What's a Logic? I'm taking a logic course, and so far we've seen


*

*Logical Symbols

*Logical Axioms

*Rules of Inference

*Non-Logical Symbols

*Non-Logical Axioms


Now, I'm confused about definitions.


*

*Alphabet/Vocabulary = 1+2

*Logic = 1+2+3 ?

*Signature/Language $L$ = 4 OR Logic (implicitly)+4 ?

*Theory = 5 OR Logic (implicitly)+5 ?

*$L$-Theory = 4 (implicitly)+5 OR Logic (implicitly)+4 (implicitly)+5 ?


Also, I've seen Logic being defined as an ordered pair consisting of a function mapping alphabets to sets of formulae and a so-called satisfaction relation... or by means of institutions in category theory... I'm just confused as to what a logic ultimately really is, because even the most rigorous logic texts I could find begin their definitions at languages and vocabularies. Or, if not a universal definition, maybe at least what a logic consists of, i.e. foregoing technicalities, what needs to be defined to have a certain logic. 
Perhaps the difficulty is that Rules of Inference are more Proof-theoretic and the other stuff more Model-Theoretic, but maybe you can help me anyway, it'd be much appreciatetd.
 A: We start with Symbols : either logical : $\to, \lnot, \land$ etc. (the connectives), the quantifiers and equality ($=$) and some auxiliary symbols (like parentheses).
In addition we need propositional variables : $p_0,p_1,\ldots$ for propositional logic as well as predicate symbols : $P,Q,\ldots$ and individual variables : $x_0,x_1,\ldots$ for predicate logic.
This is enough for the languages of "pure" logic : propositional calculus, predicate calculus.
If we want to develop some mathematical theory, we have to enlarge the language of predicate logic with specific mathematical symbols : $\in$ for set theory and $0,s(x),+,\times$ for arithmetical theory.
Usually, we call the list of the non-logical symbols of a language : signature.
Up to now, we have considered only symbols, i.e. the alphabet of the language. In order to have a language, we need also syntax, i.e. the rules to produce finite strings of symbols that are meaningful : the so-called well-formed formulas.
For propositional logic, $(p_0 \to p_1)$ is a wff while $\land p_0$ is not.
For predicate logic, $\forall x (Px \land Qx)$ is an example of wff.
Having defined the syntax of the language, we add the semantics to it, based on the concept of interpretation.

Up to now we have covered the Language : your points 1 and 4.
In order to develop a logical calculus, we need rules of inference : at least one (usually Modus Ponens) but often more than one, as well as some (maybe zero) logical axioms : this corresponds to your point 2 and 3.
We may call propositional calculus and predicate calculus with the general term : Logic.

Finally, in order to develop a mathematical theory, we have to add to the predicate calculus some specific mathematical axioms, like e.g. first order version of Peano's axioms for arithemtic or Zermelo-Fraenkel axioms for set theory (and this is point 5).

If we denote with $\text L$ a specific language, e.g. the language of predicate calculus with the mathematical symbol $\in$, we can use $\text L$-theory to denote a theory that uses language $\text L$.


For a more abstract definition, see e.g. Heinz-Dieter Ebbinghaus et alii Mathematical logic (Springer, 2nd ed.1994), Ch.XIII.1 Logical Systems :

In the following definition of a "logical system" we collect several properties
  which are shared by the logics we have considered so far. As we are mainly interested in semantic aspects, we shall speak of a logical system as soon as we have the following: We are given, for every symbol set $S$, an "abstract" set whose elements play the role of $S$-sentences, and in addition, a relationship between structures and such sentences which corresponds to
  the satisfaction relation, and determines whether an "abstract" sentence
  holds in a structure.
Definition. A logical system $\mathcal L$ consists of a function $L$ and a binary relation $\vDash_{\mathcal L}$. $L$ associates with every symbol set $S$ a set $L(S)$, the set of $S$-sentences of $\mathcal L$. The following properties are required: [...]

