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The inadequacy of Kripke's semantical analysis of D2 and D3

Published online by Cambridge University Press:  12 March 2014

R. Routley  and
H. Montgomery
R. Routley
Affiliation:
Monash University and University of Canterbury
H. Montgomery
Affiliation:
Monash University and University of Canterbury
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Extract

Kripke's model structure for D2 is a quadruple (G, K, R, N) where N is the set of normal elements, i.e., the subset of K such that for every H in N, HRH. (See [1, p. 220, p. 211].) In fact, however, this structure provides a model for E2, as the work of Lemmon shows ([2, pp. 58–62]).

To show the inadequacy of Kripke's model structure, we show that the E2 thesis □B ⊃ B, not a thesis of D2, is valid under Kripke's modelling for D2. Suppose, for a reductio argument, that □ B ⊃ B is false for some valuation ϕ in some D2 model structure. Thenyϕ(□B ⊃ B, G) = F; so by truth-functional assignments

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 33 , Issue 4 , January 1969 , pp. 568
Copyright
Copyright © Association for Symbolic Logic 1969

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References

[1] Saul Kripke, A., Semantical analysis of modal logic. II: non-normal modal propositional calculi, Symposium on the theory of models, North-Holland, Amsterdam, 1965, pp. 206–220.Google Scholar
[2]Lemmon, E. J., Algebraic semantics for modal logics. I, this Journal, vol. 31 (1966), pp. 46–65.Google Scholar
[3]Lemmon, E. J., Algebraic semantics for modal logics. II, this Journal, vol. 31 (1966), pp. 191–218.Google Scholar