Simplified Kripke style semantics for some very weak modal logics
Keywordssimplified Kripke style semantics, very weak modal logics
AbstractIn the present paper we examine very weak modal logics C1, D1, E1, S0.5◦, S0.5◦+(D), S0.5 and some of their versions which are closed under replacement of tautological equivalents (rte-versions). We give semantics for these logics, formulated by means of Kripke style models of the form <w,A,V>, where w is a «distinguished» world, A is a set of worlds which are alternatives to w, and V is a valuation which for formulae and worlds assigns the truth-vales such that: (i) for all formulae and all worlds, V preserves classical conditions for truth-value operators; (ii) for the world w and any formula ϕ, V(⬜ϕ,w) = 1 iff ∀x∈A V(ϕ,x) = 1; (iii) for other worlds formula ⬜ϕ has an arbitrary value. Moreover, for rte-versions of considered logics we must add the following condition: (iv) V(⬜χ,w) = V(⬜χ[ϕ/ψ],w), if ϕ and ψ are tautological equivalent. Finally, for C1, D1and E1 we must add queer models of the form <w,V> in which: (i) holds and (ii') V(⬜ϕ,w) = 0, for any formula ϕ. We prove that considered logics are determined by some classes of above models.
Bowen, K.A., Model Theory for Modal Logic. Kripke Models for Modal Predicate Calculi. Dordrecht–Boston 1979: D. Reidel Publishing Company.
Bull, R.A., and K. Segerberg, “Basic Modal Logic”, pp. 1–88 in: D.M. Gabbay and F. Guenthner (eds.), Handbook of Pholosophical Logic, vol. II, Dordrecht 1984: D. Reidel Publishing Company.
Chellas, B.F., Modal Logic. An Introduction. Cambridge 1980: Cambridge University Press.
Chellas, B.F., and K. Segerberg, “Modal logics in the vicinty of S1”, Notre Dame Journal of Formal Logic 37, 1 (1996): 1–24.
Feys, R., “Les systèmes formalisés des modalités aristotéliciennes”, Revue Pilosophique de Louvain 48 (1950): 478–509. Also: R. Feys, Modal Logics. Louvain 1965: E. Nauwelaerta.
Hughes, G.E., and M.J. Cresswell, A New Introduction to Modal Logic, London and New York 1996: Routledge.
Girle, R.A., “S1≠S0.9”, Notre Dame Journal of Formal Logic 16 (1975): 339–344.
Kripke, S.A., “Semantical analisis of modal logic. II: Non-normal modal propositional calculi”, pp. 206–220 in: The Theory of Models. Proc. of the 1963 International Symbosiom at Berkley, Amsterdem 1965.
Lemmon, E.J., “New fundations for Lewis modal systems”, The Journal of Symbolic Logic 22, 2 (1957): 176–186.
Lemmon, E.J., “Algebraic semantics for modal logics I”, The Journal of Symbolic Logic 31 (1966): 46–56.
Lemmon, E.J., in collaboration with D. Scott, The “Lemmon Notes”: An Introduction to Modal Logic. Edited by K. Segerberg, no. 11 in the American Philosophical Quarterly Monograph Series. Oxford 1977: Basil Blackwell.
Lewis, C.I., and C.H. Langford, Symbolic Logic, New York, 1932.
Nowicki, M., “QL-regular quantified modal logics”, Bulletin of the Section of Logic 37, 3/4 (2008): 211–221.
Pietruszczak, A., “Relational semantics for some very weak Lemmon’s systems”. Draft (2005).
Pietruszczak, A., “On applications of truth-value connectives for testing arguments with natural connectives”, pp. 143–156 in: J. Malinowski and A. Pietruszczak (eds.), Essays in Logic and Ontology, Amsterdam/New York 2006, GA: Rodopi.
Routley, R., “Decision procedure and semantics for C1, E1 and S0.5◦ ”, Logique et Analyse 44 (1968): 468–469.
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