Mathematical logic: negation completeness = maximality? In his notes "Gödel without (too many) tears"  Peter Smith (2014), defines negation completeness in the following way:

A theory $T$ is negation complete if it formally decides every closed
   well-formed formulaof its language, i.e. for every sentence $A$, $T
 ⊢ A$  or $T ⊢ ¬A$

Does negation completeness as defined above correspond to the notion of maximality?
Here is the definition of maximality given by some lecture notes I'm studying:
A theory $T$ is maximal $iff$, for every formula $A$ of the language, either $A\in T \quad\vee\; \sim A \in T$
 A: Short answer "yes".
Longer answer: Well, it strictly speaking will depend exactly what you count as a "theory". We are interested in formally axiomatized theories here when looking at Gödel's theorems, so what we care about is that we have some axioms to play with and some logical apparatus. Suppose I give you e.g. some axioms $\Sigma$ for Peano Arithmetic, and specify the relevant deductive apparatus, defining a consequence relation $\vdash$. But what's the "theory" so defined? 
One natural line is to think of the theory  as the pair $(\Sigma, \vdash)$ comprising the axioms $\Sigma$ plus the consequence relation. Another line is to think of a theory as comprising $\Theta$ the set of consequences of those axioms via that logical apparatus.
If you think of a theory $T$ the first way, then $A \in T$ and $T \vdash A$ don't come to the same. If you think of a theory the second way, then they do. Nothing much is going to hang on this, as everything we want to say about axioms, logics, consequences, negation-completeness can still be said whichever way we choose to pin down the word "theory".
