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I have studied a little of logic (namely, FOL and propositional logic) using so-called "Hilbert style".

I've heared that there are different approachs to logics like , deductions, trees, natural deduction and getzen systems.

Are there any other approach? is there one which is "the best"? and what is the point of this variety? I've heared that there are algebraic approach to logic too. And That topology is somtimes used.

Please, Consider not only classical logic but also thins like, paraconsistent logic, relevance logics, fuzzy and many-valued logics, substructural logics and so on.

And for a beginner in logic ( non-classical logics ) Which approach is the best?

I've heared that lattice theory and universal algebra will be a must for a real understanding of logic, Is this true?

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    $\begingroup$ I think this question is likely too broad for satisfying answers. You are essentially asking for an entire introductory book worth of material. $\endgroup$ Sep 12 '14 at 10:29
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There is a whole heap of questions here, of very different kinds. For a start,

(1) There is a question about different strengths of logics which count different inferences to be valid (e.g. intuitionistic, classical, relevance, ...).

(2) There is the quite independent issue that arises, having fixed on a particular strength of logic, about different deductive styles for presenting it. For example, there is a choice of Hilbert systems, natural deduction of various kinds, tableaux or tree systems, etc.

For example, you can present classical logic in different deductive styles, but you'll end up each time with the same theorems, and the same inferences being deemed valid (that's what makes them all variant styles of classical logic, of course!). Similarly for intuitionistic logic and other strength logics.

So the questions cut right across each other: "what strength logic should you go for in what contexts, and why?" and "what style of deductive system system for logics is best for what purposes [for beginners? for proof-discovery? for reflecting the way mathematicians ordinarily reason? for easing the comparison of different strengths of logic?]".

Some of your other questions are very different again, : e.g. universal algebra comes up when we are studying serious model theory.

For some guidance on how various bits of logic are related to each other, and pointers to reading at various levels on various areas you could look at the Teach Yourself Logic Study Guide .

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I disagree with Peter Smith,

I think "Hilbert style" axiomatic proofs while more complex than the other methods are the gold standard of proof.

Especially when you are going seriously into non-classical logic.

There are other methods: but they all have serious drawbacks:

Tableux method:

Preassumnes the principle of excluded middle, no good for logics where $ (P \lor \lnot P ) $ is not a theorem.

Natural deduction:

Preassumnes the principle of contraction, so no good for logics where $ (P \to (P \to Q )) \to (P \to Q) $ is not a theorem. Also assumes the principle of weakening no good for logics where $ P \to (Q \to P )$ is not a theorem.

Sequent calculus:

Preassumnes the principle of permutation, no good for logics where $ (P \to (Q \to R)) \to (Q \to (P \to R ))$ is not a theorem.

There are ways to add limitations to the above methods to remove these built in presumptions, but why start with a method that implies allready?

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  • $\begingroup$ Natural deduction doesn't necessarily preassume CCpCpqCpq (though it will always end up as a theorem). Also, I think you've confused contraction with expansion here. That said, I agree that Hilbert-Frege axiomatic proofs are the gold standard of proof. $\endgroup$ Oct 24 '14 at 18:24
  • $\begingroup$ i corrected the formula (you were right in this) but a theorem is assumed when it always end up as a theorem, you need special arrangements to make sure it not a theorem. and that makes it all less "natural" $\endgroup$
    – Willemien
    Oct 26 '14 at 20:53

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