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In the preface to his book on logic Dirk van Dalen talks about the duality between "profane" and "sacred" logic, referring to relaxed logic and extremely formalized logic. He then explains his book will be more of the relaxed kind.

I've read the introduction of a handful of books on logic and this is a common theme, they all say "this can be done very very formally, but we won't do it here". I feel, however, that due to my interest in mathematical philosophy (related to the fact that I'm beginning to study independence proofs) I'm in need of one of these overly formal books on logic that so many authors want to avoid writing.

Can somebody give me a recommendation?

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    $\begingroup$ Whitehead and Russell's Principia Mathematica ? $\endgroup$ – Guest Jun 1 '16 at 18:17
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    $\begingroup$ I'm not sure there is a modern text which has an extremely high level of formality. It would be like a book that was supposed to introduce someone to programming being in machine-code. Usually the "informal" parts involve things like dropping extra parenthesis, not having a fixed set of variables for everything in the book, perhaps sometimes treating a connective as a defined symbol, and sometimes as a primitive one etc etc. $\endgroup$ – James Jun 1 '16 at 18:18
  • $\begingroup$ You can find W&R's Principia online, the section on mathematical logic starts here. $\endgroup$ – pjs36 Jun 1 '16 at 18:22
  • $\begingroup$ @James That's a useful comment. I don't really know how "naive" the logic I'm studying is, and how much more formal "sacred" logic is. How can I tell if I'm being naive or not? $\endgroup$ – JKEG Jun 1 '16 at 18:27
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    $\begingroup$ If your interest is in set theory and independence, I would imagine you'll eventually want to learn forcing. It is my understanding that working set theorist like to ignore the formal part of what they are doing as much as possible because the details of forcing are very complicated. In my own subject (recursion theory) I have never once, in my whole live, explicitly written out a Turing machine program to do anything, I just give high level descriptions of algorithms in English. $\endgroup$ – James Jun 1 '16 at 18:46
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Try Alfred Tarski's Introduction to Logic and to the Methodology of Deductive Sciences. His notation is dated but it's still a good source. And the book is very cheap.

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Some suggestions:

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