Generalized negative translations and interpolation Walter Carnielli 2,3 Jo˜aoRasga 1,3 Cristina Sernadas 1,3 1 DM, IST, TU Lisbon, Portugal 2 CLE and IFCH, UNICAMP, Brazil 3 SQIG - IT, Portugal Abstract A general technique generalizing negative translations is presented for proving interpolation of a deductive system. Negative translations are used as an illustration of the technique. AMS Classification: 03C40, 03F03, 03B22 Keywords: negative translation, interpolation 1 Introduction The well-known family of double-negation (Gentzen-G¨ odel-Kolmogorov) transla- tions from classical logic into intuitionistic logic, also called negative translations, have been used for proving that classical and intuitionistic logics have the same consistency strength (see the work of Glivenko [5] for the propositional case and the work of G¨odel [6] and Gentzen [4] for the predicate case). The general schema of a negative translation g of classical into intuitionistic logics is as follows: (i) α ⇔ g(α) is proved in classical logic; (ii) g(α) ⇔ (¬¬ g(α)) is proved in intuitionistic logic; and (iii) if α is proved in classical logic if and only if g(α) is proved in intuitionistic logic. Herein we propose a technique based on a generalized schema of a negative translation between two deductive systems which allows proving interpolation in a general setting extending some results in [2] (for a critical view on translations see [3]). The technique here developed is not restricted to classical and intuitionistic deductive systems although the inspiration is clear from them. In this context, a deductive system includes a signature composed of sets of symbols used to define the set of formulas and a consequence relation between sets of formulas and formulas. In Section 2 the generalized schema is presented and a sufficient condition is proved for a deductive system to have interpolation. In Section 3, we give an instantiation of the generalized schema for classical and intuitionistic deductive systems. The negative translation used slightly differs from the original one since it is tailored for interpolation. Moreover, in Section 4, we also discuss the importance of this schema for proving the preservation of interpolation by combinations of deductive systems. 2 Generalized schema for interpolation Consider deductive systems D, D ′ and D ∗ with signatures C, C ′ and C ∗ and con- sequence relations ⊢ D , ⊢ D ′ and ⊢ D ∗ respectively, such that C ∗ = C ∪ C ′ and