of true when interpreted in a model.
Then when we set-up a proof system : we choose (zero or more) axioms and some (at least one) rules of inference; with an Hilbert-style proof system, typically we have modus ponens. 
We call theorems of the proof system all the formulae that are deducible, i.e. “produced” starting from the axioms with a finite number of applications of the rules of inference.
There are a lots of different way to set up a proof system; but the basic properties we are requesting to it (the nice properties) are the following : 

1) it  must be sound : assuming a model for our axioms (so that they are true in it), we want that all the theorems must be true in that model (i.e. rules of inference must preserve truth); 
2) it must be adequate :  we want that all the logical consequences of our axioms must be deducible from the axioms (this second properties is usually called completeness of the proof system).

I prefer to speak of adequcy because the interplay of Godel's Completeness Theorem and of Godel's (first) Incompleteness Theorem give us that some formal system, like first-order arithmetic (based on f-o Peano's Axioms) is based on a complete proof system and it is incomplete.

3) Consistency is also nice, but we may have consistent theories that prove false theorems.

