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I'm familiar with textbooks on logic, proof techniques, and sets. But I have yet to encounter a textbook that dives into the language used w/ definitions and sentence structure used in proofs, for example(when to use the word "claim" in a proof and its meaning and what you have to deduce by saying that, or what does it mean to use the word "fix" on an element and its meaning. Or when should you use "without loss of generality" and its meaning, you get the picture. In other words, proof writing

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    $\begingroup$ Without loss of generality we can safely say it's all contextual :P $\endgroup$ – Zelos Malum Nov 3 '15 at 8:19
  • $\begingroup$ I think one learns the 'language' of proofs just like any other language: by imitation. It helps if you use textbooks that are written with clarity and elegance. Also, if you have instructors who take the time to give constructive criticism of your proofs, indicating incorrect or confusing choices of words, unnecessary repetition, gaps in logic, and so on. $\endgroup$ – BruceET Nov 8 '16 at 5:48

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