I want to learn axiomatic (Hilbert style ) logic. not just a book that says that it exist and is an good way to proof theorems.

What is a good book to learn and practice this method?

would like:
- a book published after 2000
- not limited to a particular axiomset or set of connectives
- lots of examples.

What I especially would like is a book that teaches how to transform Natural Deduction or Sequent Calculi style proofs to axiomatic Hilbert Style proofs. (I know it is a complex subject, it depends on the axioms and is not even always possible)

At the moment I am studying Bergmann's "An introduction to many valued and fuzzy logic" that uses this style of proof just because other proof styles are either invalid or even more complicated for this type of proof

  • $\begingroup$ A good book for this topic is Kleene's "Introduction to metamathematics", but it is oldish and you may find the style difficult as an introduction. A good possibility is "Fundamentals of mathematical logic" by Peter Hinman, though the book covers much more than this topic. Or you may want to visit Peter Smith's webpage, Logic Matters. $\endgroup$ – Andrés E. Caicedo Aug 12 '13 at 17:15
  • $\begingroup$ It might come as some use to consider thinking of each axiomatic system as having its own methods to some extent, since different Hilbert systems have different derivable rules of inference. Even when you can derive the same theorem set in principle, derivable rules of inference can change the ease with which a given theorem can get proved. E. G. for {CCpqCCqrCpr, CCNppp, CpCNpq} you can prove Cpp using the derivable rule "from C$\alpha$$\beta$, C$\beta$$\gamma$, infer C$\alpha$$\gamma$" and making just one substitution. For {CpCqp, CCpCqrCCpqCpr, CCNpNqCqp} you need to work differently. $\endgroup$ – Doug Spoonwood Aug 13 '13 at 19:17
  • $\begingroup$ This question has lead me to look at L. H. Hackstaff's 1966 System of Formal Logic. A reviewer said "An excellent reference. Shows the properties of a great many formal logic systems (such as standard PC and intuitionist PC) by building them as extensions of simpler systems with more easily proven properties." Is the age of the book really a relevant consideration here? Does it affect the content or usability or general value of the book? Maybe it is, maybe I've missed something, I don't know. $\endgroup$ – Doug Spoonwood Aug 21 '13 at 5:08
  • $\begingroup$ If you send me your email address (mine is at my profile), I can send you a manuscript-in-progress of mine that may be of help. $\endgroup$ – Dave L. Renfro Aug 27 '13 at 18:25
  • $\begingroup$ "What I especially would like is a book that teaches how to transform Natural Deduction or Sequent Calculi style proofs to axiomatic Hilbert Style proofs." There exist different ways to do this, because there exist different ways to derive the rules needed for a deduction metateorem. If you use CpCqp and CCpCqrCCpqCpr to get to the rules $\alpha$ $\rightarrow$ C$\beta$$\alpha$ and {C$\alpha$C$\beta$$\gamma$, C$\alpha$$\beta$} $\rightarrow$ C$\alpha$$\gamma$, when changing a derivation of {C$\psi$$\phi$, $\psi$} $\vdash$ $\phi$ into one of ... $\endgroup$ – Doug Spoonwood May 13 '14 at 14:42

I've started studying L. H. Hackstaff's 1966 book Systems of Formal Logic. It looks useful for this purpose and seems to have fundamentally sound advice. Its systems don't have a rule of simultaneous substitution as a primitive rule of inference or involve axiom schema. If more systems did this it might actually make it possible to teach the axiomatic method in logic at a lower level than presently done.

It has sections on the axiomatic method which says things like:

  1. Know (the major ones/some of the major ones) the derivable rules of inference of the logical systems by heart.
  2. Know the axioms of the logical system by heart. Otherwise you'll have to refer to the axioms constantly when trying to construct a proof.
  3. Know the rules of inference by heart.
  4. Know the problem.
  5. Know the evidence.

Unlike many other books I have seen it also has plenty of examples axiomatic proofs (with the abbreviations for wffs adopted by the text).

As an example of how to prove something using just one variable at a time substitution. Suppose our first thesis is that of suffixing or reverse hypothetical syllogism.

Our language here has only has variables {a, b, c, ..., f} and subscripted variables if necessary. Our meta-language will happen with {p, ..., z} so we can talk about rules of inference.

1 C Cab C Cca Ccb

If we used condensed detachment would tell us that the next most general result of this system will be C CdCab C d C Cca Ccb. Can we prove this using only single variable at a time substitution and detachment?

1 has form Cxy, and so does Cab. So, since we only have one thesis, it seems to try to make or find some form of CCabCCcaCcb such that we can detach something. How might we do this? Well the antecedent of 1 is Cab and the antecedent of Cab is a. So, we need to substitute something for a such that a has form Cxy. Would Cab work?

2? C CCabb C CcCab Ccb.

We want to have "b" in one have the form CCcaCcb. But, the rule of uniform substitution happen tells us we have to substitute for all propositional variables. So, now if we used 2 to substitute something, we'd have to substitute the b in the first Cab. But, then we'd have to make even more substitutions. Do we have a simpler path possible here?

