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So I have been learning Hilbert style FOL from "Introduction to Mathematical Logic" but some people say that it is impractical. On the other hand , I want to study axiomatic set theory , real-analysis etc using a FOL deductive system. So I have been trying to learn Fitch-style Natural Deduction for Propositional and Predicate Calculus.

The textbook I am trying to find should have the following things:

(1)FOL syntax.
(2)A logical deductive system (Natural Deduction fitch style).
(3)Conventional and self contained.
(4)Should be Concise.
(5)Only requires pen and paper.

I have found one book "Language, Proof and Logic" but I found it a bit too wordy for It has a lot of exercises which requires the programmes provided with the paid version of the book.I have the free version,So I might miss out on a lot of things I could have learned.I haven't been able to find any other book about this subject.Does anyone know of a textbook where I can learn this kind of Deductive system to use for all mathematics?

Kripke Platek
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  • If you want to master the basic methods of mathematical proof, may I humbly suggest my DC Proof 2.0 freeware with accompanying interactive tutorial that is downloadable from my homepage: http://www.dcproof.com It is based on a form of natural deduction that is implicitly used in most math textbooks. It is not standard FOL or Fitch-style. – Dan Christensen Jul 26 '21 at 14:51
  • You can search for F.B.Fitch, Symbolic Logic: An Introduction in libraries. – Mauro ALLEGRANZA Sep 23 '21 at 14:30
  • but there are many textbooks with ND: van Dalen as well as Chiswell & Hodges – Mauro ALLEGRANZA Sep 23 '21 at 14:32
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    @MauroALLEGRANZA: The asker was looking for a textbook with a practical deductive system. So, unfortunately, almost all logic texts including those two you mentioned are useless for this purpose, because they are books that study logic rather than teach how to use logic to do mathematics. For example, van Dalen's system starts with only ∧,→,⊥,∀ as primitive and defines the others in terms of those. That's good for studying FOL (less cases), but bad for using FOL. – user21820 Sep 24 '21 at 14:19
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    Also, C&H is wrong in their claim "Sometimes in mathematics one would like to allow structures with empty domains, but these occasions are too few to justify abandoning natural deduction.", because there are ND systems and Fitch systems that work for empty structures. In fact I even recommend my Fitch-style system over other systems. And here is a sequent-style counterpart. – user21820 Sep 24 '21 at 14:20
  • Oh by the way: https://math.stackexchange.com/posts/comments/4272361. – user21820 Jun 07 '22 at 22:45
  • Fitch style (and flag style) natural deduction really is just a linearized representation of usual natural deductions, plus some variations on the how open assumptions are defined (set vs multiset vs sequence) and discharge conventions (full discharge? single discharge? optional discharge?). – Poscat Apr 07 '24 at 08:33
  • I would recommend starting with an introductory proof theory book, since those books actually mention natural deduction and the first order logic used usually has all the connectives (cause intuitionistic logic). They also usually teach you how to use natural deduction to prove stuff, though those are usually very general schematic derivations. For example, the book An Introduction to Proof Theory (978-0192895943) would be an excellent choice. – Poscat Apr 07 '24 at 08:33
  • @Poscat: The fact that those books only teach you with toy examples demonstrates that they are not actually practical systems. In contrast, the system I have linked to is a system that can be used for real mathematics. The asker finished learning the core part of that system over a few months. Anyone who learns my system can complete these exercises, which provide stepping stones to actual full-scale mathematics within the system (perhaps plus a bit of syntax sugar). – user21820 Apr 07 '24 at 12:35
  • I haven’t found such a book. I used the information mostly in Van Dalen to create my own. Note that fitch style natural deduction (or tree style natural deduction or Hilbert style FOL) is not suitable for writing general mathematical proofs; it’s much too tedious. In my opinion, however, understanding Fitch style natural deduction does help to better understand proof techniques used in general maths. – Porky Sep 01 '24 at 13:02
  • @Porky: What do you mean by too tedious? Have you tried using my Fitch-style system (see the link above), which was specifically designed to be practical? A bit of syntactic sugar is needed (e.g. chaining relations), but all mathematics can be done within my system without much more difficulty than an actual rigorous proof would entail. – user21820 Sep 01 '24 at 18:44

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forall x checks all the points in your list. It doesn't discuss examples from concrete mathematical theories, but it introduces the formalism generally enough to be applicable to set theory and the like, and is well-written i.m.o., and completely free.

Natalie Clarius
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  • Please don't recommend forallx next time as it causes serious conceptual misunderstandings. I also didn't know until multiple people came to me with questions about that book. If you find another free introductory logic text with a Fitch-style deductive system, please let me know as I could not find any except LPL. I also have my own, but it is not for really beginner-level consumption, though I use it in my teaching. – user21820 Jul 30 '21 at 13:43
  • Thanks for letting me know, I'll have a look at the chat messages you linked. – Natalie Clarius Jul 30 '21 at 14:20
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    The book is open source, and the sections of the book causing the misunderstanding can be edited on the github page if you think those paragraphs can be improved. – saolof Jul 25 '23 at 07:57