Abstract
We follow along the line of research begun with the pioneering work Maksimova and Rybakov (1974) which has initiated comparative analysis of the lattices of normal extensions of S4 and of intermediate logics. Grounding on the Gödel–McKinsey–Tarski embedding, for any S4-logic, we define the notion of \(\tau \)-decomposition and that of a modal component of the logic. Then, we investigate conditions, when a modal logic has its least modal component.
Dedicated to Larisa L. Maksimova, teacher and friend.
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Notes
- 1.
See also Gödel (1986), pp. 296–302, where the translation into English and commentary by A. Troelstra are provided.
- 2.
- 3.
As is indicated in the introductory part of Maksimova and Rybakov (1974), Sects. 1, 2, and 4 were written by V. Rybakov and Sect. 3 by L. Maksimova.
- 4.
We note that Grzegorczyk’s original axiomatization of \(\mathbf {Grz}\) was different from the one given above. The equivalence of the two axiomatizations was proved in Segerberg (1971), vol. 2.
- 5.
In Muravitsky (2006), I mistakenly used the term dense instead of convex.
- 6.
This proposition answers in the affirmative the question of Problem 9.2 in Muravitsky (2006).
- 7.
There are examples, when \(Y\subset X\) but d(X, Y) is undefined. For instance, \(\mathbf {Dum}\subset M_{0}\), where \(\mathbf {Dum}=\mathbf {S4}+\Diamond \Box p\rightarrow (\Box (\Box (p\rightarrow \Box p)\rightarrow p)\rightarrow p)\), and \(d(M_{0},\mathbf {Dum})\) is undefined.
- 8.
Actually, \(d^{*}(M)\) lies in the nth M-slice which is a proper subset of the nth S-slice.
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Muravitsky, A. (2018). Lattice NExtS4 from the Embedding Theorem Viewpoint. In: Odintsov, S. (eds) Larisa Maksimova on Implication, Interpolation, and Definability. Outstanding Contributions to Logic, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-319-69917-2_10
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