General frames for relevant modal logics. (English) Zbl 1071.03009

Seki, Takahiro

Notre Dame J. Formal Logic 44, No. 2, 93-109 (2003).

Summary: General frames are often used in classical modal logic. Since they are duals of modal algebras, completeness follows automatically as with algebras but the intuitiveness of Kripke frames is also retained. This paper develops basics of general frames for relevant modal logics by showing that they share many important properties with general frames for classical modal logic.

MSC:

03B45	Modal logic (including the logic of norms)
03B47	Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
03G25	Other algebras related to logic

Keywords:

relevant modal logic; Routley-Meyer semantics; relevant modal algebra; general frame; descriptive frame; duality

References:

[1]	Blackburn, P., M. de Rijke, and Y. Venema, Modal Logic , vol. 53 of Cambridge Tracts in Theoretical Computer Science , Cambridge University Press, Cambridge, 2001. · Zbl 0988.03006
[2]	Brink, C., ”\(\mathbf R^ \neg\)”-algebras and \(\mathbf R^ \neg\)-model structures as power constructs, Studia Logica , vol. 48 (1989), pp. 85–109. · Zbl 0695.03006
[3]	Celani, S. A., ”A note on classical modal relevant algebras”, Reports on Mathematical Logic , vol. 32 (1998), pp. 35–52. · Zbl 0941.03067
[4]	Chagrov, A., and M. Zakharyaschev, Modal Logic , vol. 35 of Oxford Logic Guides , The Clarendon Press, New York, 1997. · Zbl 0871.03007
[5]	Davey, B. A., and H. A. Priestley, Introduction to Lattices and Order , Cambridge Mathematical Textbooks. Cambridge University Press, Cambridge, 1990. · Zbl 0701.06001
[6]	Došen, K., ”Duality between modal algebras and neighbourhood frames”, Studia Logica , vol. 48 (1989), pp. 219–34. · Zbl 0685.03013
[7]	Dunn, J. M., ”Relevance logic and entailment”, pp. 117–224 in Handbook of Philosophical Logic. Vol. 3: Alternatives to Classical Logic , edited by erseeditorsnames D . M. Gabbay and F. Guenthner, vol. 166 of Synthese Library , D. Reidel Publishing Company, Dordrecht, 1986. · Zbl 0603.03001
[8]	Goldblatt, R., Mathematics of Modality , vol. 43 of CSLI Lecture Notes , Center for the Study of Language and Information, Stanford, 1993. · Zbl 0942.03516
[9]	Routley, R., V. Plumwood, R. K. Meyer, and R. T. Brady, Relevant Logics and Their Rivals. Part I. The Basic Philosophical and Semantical Theory , Ridgeview Publishing Company, Atascadero, 1982. · Zbl 0579.03011
[10]	Seki, T., ”A Sahlqvist theorem for relevant modal logics”, Studia Logica , vol. 73 (2003), pp. 383–411. · Zbl 1027.03020
[11]	Urquhart, A., ”Duality for algebras of relevant logics”, Studia Logica . Special issue on Priestley duality, vol. 56 (1996), pp. 263–76. · Zbl 0844.03032