Well, maybe we could adopt a procedural technique to vary substitutions such that when substituting for variables, more variables appear than in the original wff. So, let's try substituting a with Cad in 1. Then we obtain 2:

2 C CCadb C CcCad Ccb.

Now we want to have some form of 1 become the antecedent. So we want something like CCbaCbd. Oops, I didn't follow my own technique. I mean 2 b/CCeaCed yields 3

3 C CCadCCeaCed C CcCad CcCCeaCed *

Now we want CCabCCcaCcb to match 3 (this exercise, surprising to me, I find kind of hard, because I know the form CCqrCCpqCpr by heart, but CCabCCcaCcb I don't know by heart). So, 1 b/d yields

4 CCadCCcaCcd

Now 4 c/e yields 5

5 C Cad C Cea Cea

Now we can detach 6 from 3 and 5:

6 C CcCad C c C Cea Ced We want the following:

G C CdCab C d C Cca Ccb

So, 6 c/d yields

7? C CdCad C d C Cea Cdb

No, that won't work since substitutions have to happen uniformly. b does not appear in 6. So, if we substitute d for b first, we can still apply substitutions later. So, 6 d/b yields

7 C CcCab C c C Cea Ceb Now 7 c/d yields

8 C CdCab C d C Cea Ceb Now 8 e/c yields

9 C CdCab C d C Cca Ccb

This proof isn't unique since I might have made other substitutions first. Let's look at 3 again:

3 C CCadCCeaCed C CcCad CcCCeaCed

The Detachment Theorem gives us a tip or hint that this from a wff of the type CCadCCeaCed, another wff of the type CcCad, we may infer CcCeaCed. Do we have a more general rule? Let's back up to 1

1 C Cab C Cca Ccb

But this time let's substitute b with CCeaCed first. We then obtain

10 C CaCCeaCed C Cca CcCCeaCed.

Along with The Detachment Theorem this gives us a clue about more general rules of inference than 3 would. So, we have

CpCCqpCqr => CC s p C s CCqpCqr (s does not appear on the left-side of the statement of the rule)

CpCCqpCqr, Csp => CsCCqpCpr

CpCCqpCqr, Csp, s => CCqpCpr

CpCCqpCqr, Csp, s, Cqp => Cpr and

CpCCqpCqr, Csp, s, Cqp, p => r

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  • $\begingroup$ Nice book it is from 1958 so older than i wanted and other than Doug describes it is in (normal) infix notation still it is the only book i know that comes close to what i wanted. $\endgroup$ – Willemien Aug 27 '13 at 10:39

Are you looking for a book on Hilbert axiomatic systems in particular, or just a book on proof theory more generally? There are many different kinds of proof systems, apart from the axiomatic Hilbert-style systems. In particular, proof theory benefits most from the sequent calculus presentation. If you're only interested in Hilbert systems, then I don't know of any resources that focus exclusively on that. If you're just interested in proof theory, there's an array of books that would suit most of your needs.

The thing about proof theory books is that, most of the time, they focus mostly on two logics: classical and intuitionistic. Those, it seems, have been the most influential in developing the course of proof theory. Still, to understand proof theory for other logics, you need to understand some of the basics for proof theory in classical and intuitionistic logic first. Negri and von Plato's Structural Proof Theory is a good place to start. Takeuti's Proof Theory develops proof theory from a different angle, focusing on arithmetic and consistency proofs. For something a little faster paced, look at Troelstra and Schwichtenberg's Basic Proof Theory. All of these are written post-2000.

If you've got some understanding of sequent calculi, then the book you're looking for is definitely Greg Restall's An Introduction to Substructural Logics. This is a fantastic book (one of my favorites actually), and I keep going back to it again and again. Only the first 7 chapters talk about proof theory, but he talks about proof systems with fewer connectives, nonclassical proof systems, modal proof systems, etc., and gives plenty of good examples. The cool thing about the substructural viewpoint is that it's an amazingly elegant way to characterize a lot of nonclassical and many-valued logics in one fell swoop. Again though, to fully appreciate it, you should be sure you know some sequent calculus before reading this book.

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  • $\begingroup$ No I am looking especiallyfor a book on hilbert style axiomatic proofs (which is good for either classical , intuitionistic and relevant logics) $\endgroup$ – Willemien Aug 12 '13 at 19:37
  • $\begingroup$ Ah, alright then. Well, all these proof systems are good for any of those logics as well, it's just Hilbert systems are usually hard to manipulate. An easy natural deduction or sequent proof can very easily become a long and tedious Hilbert proof. I'd be interested to see if there is a book focusing exclusively on techniques for dealing with Hilbert systems though. $\endgroup$ – Alex Kocurek Aug 12 '13 at 20:04
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    $\begingroup$ Before I made this post and saw your question math.stackexchange.com/questions/374937/… I tried to make a Answer but realised it was all more complex then I assumed and my answer was far from complete (so never published) and that led to this question $\endgroup$ – Willemien Aug 13 '13 at 8:16

Try Cori & Lascar "Mathematical Logic" or check out a course book of some University.

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    $\begingroup$ I looked at amazon to the index of this book and it looks like the book has only a very few pages on proofs and proofconstruction themselves (pages 193-200) or am i mistaken? $\endgroup$ – Willemien Aug 12 '13 at 9:32
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    $\begingroup$ @Willemien No, you are not mistaken. This book does not cover the material you are asking for in any depth. In fact, this is one of the points of their approach. $\endgroup$ – Andrés E. Caicedo Aug 12 '13 at 17:04

A classic presentation of logic via an axiomatic presentation -- rather than, say, via a natural deduction system -- is Elliot Mendelson's Introduction to Mathematical Logic (5th edition 2009). This is the book I've always recommended to students wanting to see an old-school axiomatic system being put to work.

The first edition of Mendelson's book is almost 50 years old, however. For a modern book which is axiom-based you could try the admirable Christopher C. Leary’s A Friendly Introduction to Mathematical Logic (Prentice Hall, 2000).

  • $\begingroup$ @Peter_Smith , thanks for your sugestions, but both books are more about proofs and what it all means(mathematical logic) than about how to make proofs and how to prove theorems, thinking about what I would like: it would be something like "Logic primer " by Allan and Hand, mitpress.mit.edu/books/logic-primer but then for Hilbert style axiomatic logic. also maybe I should let go of the publication date limit. $\endgroup$ – Willemien Aug 23 '13 at 10:21

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