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In general, the standard turnstile is used to represent a/the consequence relation in standard monotonic logic, whereas the wavy turnstile is used for the consequence relation for non-monotonic logic.

I don't know if this will be at all helpful, but when I was studying non-monotonic logic, I used to think of the wavy turnstile as being similar to the approximately-equals sign - if a = b, then a is always equal to b, but if a ≈ b then there might be some future circumstance in which we cannot consider them to be similar.

In terms of reading material, I can recommend the textbook that we used, D Makinson's Bridges from Classical to Nonmonotonic Logic. I used it as part of a class, but I think Makinson does a great job of explaining concepts and it wouldn't be too bad to go through solo. Also, assuming your professor isn't a total nightmare, I'm sure she will be happy to help you when you get stuck - they love when students make an extra effort to engage with the material.

Ooh, I'm studying this stuff right now! Check out this paper by Kripke for a semantic approach to truth inside a theory. That sparked a lot of interest in the area, so check out this book or just read this page, both by Volker Halbach, for a great outline of the different theories that have emerged to capture truth in a theory.

Basically, the whole idea depends on your notion of truth. Someone correctly mentioned below that you cannot define truth inside a theory (a predicate T such that for every statement P in the language you have the following equivalence: T(P) <-> P). This said, people have developed different axiomatizations of a truth predicate that enables certain qualities we think truth should have (that it is compositional for example, so that T(p&q) <-> (T(p)&T(q))). These each have their pitfalls, though; this paper by Hannes Leitgeb outlines nice qualities that a truth predicate should have (but cannot have simulataneously).

There are some really interesting results in the literature and I'd be happy to link you to further papers/answer questions if you're interested.

> Unfortunately, since that last class, I've fallen out of it and I'm not entirely sure how to get back in. I'm not very good at teaching myself things.

I think that self-studying is a skill. And just like any other skill, you become better at it the more and the better you practice it. If you aren't very good at it yet, then you probably just haven't done it much, or perhaps you haven't done it properly.

If you don't know where to start developing the skill, I highly recommend reading the article The Making of an Expert (PDF) by K. Anders Ericsson, published in the Harvard Business Review. It is a concise introduction to Ericsson's research on acquiring expertise, full of valuable insights. Some of the more useful and relevant ones are the importance of deliberate practice in acquiring expertise, how long it actually takes to become proficient in a field of expertise, and the fact that the final stage in acquiring expertise involves no instructors (i.e. it is characterized by self-studying).

I also believe How to Read a Book by Mortimer J. Adler to be useful in developing this skill. This book describes the difference between present teachers, like the ones you can interact with in an educational institution, and absent ones, such as the authors of books. It then lists a number of very useful general guidelines on how to approach learning from these absent teachers, followed by some more specific ones describing how to approach different kinds of reading matters. It is essentially a self-studying guide.

And since this is /r/logic and you expressed an interest in getting back into the subject, my final recommendation is A First Course in Mathematical Logic by Patrick Suppes and Shirley Hill, which is an exceedingly lucid, accessible, elementary and rigorous introduction to logic. It is very well-suited for self-studying and might be a useful refresher, although depending on the courses you've taken and how much you recall from them, it may be too elementary for you. I posted a more detailed description of the book in a different thread on here a few days ago.

There are probably a couple boolean logic ones? I haven't played a lot of logic games. I used to play a game called tis-100 which is a game about a weird parallel assembly type language that I found pretty fun, it has some logic elements to it. It looks like there are a few logic games on the android playstore but I can't vouch for any specifically.

I know a couple books that looked kind of fun:

https://www.amazon.com/Mock-Mockingbird-Raymond-Smullyan/dp/0192801422?SubscriptionId=AKIAILSHYYTFIVPWUY6Q&tag=duckduckgo-ffab-20&linkCode=xm2&camp=2025&creative=165953&creativeASIN=0192801422

Some of the recommended ones for this book that popped up for me looked cool as well.

Dover has some cool looking recreational logic books.

You can also always try and make new formulas to work on for yourself by using chapters from topics that you already covered as inspiration.

So if you know propositional logic then you can make some propositional arguments and try to prove or refute them for yourself.

I learned logic from 'A concise introduction to Logic.' by hurley

Do you have some idea of the type of logic you want to learn? an introduction into modern logic usually encompases two aspects: propositional (sentential) and predicate (first-order) logic. Once you learn these, you can learn other types of fun logic like metalogic, modal logic, and maybe even set theory. There's also Aristotelian logic (i.e. syllogisms and the square of opposition.) I learned this, but i never use it so i wouldn't recommend learning it (although the text i provided has chapters on both)

Overall, I think it's a great text and you can easily learn everything in the book on your own without a teacher. The book's pricey, but it's worth it if you're serious about learning Logic. I still use it whenever i want to learn what a specific term is called, remember a rule of inference, predicate logic restriction, etc.

http://www.amazon.com/Concise-Introduction-Logic-Patrick-Hurley/dp/0495503835

i've found that this is a pretty good resource if you want a written introduction online:

http://www.cs.odu.edu/~toida/nerzic/content/logic/intr_to_logic.html

I really liked Irving Copi's Introduction to Logic. I don't know if its the best for self-learners per se but over all its just a great logic textbook and really helped me out. Also, Irving Copi studied under Bertrand Russell while at the University of Chicago so there's some bonus points right here.

