By Luiz Carlos Pereira, Edward Hermann Haeusler, Valeria de Paiva
This selection of papers, celebrating the contributions of Swedish philosopher Dag Prawitz to evidence idea, has been assembled from these offered on the typical Deduction convention geared up in Rio de Janeiro to honour his seminal learn. Dag Prawitz’s paintings types the root of intuitionistic variety idea and his inversion precept constitutes the root of most up-to-date debts of proof-theoretic semantics in good judgment, Linguistics and Theoretical desktop Science.
The variety of contributions contains fabric at the extension of average deduction with higher-order principles, rather than higher-order connectives, and a paper discussing the appliance of usual deduction principles to facing equality in predicate calculus. the quantity keeps with a key bankruptcy summarizing paintings at the extension of the Curry-Howard isomorphism (itself a derivative of the paintings on common deduction), through equipment of class concept which have been effectively utilized to linear good judgment, in addition to many different contributions from very popular gurus. With an illustrious crew of participants addressing a wealth of issues and functions, this quantity is a useful addition to the libraries of teachers within the a number of disciplines whose improvement has been given additional scope by means of the methodologies provided through typical deduction. the amount is consultant of the wealthy and sundry instructions that Prawitz paintings has encouraged within the zone of typical deduction.
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Extra info for Advances in Natural Deduction: A Celebration of Dag Prawitz's Work
Logic Journal of the IGPL, 10, 299–337. 43. Tesconi, L. (2004). Strong Normalization Theorem for Natural Deduction with General Elimination Rules. Ph. D. thesis: University of Pisa. 44. Wansing, H. (1993). The Logic of Information Structures. Berlin: Springer (Lecture Notes in Artificial Intelligence, Vol. 681). Revisiting Zucker’s Work on the Correspondence Between Cut-Elimination and Normalisation Christian Urban Abstract Zucker showed that in the fragment of intuitionistic logic whose formulae are build up from ∧, → and ⇒ only, every reduction sequence in natural deduction corresponds to a reduction sequence in the sequent calculus and vice versa.
Figure 6 shows the standard translation, which appeared first in Prawitz . It translates inductively a sequent proof so that right-rules are mapped to introduction rules on the root of natural deduction proofs, and left-rules to elimination rules at the top (except ≤ L and ∨ L which translate like right-rules). It is not hard to show that this translation is onto, but a proof is omitted. First we show that typed terms translate to typed terms. Proposition 2 For all M ∈ S, if the sequent α α ◦ |M| : B is derivable and vice versa.
To see this, we have to observe that every formula C, which is a conclusion of a (generalizedHL ) elimination inference and at the same time major premiss of a (generalizedHL ) elimination inference can be eliminated, as the following example demonstrates, which shows that corresponding segments10 of formulas of this kind are shortened: 1 [E] D1 D →E (→ ESL ) (→ ESL ) D2 D3 D A →B A →B 2 [B] 1 D4 A D5 C C reduces to 2 [B] 1 [E] D1 D →E (→ ESL ) D2 D D3 (→ ESL ) A → B C 2 D4 A C 1 D5 C 2. Here it is assumed that below A → B there is no formula of the incriminated kind (in particular, C is not of that kind).