David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Journal of Symbolic Logic 64 (3):1324-1338 (1999)
A grafted frame is a new kind of frame which combines a modal frame and some relevance frames. A grafted model consists of a grafted frame and a truth-value assignment. In this paper, the grafted frame and the grafted model are constructed and used to show the completeness of S1. The implications of S1-completeness are discussed. A grafted frame does not combine two kinds of frames simply by putting relations defined in the components together. That is, the resulting grafted frame is not in the form of $\langle W, R, R'\rangle$ , or more generally, in the form of $\langle W, R, R', R'',...\rangle$ , which consists of a non-empty set with several relations defined on it. Rather, it resembles the construction of fibering proposed by D. M. Gabbay and M. Finger (see  and ). On a grafted frame, some modal worlds, which belong to the initial modal frame, are attached by some relevance frames. However, these two semantics have important differences. Consider the combined semantics involving semantics of relevance logic and modal logic. A fibred model and a grafted model proposed in this paper differ in the following respects. First, a fibred model is constructed from a class of modal models and a class of relevance models. A grafted model consists of a grafted frame and a truth-value assignment, where the grafted frame is constructed from a modal frame and some relevance frames, and the assignment is a union of a modal truth-value assignment V M and some relevance truth-value assignments V R . V M (V R ) defined in this paper is not the same as the assignment contained in a modal (relevance) model. Second, in a fibred model each relevance world is associated (or fibred) with a modal model and each modal world with a relevance model. To be the grafted frame on which a grafted model is based, it is enough to have some modal worlds attached by some relevance frames. Moreover, no relevance world is associated with a modal frame in the grafted frame. Third, fibred models are intended to provide an appropriate semantics to combined logics. Grafted frames and grafted models are inspired to characterize S1, which, containing only one modality □, is not a combined logic. It is shown in this paper that S1 is sort of a meta-logic of the intersection of S0.4 and F, where S0.4, a new system proposed in this paper, is in turn a meta-logic of the relevance logic
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
|Through your library||Configure|
Similar books and articles
I. L. Humberstone (1990). Expressive Power and Semantic Completeness: Boolean Connectives in Modal Logic. Studia Logica 49 (2):197 - 214.
André Fuhrmann (1990). Models for Relevant Modal Logics. Studia Logica 49 (4):501 - 514.
W. J. Blok (1979). An Axiomatization of the Modal Theory of the Veiled Recession Frame. Studia Logica 38 (1):37 - 47.
Bruce M. Kapron (1987). Modal Sequents and Definability. Journal of Symbolic Logic 52 (3):756-762.
Giovanni Sambin (1999). Subdirectly Irreducible Modal Algebras and Initial Frames. Studia Logica 62 (2):269-282.
R. Hirsch, I. Hodkinson & A. Kurucz (2002). On Modal Logics Between K × K × K and $S5 \Times S5 \Times S5$. Journal of Symbolic Logic 67 (1):221 - 234.
D. M. Gabbay (1996). Fibred Semantics and the Weaving of Logics Part 1: Modal and Intuitionistic Logics. Journal of Symbolic Logic 61 (4):1057-1120.
Robert Goldblatt (2006). Maps and Monads for Modal Frames. Studia Logica 83 (1-3):309 - 331.
Added to index2009-01-28
Total downloads2 ( #254,287 of 1,004,782 )
Recent downloads (6 months)1 ( #64,743 of 1,004,782 )
How can I increase my downloads?