Linear Kripke Frames and G¨odel Logics ∗ Arnold Beckmann † University of Wales, Swansea Singleton Park Swansea SA2 8PP, UK a.beckmann@swansea.ac.uk Norbert Preining ‡ Universit`a di Siena 53100 Siena, Italy preining@logic.at August 22, 2006 Abstract We investigate the relation between intermediate predicate logics based on countable linear Kripke frames with constant domains and G¨odel logics. We show that for any such Kripke frame there is a G¨odel logic which coincides with the logic defined by this Kripke frame on constant domains and vice versa. This allows us to transfer several recent results on G¨odel logics to logics based on countable linear Kripke frames with constant domains: We obtain a complete characterisation of axiomatisability of logics based on countable linear Kripke frames with constant domains. 1 Furthermore, we obtain that the total number of logics defined by countable linear Kripke frames on constant domains is countable. 1 Introduction Kripke frames as possible semantics for modal logics were introduced by S. A. Kripke in the late fifties and early sixties. While the origin of this notion is disputable, the influence of the possible world interpretation has been enormous. This new type of semantics provided an attractive model theory that seemed more manageable than the previous algebra-based semantics. One of the reasons for its early success was that well known logical systems, like S4, S5 and Intuitionistic Predicate Logic, were shown to be characterised by natural first-order properties of their frames. Kripke himself in [Kri65] used these frames to prove the completeness of Intuitionistic Predicate Logic. For a more detailed presentation of these and related topics see [Gab81, Gol03]. Detailed studies of intermediate predicate logics based on linear Kripke frames have been carried out by several researchers. For example, the general structure of linear Kripke frames and their logics is discussed in [Ono88], the logics defined by Kripke frames determined by ordinals on constant domains are analysed in [MTO90], and the logics based on Kripke frames R and Q with constant domains are determined in [Tak87b]. * The results of this paper have been presented at the ESF Exploratory Workshop ”The Chal- lenge of Semantics” in Vienna, July 2004. † Supported in part by FWF-grants #P17503-MATH and #P16539-N04 of the Austrian Science Fund. ‡ Supported by the European Union under EC-MC 008054 and FWF-grant #P16539-N04 of the Austrian Science Fund. 1 Skvortsov [Skv05] recently has announced similar results on the characterisation of axiomati- sability. 1