Bulletin of the Section of Logic Volume 19/1 (1990), pp. 21–23 reedition 2005 [original edition, pp. 21–24] A. V. Chagrov and M. V. Zakharyashchev ON HALLD ´ EN-COMPLETNESS OF INTERMEDIATE AND MODAL LOGICS A logic L is said to be Halld´ en-complete (or Halld´ en-reasonable) if for any formula A ∨ B provable in L, where A and B have no variables in common, L ⊢ A or L ⊢ B. A. Wro´ nski [4] has obtained the algebraic equivalents of Halld´ en- completeness for intermediate and modal logics. J. van Benthem and I. Humberstone [2] have given in semantic terms a sufficient condition for Halld´ en-completeness in normal modal logics; it is unknown whether the condition is necessary. In this paper we deal with the problem of deciding, given an axiomati- zation of intermediate logic or normal modal logic containing S4, whether the logic is Halld´ en-complete. Recently we have shown [1] that the disjunction property of intermedi- ate logics is undecidable. (Recall that an intermediate logic L is said to have disjunction property if L ⊢ A or L ⊢ B whenever L ⊢ A ∨ B; the definition of disjunction property for modal logics is as follows: L ⊢ A ∨ B ⇒ L ⊢ A or L ⊢ B.) It is obvious that for intermediate logics the disjunction property im- plies Halld´ en-completeness (compare with Theorem 6 below). With the help of this fact we have proved in [1] the following Theorem 1. Halld´ en-completeness in intermediate logics is undecidable (i.e. no algorithm exists which is capable of deciding given a formula A, whether or not logic Int + A is Halld´ en-complete). Undecidable are also such properties of intermediate logics as finite model property, decidability (and some others; see [1]). These properties as well as the disjunction property are preserved under transferring from