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There are few well-know styles of logical systems (LS): Hilbert-style, Sequent-style, Natural deducton style. And such lesser-know styles as Gottlob-Frege two-demensional notation, systems with graphical syntax etc. (In wikipedia)

And I have some questions:

  1. For example, is the Resolution method (language of clauses and resolution rule) is another style of LS? Or is it (resolution method) something other than style of LS?

  2. Is the Lambda-calsulus is another style of LS? Or one can consider it as kind of semantics? How Lambda-calculus relate to logical systems?

  3. And finally, how one can modify logical systems (classical LS)? For example, One can otherwise interpret the logical connectives (the other semantics of logical connectives) and as a result get a constructive logic; or one can introduce anoter logical operation, and as a result get a modal logic; or one can modify type of quantifiers and as a result get a high-order logics; etc.

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Resolution can be thought as a proof system for a restricted class of formulas.

Lambada calculus is a model of computation. It is related to proof systems because of Curry-Howard isomorphism between proofs and lambda calculus terms.

I am not sure what is the third question. You have already listed some modifications.

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Third queston deal with nonclassical logics. How one can obtain new nonclassical logic from classical? Exactly, what kinds of nonclassical logics are exists? – user7529 May 5 '11 at 10:00
There are too many, there model theoretic logics, there are poor theoretics logics, those coming from philosophy, linguistics, AI, ... the question is too broad. Just check the Wikipedia article for non-classical logics. – Kaveh May 5 '11 at 18:42

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