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 ! :)
There are two very different kinds of question here:
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.
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