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.

I don't know if category theory is your thing but maybe you'd be interested in categorical logic?
http://www.amazon.com/Introduction-Higher-Order-Categorical-Cambridge-Mathematics/dp/0521356539

that book goes a lot into various ways to study logics inside categories and indeed categories as certain kinds of deduction systems. It also has a nice introduction to toposes which are kinds of categories which have a kind of inner logic (usually multivalued and always intuitionistic). It links nicely to some newer branches of math too.

If you're unfamiliar with categories they are abstractions of collections of mathematical structures and the functions between them(eg. sets and functions, or vector spaces and linear functions).

Well I learned from these (undergraduate level):

The Power of Logic by Howard-Snyder, Howard-Snyder and Wasserman

and

Methods of Logic by Willard Van Orman Quine

I highly recommend both but Methods is not a good place to start. Excellent once you can handle yourself though. Unfortunately The Power of Logic is somewhat expensive.

By the way, here's an excellent online resource that you may find helpful.

For basic logic (first-order, classical) these are excellent textbooks...

• Goldrei, Propositional and Predicate Calculus
  
• Enderton, A Mathematical Introduction to Logic
  
• Smullyan, A Beginner's Guide to Mathematical Logic
  
  I should also mention that there are some free, open source textbooks on standard logic (good books, written by scholars in the field) such as the following...
  
  
• "Forall X" (Cambridge Edition)
  
  And if you want to go beyond classical logic, these are excellent textbooks...
  
  
• Priest, An Introduction to Non-Classical Logic
  
• Van Benthem, Modal Logic for Open Minds

Hey there.

You mention that your brother is bright -- how bright exactly? First of all, Computability and Logic is quite an advanced book that is typically aimed towards 2nd year logic students, and is usually for students who have taken a rigorous discrete mathematics course in their first year.

It delves quite deeply into the theory of logic and the philosophy of mathematics and would not be suited as a light exercise book for someone unless they have taken a math-heavy first-year logic course and are planning on taking up electrical engineering or something of the sort.

As for Hurley's book, a Concise Intro to Logic, well, this is on the other side of the spectrum -- it is very watered down compared to other logic readings, and pales in comparison (to most other introductory logic books) with regard to depth and breadth on formal logic.

It's usually aimed at first-year philosophy students who are taking introductory courses in logic or critical thinking, and most of it is simply rote-learning certain forms of argument as well as a lot of "quick and dirty" techniques which mimic that of a dry maths textbook. If you're looking for an interesting exposition into logic, then this book is certainly not it -- it would serve better as a high-school introduction for logic, and if prescribed to anyone older, would be very lackluster.

Here are some suggestions for you:

• Raymond Smullyan's A Beginner's Guide to Mathematical Logic
  
  
• Nicholas JJ's Logic: The Laws of Truth
  
  
• Peter Smith's An Introduction to Formal Logic
  
  
• RL Simpson's Essentials of Symbolic Logic
  
  If you're looking for mathematically-oriented logic puzzles, then you should check out Smullyan's series of puzzle books. You can find them on Amazon.
  
  
  
  
  

If you're interested in non classical logics I'd recommend "An introduction to non classical logic" by Graham Priest (it has modal logic and other very interesting non-classical logics). It's a good overview of the field.

For denser subjects in classical logic like computability, Turing machines, Gödel theorems, proofs for compactness, correctness, completeness, etc.; I'd go for a classical work by now: "Computability and logic" by Boolos, Burgess & Jeffrey. It's not an easy reading though.

Greg Restall's Logic: an Introduction is an excellent introductory text, I think. It not only covers propositional and predicate logic well, it also shows some of the limitations of those systems, and thus why extensions to them might be useful for some purposes. Very easy to read, too.

I'm just a beginner but Peter Kreeft's book on Socratic Logic is very good. I've learned a lot from this introductory book. He's very effective at communicating rather complex concepts with simple language. There are also a plethora of exercises in the book at the end of every section.

https://www.amazon.com/Socratic-Logic-Questions-Aristotelian-Principles/dp/1587318083

Sol Feferman is one of the greatest logicians of the second half of the 20th century, and quite a good writer. I haven't read his Gödel biography, but you can rest assured that, as far as its mathematical and philosophical content is concerned, it is of the very highest quality; he actually knew Gödel in person, though not very well, as he was a shy graduate student back then in Princeton. One of the few close friends of Gödel that wrote about him was Hao Wang. You might want to take a look at his writings, e.g. https://www.amazon.com/Reflections-Kurt-G%C3%B6del-Hao-Wang/dp/0262730871.

Consider Raymond Smullyan's A Beginner's Guide to Mathematical Logic. It has some history of logic mixed in with pretty good coverage of propositional and first order logic, as near as I can tell. Lots of exercizes, which helps me personally.