Geodesic
New Modification of the Subformula Property for a Modal Logic
Bulletin of the Section of Logic, Tome 49 (2020) no. 3, pp. 255-268.

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A modified subformula property for the modal logic KD with the additionalaxiom □ ◊(A ∨ B) ⊃ □ ◊ A ∨ □ ◊B is shown. A new modification of the notion of subformula is proposed for this purpose. This modification forms a natural extension of our former one on which modified subformula property for the modal logics K5, K5D and S4.2 has been shown ([2] and [4]). The finite model property as well as decidability for the logic follows from this.
Keywords: subformula property, modal logic, sequent calculus, scope of □ 
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Takano, Mitio. New Modification of the Subformula Property for a Modal Logic. Bulletin of the Section of Logic, Tome 49 (2020) no. 3, pp. 255-268. http://geodesic.mathdoc.fr/item/BSL_2020_49_3_a2/

[1] G. E. Hughes, M. J. Cresswell, A new introduction to modal logic, Routledge (1996), DOI: http://dx.doi.org/10.4324/9780203028100

[2] M. Takano, A modified subformula property for the modal logics K5 and K5D, Bulletin of the Section of Logic, vol. 30(2) (2001), pp. 115–122.

[3] M. Takano, A semantical analysis of cut-free calculi for modal logics, Reports on Mathematical Logic, vol. 53 (2018), pp. 43–65, DOI: http://dx.doi.org/10.4467/20842589RM.18.003.8836

[4] M. Takano, A modified subformula property for the modal logic S4.2, Bulletin of the Section of Logic, vol. 48(1) (2019), pp. 19–28, DOI: http://dx.doi.org/10.18778/0138-0680.48.1.02