(Part 2) Best products from r/logic

The best introductory logic text you will ever find: Logic: Techniques of Formal Reasoning, 2nd Edition Donald Kalish, Richard Montague.
This book is especially good if you have done any programming. The structure of main and sub-proofs corresponds to main program and subroutine calls. You can pick up a used copy for around $23 here: https://www.abebooks.com/book-search/author/kalish-montague-mar/ and you can see the table of contents here: https://www.powells.com/book/logic-techniques-of-formal-reasoning-9780195155044 (but, obviously, don't buy it for $133!)

For meta-theory, take a look at: Metalogic: An Introduction to the Metatheory of Standard First Order Logic by Geoffrey Hunter, https://www.amazon.com/Metalogic-Introduction-Metatheory-Standard-First/dp/0520023560. This book explains things in a clear way using ordinary English, before setting out the proofs.
And, if you are interested in model theory, take a look at Model Theory by C.C. Chang and H. Jerome Keisler, https://www.amazon.com/Model-Theory-Third-Dover-Mathematics/dp/0486488217 and you should get a good idea of what additional mathematics you might want to pursue.

Well there's Tennant's new book, Core Logic. I haven't read it, but I hope to convince a couple of my colleagues to join me in doing so this year.

The Stanford Encyclopedia of Philosophy and the Internet Encyclopedia of Philosophy are free sources. Most books I recommended are pretty cheap and worth having a physical copy. For instance, Forever Undecided is just $12 for a new copy, less than $3 for an used hardcover. But, if price is too impeding for you, you can always find a pdf copy on the internet.

Stoll's Set Theory and Logic is excellent (albeit a tad old). I particularly like that he devotes a whole chapter to Boolean algebra--an in-depth investigation of a complete axiomatic system with deep ties to the logics covered earlier in the book.

I can recommend two books which I have read recently.

1. An Introduction to Mathematical Logic is more structured and formal description of logic.
   
2. [Introduction to Logic] (http://www.amazon.com/Introduction-Logic-Methodology-Deductive-Mathematics/dp/048628462X/ref=sr_1_7?ie=UTF8&qid=1449702263&sr=8-7&keywords=mathematical+logic) gives more insights and helps to get a big picture of logic.
   I enjoyed both of them a lot and going to read them again.

Logical Labyrinths by Raymond Smullyan would serve as a friendly introduction to the subject. Smullyan frequently deploys his famous logic puzzles throughout the text to teach a variety of lessons. And a word of warning: friendly doesn't mean easy! Smullyan will push your buttons and make you think deeply about tricky concepts. This one is very rewarding and comes highly recommended.

In layman's terms, incompleteness:

Imagine an actor, like Arnold Schwarzenegger, is playing some role in a fun, not so serious movie. At one point, Arnold's character says, "Even Arnold Schwarzenegger couldn't pull this off!"

So if you believe what the character is saying, then you believe the actor couldn't actually be in the movie they are in and said what they said. But then you couldn't have believed what the character was saying, since it was that actor themself making the claim. But that was what the character claimed to begin. Paradox. (this example via)

The same goes for some mathematical systems:

If they can talk about themselves in a certain way, then they can talk about what they can or can't do (prove). By constructing a particular self-referential statement within the system about what can be done, it ends up in paradox. We interpret this to mean that there is a limit to what can be consistently proved in these systems.

Since mathematical logic has split after the Second World War in four or five branches, it is uncommon for more advanced texts to have a broad focus on logic in the general sense like beginner's books.

That being said, the most 'advanced' (and quite recent) logic book that attempts at doing justice to the main (though pretty introductory) results of all the major branches of logic is Hinman's Fundamentals of Mathematical Logic (it claims to be an update on Shoenfield's Mathematical Logic, which has some more info on proof theory).

In the spirit of the answers below, where 'advanced' is taken to mean most unreadable or with the most prerequisites, I would nominate another of Shelah's books, Classification Theory.

One of the best and most comprehensive sources for a historical narrative is Kneale & Kneale, The Development of Logic.

Barwise and Etchemendy is pretty good for a first exposure: http://www.amazon.com/Language-Proof-Logic-2ND-Edition/dp/1575866323

You will be interested in Peter Smith's Logic Matters site. For comic relief, check out Logicomix.

If inconsistent, paraconsistent, and otherwise Brazilian logics aren't already your bag, my usual recommendation is Priest's Introduction to Non-Classical Logic.

I would, of course, love to hear others' suggestions.

This is how I learned logic, for computer science.

First chapter of this Discrete mathematics book in my discrete math class

https://www.amazon.ca/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328


Then, using The Logic Book for a formal philosophy logic 1 course.
https://www.amazon.ca/Logic-Book-Merrie-Bergmann/product-reviews/0078038413/ref=dpx_acr_txt?showViewpoints=1


The second book was horrid on itself, luckily my professor's academic lineage goes back to Tarski. He's an amazing Professor and knows how to teach...that was a god send. Ironically, he dropped the text and I see that someone has posted his openbook project.

The first book (first chapter), is too applied I imagine for your needs. It would also only be economically feasible if well, you disregarded copyright law and got a "free" PDF of it.

This is not an online resource, but this book is good if you can find it.

https://www.amazon.co.uk/Logic-Trees-Introduction-Symbolic/dp/0415133424

> (p ≡ q) :: [(p ⊃ q) ∧ (q ⊃ p)]

How would you derive T ⊃ H and H ⊃ T?

Edit: I would suggest this book instead. IMO Hurley contains way too much fluff trying to justify its ridiculous cost. 300 pages to get to truth tables and connectives is a joke.

Edit: Section 7.5 should help you.

Well, to answer the question "is it logical that both can be correct?" Sure! There are logics that allows contradictions to be designated, so it's "logical" in that it's perfectly acceptable within the rules of at least one logic that "p ^ ~p" is true.

As far as the applicability of those logics to reality, which might be another aspect to the question rather than a new question, the Liar Sentence and phenomena in the boundary area of vague predicates have been put forth as examples of things that actually are contradictory, and so would be accurately modeled by logics that tolerate contradictions.

http://www.amazon.com/The-Law-Non-Contradiction-Graham-Priest/dp/0199204195

That book there is highly relevant.

This is for a general study guide for logic. Very solid.

http://www.logicmatters.net/tyl/

and I think you can give one of these two a try if you find "Computability and Logic" difficult.

https://www.amazon.com/Metalogic-Introduction-Metatheory-Standard-First/dp/0520023560

or

https://www.amazon.com/Friendly-Introduction-Mathematical-Logic/dp/1942341075