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I would like to receive a book recommendation about first order logic. One might say that there are plenty of references in the internet about this topic but finding a book is difficult for me because of two reasons:

  1. I have no understanding in this topic at all - I am complete beginner. To see why this is difficulty, I wanted to consider Bourbaki as they always have nice historical notes that help me to understand the "logic" behind definitions in logic, for example, but then I (luckily) found a lot of discussions that the text of Bourbaki about logic is shown to be refuted.

  2. There are too many approaches in first order logic. Some books start with basic set theory and then do logic. Some books start with logic to allow reader understand set theory written in first order logic. Some books give one axioms and rules of inference, some give others. This makes me confused as to which kind of approach should I take?

Another information about me that could help me to find a book is that I am not really interested in mathematical logic in a deep way. I want to learn it to be sure that I prove theorems correctly and express myself in mathematics clear enough so that I do not make some mistakes due to my intuition.

My goal is to read Halmos "Naive set theory" which is written in somewhat informal language. I would like to be able to transfer informal language into formal first order logic language.

My specific request would be for a book that contains basic definitions, uses philosophy that "first order logic is before set theory" and most importantly gives rules of inference for mathematical proofs. At this moment in my education I prefer Hilbert style (if this is the correct name) where rule of inference is only modus ponens. With natural deduction I feel that the problem for me is that there are too many rules of inference to remember and that might confuse me even more. Maybe this is not true, I don't know. I am not interested in very deep applications as Godel theorems or model theory, or something similar. Of course, it would be nice if there was some basic explanation why these things (such as Godel theorems) are important and interesting, but nothing more than that.

I hope this question does not count as duplicate as I have tried to explain the current specific state of my education and future goals.

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You could start from this Study Guide to logic books as a pointer to some of the best that are available, and then you can pick what suits you.

Chapter 3 is the key one for you. The warmly recommended Friendly Introduction ... is now freely available as a PDF download, courtesy of the authors.

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