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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 ! :)

Zev Chonoles
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Mathlover
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  • and i found this definition from an article that : A formal language consists of a set of symbols together with a set of rules for forming \grammatically correct" strings of symbols in this language. In this context, what does the author mean by "set"? Moreover, how could one say "infinite" in logic? In ZFC, axiom of infinite assures we can say "infinite", but in logic, how does one so? – Mathlover Feb 03 '14 at 20:06
  • You may want to check out the answers here: http://math.stackexchange.com/questions/140681/where-to-begin-with-foundations-of-mathematics – GovEcon Feb 03 '14 at 20:09
  • Pick up Enderton's A Mathematical Introduction to Logic. Mendelson's Introduction to Mathematical Logic is also standard. Once you're done with those, continue with Shoenfield's Mathematical Logic. Peter Smith has a useful guide for these things, so just google "peter smith logic". – Hunan Rostomyan Feb 03 '14 at 20:16
  • @JordanMahar Thank you, the link is really helpful :) – Mathlover Feb 03 '14 at 20:21
  • @HunanRostomyan Peter Smith recommended me his text last year, but i found really hard to read his text, since his text assumes readers have logical background. The text was : Godel's Incompleteness Theorem – Mathlover Feb 03 '14 at 20:23
  • If you prefer a "mathematical" perspective on logic, I think you can try with Richard Kaye, The Mathematics of Logic (2007). If you want a "philosophical" perspective on mathematical logic (and not philosophical logic), I suggest you Yuri Manin, A Course in Mathematical Logic for Mathematicians (2010). – Mauro ALLEGRANZA Feb 03 '14 at 20:27
  • Mathlover, Peter Smith's introduction to Goedel's theorems is extremely readable when compared to the alternatives, such as Boolos & Jeffrey & Burgess. It really doesn't get any more accessible, in my humble opinion. But leave proof theory alone for now and just study some axiomatic set theory and mathematical logic with Enderton's texts. There are lots of alternatives, but you gotta start somewhere and Enderton is an easy and tested choice. – Hunan Rostomyan Feb 03 '14 at 20:28
  • @Jon The reason i disliked texts starting with propositional logic is that i have seen (i don't remember what text) that truth values of propositional logic is not anymore axioms but rather theorems in some general context. If propositional logic is more 'deeper' than any other logic, why would i not start with it? Is propositional logic more general than first-order logic? – Mathlover Feb 03 '14 at 20:31
  • @HunanRostomyan It wasn't to me, at least, since he sometimes uses notions not defined in his text, to define another notions. I hope an author specifies what are undefined notions and what are defined notions. Anyway, thank you for the recommendation! I'll check Enderton's text in a library :) – Mathlover Feb 03 '14 at 20:37
  • If you want to skip propositional logic you may try out Ebbinghaus / Flum / Thomas: Mathematical Logic, where first-order logic takes center stage from a model theoretic-point of view with many applications to mathematical structures. I like its presentation of first-order semantics, but its proof-theory (a really strange sequent calculus) is quite awkward and the completeness proof more complicated than necessary. General properties of logics are studied in universal logic, where a logic is taken to be a (recursive) set together with a relation between its subsets and its elements. – Jon Feb 03 '14 at 20:37

4 Answers4

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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

Peter Smith
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  • What is the difference between "philosohical" and "mathematical" logic? – Trismegistos Feb 04 '14 at 09:37
  • Well, for an example, compare (A) the mathematical investigation of intuitionistic logic with (B) the philosophical task of assessing various considerations which purport to show that, in certain domains, we are only justified in using intuitionistic modes of reasoning. These are different enterprises, even if each may throw light on the other. – Peter Smith Feb 04 '14 at 12:08
  • B) sounds interesting. What are examples of domains for which you are only justified to use intuitionistic approach and how it is justified? – Trismegistos Feb 04 '14 at 12:35
  • @Trismegistos - I think that can be of your interest to see at (some portion of) the philosophical writings of Michael Dummett, for application of an "ituitionistic" approach applied to the theory of meaning, for example. Or you can see the "constructive" approach to mathematics of Errett Bishop, for an application of "traditional" logic to mathematics motivated by the "intuitionistic" atttention to "effective" proofs and results. – Mauro ALLEGRANZA Feb 04 '14 at 13:14
  • @Trismegistos - Check out smooth infinitesimal analysis (there's a book by John L. Bell about it). The theory becomes inconsistent if you work in classical logic, but it's consistent if the underlying logic is intuitionistic. – Nagase Nov 19 '15 at 14:12
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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.

user27887
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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.

Forgottenscience
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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.

mamabergy
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  • Oh and get the latest edition grey and white cover....not the green and blue one. Both cover same material different formatting. Grey is the one I review here. – mamabergy Nov 19 '15 at 12:18