Confused about the use of the term "logic" I am trying to gain a deeper understanding of logic, especially proof theory in its many forms, and I am curious about the common use of the term "Logic".  It is often confusing what the precise semantics of "logic" are in any isolated reading of the many descriptions of the topic area.  For instance, the term "Propositional Calculus" and "Propositional Logic" are sometimes used interchangeably.  Confusing things more, "Propositional Logic" and "Classical Logic" are used interchangeably.  
From what I can intuit, it seems that there are a few popular logics; Classical, Intuitionistic, and Linear.  Each with differing definitions of the semantics of truth and falsehood.  Then there are algebras for logic; propositional, relational, first order, and second order.  Lastly (at least for the distance i am currently willing to travel) there are proof calculi; sequent calculus, natural deduction, and axiomatic (i.e. Hilbert style).
Am I way off here?  Are there any resources for clarifying this (please don't reply with wikipedia because that is causing some of the difficuties)?
Thanks in advance.
Chuck
 A: I think you mean "predicate logic" (which has quantifiers and equality) rather than "propositional logic" (which has only boolean connectives). The reason "classical logic" is often called "predicate logic" is because the ones using the term are not even talking about non-classical logic so there is no need to specifically say "classical predicate logic". And Intuitionistic logic uses exactly the same language as classical logic but with different inference rules.
But what you need is to start with learning one logic first otherwise you're going to get confused very quickly, just like you'll get a stomachache if you try all the kinds of apples in the world all at one go. Naturally, the first logic to study should be classical first-order logic. I suggest the first two references in this post, which cover the intuition behind first-order logic, as well as common deductive systems (also called proof calculii) including natural deduction systems, Hilbert-style systems and the tableaux method (which is surprisingly simple).
While you are at it, always keep in mind that to study any formal system you always need to work in a meta-system. So take note what exactly is carried out within the formal system and what is carried out outside in the meta-system. I mention this because many students do not distinguish carefully between the two and get very confused. By the way, most logicians will take ZFC as their meta-system, but usually you do not need that much. You will see as you go just how much you need.
After the basics are there, it is not too hard to learn about higher-order logics and non-classical logics on your own, and you would be able to ask specific questions about those on Math SE if you cannot figure out something. Sorry for citing Wikipedia, but it is indeed a reasonable starting point once you have sufficient background knowledge. I agree it is quite useless if you don't, which is why one should go to proper texts for introductory material.
