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Extended Natural Deduction

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Book cover Natural Deduction, Hybrid Systems and Modal Logics

Part of the book series: Trends in Logic ((TREN,volume 30))

Abstract

In this Chapter there is a continual emphasis on the application of ND as a tool of proof search and possibly of automation. In particular, we take up the question of how to make ND a universal system. In order to find satisfactory solutions we compare ND with other types of DS’s. Although ND systems are rather rarely considered in the context of automated deduction they presumably accord with each other and ND systems may be turned into useful automatic proof search procedures. Moreover, even if there are some problems with the construction of efficient ND-based provers, it seems that for the widely understood computer-aided forms of teaching logic, ND should be acknowledged. A good evidence for this claim is provided by the increasing number of proof assistants, tutors, checkers and other interactive programs of this sort based on some forms of ND. Section 4.1. is devoted to the general discussion of these questions, whereas the rest of the Chapter takes up successively the presentation of some concrete, universal and analytic versions of ND for classical and free logic.

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Notes

  1. 1.

    One may consult in this matter e.g. papers collected in the monographic volume of Studia Logica (1998) devoted to automated ND.

  2. 2.

    Cf. e.g. remarks on performance of OSCAR in [216] and more substantial comparison of OSCAR and OTTER performance in [217].

  3. 3.

    Von Plato discovered that Gentzen himself has proved a normalization theorem for intuitionistic ND; he published Gentzen’s version in [213].

  4. 4.

    For example, the original proof of Prawitz is for the system with rules for \(\bot, \wedge, \rightarrow, \forall\); the solution for full first-order language was provided much later by Seldin [247] and Stalmarck [263].

  5. 5.

    The exception for classical logic is the work of Sieg mentioned above.

  6. 6.

    In fact, only special sort of such analytic applications is needed. For details see Section 3.3.

  7. 7.

    For those who prefer TS with rules introducing parameters instead of free variables the system KMP may be more convenient.

  8. 8.

    The same result may be demonstrated by showing that any β may be used to creation of new subderivations at most \(2^{n+1}\) times, where n is the number of β-formulae preceding this \(\beta \) in the derivation.

  9. 9.

    Such a solution was applied by the author in [147].

  10. 10.

    As we will see in Chapter 10 the application of cut in modal logics may lead to other problems, as well.

  11. 11.

    This is somewhat related to the strategies from resolution provers, like ordering or selection function, which are applied to reduce the number of useless inferences. But it is not possible to transfer these strategies directly to ordinary ND since it does not work on clauses – one more good reason to introduce RND in the next section.

  12. 12.

    But remember that these results should not be treated as indicating the weakness of tableaux for automated deduction in general, since the possibility of obtaining shorter proofs does not mean that the space of proof search is also smaller; sometimes it is just the opposite. We have mentioned about that already in Chapter 3.

  13. 13.

    One may find it also in [150].

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  1. Dept. Logic, University of Lódz, Kopcinskiego 16/18, 90-232, Lódz, Poland

    Dr. Andrzej Indrzejczak

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Indrzejczak, A. (2010). Extended Natural Deduction. In: Natural Deduction, Hybrid Systems and Modal Logics. Trends in Logic, vol 30. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-8785-0_4

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