Recommendation on a rigorous and deep introductory logic textbook In this post, I don't mean any word by its somewhat "mathematical or logical" meaning but just "literally".
It's been three years since I started "formal" mathematics, and now I'm familiar with set theory and formal proof.
In the meantime, I have never studied "logic" before (it's nonsense to me), so now i think it's the time to start with it.
I have asked a similar question before, and people recommended me some texts. Almost all of them started with introducing "proposition logic". I guess authors intended to introduce a rather easier example at first. I don't think it's a good way to study logic rigorously. I felt like I'm not studying mathematics when I was reading those books, but I felt like I'm reading an philosophy article, which I felt extremely uncomfortable.
Frankly, to me, it's really hard to know what people mean by logic. I have searched wikipedia, but there are so many types of logics such as propositional logic, intuition logic(?), classical logic and etc. I even found some "logics" are subcategory of other!
What is logic exactly?
I don't want to start logic with 'handy and easy' examples. I want to study logic from its core so I could answer questions like: What is "proof"? What is "truth"?
Please... please recommend me a good precise logic textbook. I'm eager to learn logic precisely... Thank you in advance ! :)
 A: There are two very different kinds of question here:

What is logic exactly? ... What is "proof"? What is "truth"?

All good questions. But famously they do not have sharp, determinate, clear, uncontentious answers. Indeed, they are characteristically philosophical questions (that fall into the purview of what is often called "philosophical logic").
Of course, a technical logic text will introduce e.g. a sharp, technical, notion of a proof-in-a-given-formal-system (the fine print can be significantly different in different texts). But what is the relation between (1) the everyday notion of mathematical proof and (2) various notions of proof-in-a-given-formal-system which aim to model mathematical proof? This is up for (philosophical) debate. Similarly for the notion of truth, and indeed for the notion of a logic. 
A "rigorous logic text" is therefore not the best place, really, to look for the discussion of the philosophical questions here. For those questions are (as it were) standing back from details in those rigorous texts and asking more general, philosophical, questions about them.

Please recommend me a good precise logic textbook.

Still, if you do want pointers to formal logic textbooks then there are a lot of suggestions, at various levels, on various areas of logic, in the Guide you can find at http://www.logicmatters.net/tyl
A: There is one logic which is the most important of all, and that is the first-order logic. I introduce it in with my site : settheory.net, though I only describe formulas, not rules of proof.
I consider the questions : What is "proof"? What is "truth"? as legitimate, having proper answers. 
The concept of proof is essentially clear, in the sense that there is a unique equivalence class of formal systems that the word "proof" may properly refer to : a deduction system for first-order logic, such that the existence of a proof of a formula as deduced from any given list of axioms, is equivalent to the non-existence of a model where the formula is false, according to the completeness theorem.
I also wrote a list of possible meanings of the concept of truth, which I found relevant.
A: I liked Robert Stoll's Set Theory and Logic, which is also a Dover and quite cheap. It covers a relatively wide range of logic topics. 
A: According to Jim Nance texts Formal Logic is the science and art of reasoning well. He starts with deductive reasoning and the standard syllogism. He clearly defines the differences of truth and validity. If you are interested in a well built, straight forward approach, check out his Introductory AND Intermediate Logic texts. Get the Teachers manuals, as these house the student manual fused with the answer key, quizzes and texts. If you are a self taught learner, you will LOVE this format! There are also dvd videos he produced in which he teaches he lesson. He also has a facebookpage that he moderates and he has answered my questions with a positive and "mentor" minded demeanor. This text is a collegic entry level logic course/advanced highschool course. I teach/tutor claasically taught eighth graders both of these texts over the course of one year. Your time would not be wasted with a study as this and you could potentially finish both texts in one semester, depending on how motivated and determined you are. I have students who have found a LOVE for the structure of Logic through this examination approach of the standard syllogism. Good luck. And have fun! Please excuse my typos! I am using my phone, stumbled across your post, and felt led to share. I must run to teach classes now.
