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I'd like to learn formal math. Preferably, though not necessarily, starting with predicate logic/first order logic rather than higher order logic. I am trying to find resources (papers, books etc.) for doing this, but I haven't found anything I really like.

There are lots of resources for predicate and first order logic, but most do not approach the topics in a very formal way. For example, many text don't seem to try to define what they mean by "variables" or mention substitution as an important concept. Tries to explicitly describe as many of the rules of the game as it can. Many texts bring up "truth tables" without having formal rules for what you're allowed to do with those tables.

Does anyone have resources that fit these criteria?

Edit: many of the answers are good and helpful, but I feel like I should add some clarifying remarks: Many texts mention that you can view math as merely manipulation of symbols. I don't doubt that this can be done, but I would like to see it done. A resource that explains the process of producing proofs explicitly in terms of manipulating symbols rather than in terms of functions, statements etc (at least without first defining these terms) would be helpful. I'd like to be able to pretend I was a person who didn't know any math and was just acting as a human computer, producing proofs. I'd like a resource that explains producing proofs like I was such a computer (not necessarily ONLY like that).

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Just a note: I don't think there are formal rules for what you can do with truth tables, apart from create them and stare at them. They're an ephemeral topic that appears early in the subject and disappears soon after. –  Carl Mummert Nov 17 '10 at 13:02

5 Answers 5

up vote 5 down vote accepted

For an undergraduate-level book, none of these three can do you wrong:

  • Mendelson, Introduction to mathematical logic
  • Enderton, A mathematical introduction to logic
  • Boolos, Burgess, and Jeffrey, Computability and logic

They certainly go into the details, and they will leave you in a position where you could go further. If you can only look at one, try Mendelson.

An older text, which is now reprinted by Dover, is Kleene's Mathematical logic. It is also very thorough, and has the advantage of being inexpensive. (Be aware that Kleene's Introduction to metamathematics is a completely different book, which is much more advanced.)

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There is a good list on this website. I can personally recommend the Walicki notes (free). It includes a good introduction to modal logic, so that's always nice.

Of the books, Geoffrey Hunter's Metalogic is very good and covers all bases.

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Hunter's book is actually quite idiosyncratic, and I (personally) wouldn't recommend it for anyone who is going to do any additional work in logic. –  Carl Mummert Nov 17 '10 at 12:47
    
Thanks for the link :) –  John Salvatier Nov 18 '10 at 4:40

Aside from general wikipedia and searches from google I would advise going to either the public or collegiate library. When I wanted to learn more about that I got "Introduction to Mathematical Structures and Proofs" by Larry J. Gerstein. It was in a shelf of free books outside my college's library.

In regard to the truth tables they should have. You will see they are generated from the ideas of Boolean algebra. In fact, http://en.wikipedia.org/wiki/Truth_tables explains it quite well. I believe the necessary part is to do problems, and for that all you need is to go through definitions and see their consequences.

As for texts for important concepts you may wish to pick up a book on the philosophy of mathematics, though you can see simply about variables from http://en.wikipedia.org/wiki/Variable_%28mathematics%29. Some can be rather dense, I would suggest an introduction. I am reading "The Philosophy of Mathematics" by W.D. Hart. It is a collection of short essays pretty much on important philosophical issues and notions in mathematics.

Have fun.

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May I humbly suggest some educational freeware I have developed to teach formal logic and basic proof-writing skills. It is a downloadable, PC-based proof assistant (editor and logic-checker). Being a computer program, it is of necessity a purely formal system.

Included with the download is a tutorial that can be used as an introduction to formal mathematics. It includes ten worked examples of formal proofs, plus exercises with hints and full solutions.

For testimonials, and a free full-function download, visit my website: http://www.dcproof.com

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Seems like it might be interesting, but 1) you need more screenshots 2) your contact link is broken or something. –  John Salvatier Nov 18 '10 at 4:22
    
There are screenshots on the Features page. Also click on "View table of contents and excerpt" on that page. I just tested the contact link. It is a link to my gmail account. It seems to be working now. Thanks for the feedback. –  Dan Christensen Nov 29 '10 at 16:24

If you're still looking for more resources here, you might want to look at

  1. The Schuam's Outline of Logic which uses a Fitch-Jaskowski style natural deduction calculus (though it doesn't call it such), though admittedly the rule of substitution doesn't get used much here.

The next four axiomatically develop classical logic:

  1. Jan Lukasiewicz's Elements of Mathematical Logic.
  2. A. N. Prior's Formal Logic.
  3. The Metamath site.

I'd think the Metamath site closest to what you seek.

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