Different kinds of systems I got interested in learning more about Logic, recently.The first thing i noticed is that this topic is a lot bigger than i expected. As i'm trying to make a sense of it all ( seeing the big picture ) before delving into it, i'm having a lot of questions and i thought of asking help to people that are already acquainted with it.    
I'm easily grasping some concepts ( like formal language, language grammar, rules of inference, axioms, theorems ) and they seem to have a common definition over many authors.
Other terms are giving me a really hard time.My question is :      
Does the following terms have agreed definitions among logicians ?
Formal System, Logical/Logic System, Proof Systems, Axiom System,Natural Deduction System
If not all of them are universally defined, is there some of them that are pretty much agreed ?       
If it latter answer is positive, I would like to know where ( books/sites/resources) i could find information regarding the definition of those terms that are almost completely agreed.   ( Wikipedia doesn't seem to cover it all ).  
Thanks a lot in advance
 A: The latter three you give are certainly common enough, I would think, that you could use them and logicians would know what you meant. These three terms all belong to a field called proof theory, and there are several introductory texts (see third paragraph). However, "formal system" and "logical system" are just way too broad of terms to have an explicit definitions. If anything, it's a philosophical question what counts as a logical system, or what counts as a formal system, and certainly there will be shades of grey in between.
But the fact of the matter is that, ultimately, people will use a variety of notations/conventions, and you'll inevitably find disparate uses of the same term coming from different authors. Your best option, I would think, is to just pick something and stick with it when you're first learning, and not worry about it. Afterwards, when you're more comfortable with the notions, you can decide for yourself what it makes sense to call them.
You may have come across this already, but Peter Smith has a nice "Teach Yourself Logic" guide with book recommendations. The terms that you mentioned seem mostly to involve proof-theoretic terms. The basic idea is that "axiom systems" (or "Hilbert systems") and "natural deduction systems" are two different kinds of proof systems. My opinion: if you want to learn more about proof systems, and you've already seen a bit of introductory mathematical logic, I would say Negri and von Plato is a good place to start. If you're completely new to logic, I would recommend reading some Barwise and Etchemendy to get a feel for how natural deduction systems work, and then reading Enderton's introduction to mathematical logic.
