Geodesic
Some Logics in the Vicinity of Interpretability Logics
Bulletin of the Section of Logic, Tome 53 (2024) no. 2, pp. 173-193.

Voir la notice de l'article provenant de la source Library of Science

In this paper we shall define semantically some families of propositional modal logics related to the interpretability logic 𝐈𝐋. We will introduce the logics 𝐁𝐈𝐋 and 𝐁𝐈𝐋^+ in the propositional language with a modal operator □ and a binary operator ⇒ such that 𝐁𝐈𝐋⊆𝐁𝐈𝐋^+⊆𝐈𝐋. The logic 𝐁𝐈𝐋 is generated by the relational structures lt;X,R,N gt;, called basic frames, where lt;X,R gt; is a Kripke frame and lt;X,N gt; is a neighborhood frame. We will prove that the logic 𝐁𝐈𝐋^+ is generated by the basic frames where the binary relation R is definable by the neighborhood relation N and, therefore, the neighborhood semantics is suitable to study the logic 𝐁𝐈𝐋^+ and its extensions. We shall also study some axiomatic extensions of 𝐁𝐈𝐋 and we will prove that these extensions are sound and complete with respect to a certain classes of basic frames. Finally, we prove that the logic 𝐁𝐈𝐋^+ and some of its extensions are complete respect with the class of neighborhood frames.
Keywords: interpretability logic, Kripke frames, neighbourhood frames, Veltman semantics 
@article{BSL_2024_53_2_a2,
     author = {Celani, Sergio A.},
     title = {Some {Logics} in the {Vicinity} of {Interpretability} {Logics}},
     journal = {Bulletin of the Section of Logic},
     pages = {173--193},
     publisher = {mathdoc},
     volume = {53},
     number = {2},
     year = {2024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BSL_2024_53_2_a2/}
}
TY  - JOUR
AU  - Celani, Sergio A.
TI  - Some Logics in the Vicinity of Interpretability Logics
JO  - Bulletin of the Section of Logic
PY  - 2024
SP  - 173
EP  - 193
VL  - 53
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BSL_2024_53_2_a2/
LA  - en
ID  - BSL_2024_53_2_a2
ER  - 
%0 Journal Article
%A Celani, Sergio A.
%T Some Logics in the Vicinity of Interpretability Logics
%J Bulletin of the Section of Logic
%D 2024
%P 173-193
%V 53
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BSL_2024_53_2_a2/
%G en
%F BSL_2024_53_2_a2
Celani, Sergio A. Some Logics in the Vicinity of Interpretability Logics. Bulletin of the Section of Logic, Tome 53 (2024) no. 2, pp. 173-193. http://geodesic.mathdoc.fr/item/BSL_2024_53_2_a2/

[1] P. Blackburn, M. de Rijke, Y. Venema, Modal Logic, no. 53 in Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, Cambridge (2001) | DOI

[2] G. Boolos, The logic of provability, Cambridge University Press, Cambridge (1995).

[3] S. A. Celani, Properties of saturation in monotonic neighbourhood models and some applications, Studia Logica, vol. 103(4) (2015), pp. 733–755 | DOI

[4] B. F. Chellas, Modal logic: an introduction, Cambridge University Press, Cambridge (1980).

[5] D. d. Jongh, F. Veltman, Provability logics for relative interpretability, [in:] P. P. Petkov (ed.), Mathematical logic, Springer, Boston, MA (1990), pp. 31–42 | DOI

[6] J. J. Joosten, J. M. Rovira, L. Mikec, M. Vuković, An overview of Generalised Veltman Semantics (2020), arXiv:2007.04722 [math.LO].

[7] E. Pacuit, Introduction and Motivation, [in:] Neighborhood semantics for modal logic, Springer, Cham (2017), pp. 1–38 | DOI

[8] R. Verbrugge, Generalized Veltman frames and models (1992), manuscript.

[9] A. Visser, Interpretability logic, [in:] P. P. Petkov (ed.), Mathematical logic, Springer US, Boston, MA (1990), pp. 175–209 | DOI

[10] M. Vukovic, Some correspondences of principles in interpretability logic, Glasnik Matematicki, vol. 31 (1996), pp. 193–200.