Priestley Duality, a Sahlqvist Theorem and a Goldblatt-Thomason Theorem for Positive Modal Logic

SERGIO ARTURO CELANI AND RAMON JANSANA

LOGIC JOURNAL OF THE IGPL (PRINT)

Oxford University Press

Lugar: Oxford; Año: 1999 vol. 7 p. 683 - 715

1367-0751

In [12] the study of Positive Modal Logic (PML) is initiated using standard Kripke semantics andthe  positive  modal  algebras  (a  class  of  bounded  distributive  lattices  with  modal  operators)  are  introduced.  The minimum system of Positive Modal Logic is the (∧, ∨, 2, 3, ⊥, >)-fragment of the local consequence relation defined by the class of all Kripke models.  It can be axiomatized by a sequent calculus and extensions  of it can be obtained by adding sequents  as new axioms.   In [6]  a new semantics for PML is proposed to overcome some frame incompleteness problems discussed in [12].   The frames of this semantics consists of a set of indexes,  a quasi-order on them and an accessibility relation. The models are obtained by using increasing valuations relatively to the quasi-order of the frame.  This semantics is coherent with the dual structures obtained by developing the Priestley duality for positive modal algebras, one of the topics or the present paper, and can be seen also as arising from the Kripke semantics for a suitable intuitionistic modal logic. The present paper is devoted to the study of the mentioned duality as well as to proving some d-persistency results as well as a Sahlqvist Theorem for sequents and the semantics proposed in [6].  Also a Goldblatt-Thomason theorem that characterizes the elementary classes of frames of that semantics that are definable by sets of sequents is proved.