Some Logics in the Vicinity of Interpretability Logics

Sergio A. Celani

doi:10.18778/0138-0680.2023.26

Authors

Sergio A. Celani Universidad Nacional del Centro and CONICET, Departamento de Matemática, Tandil, Argentina https://orcid.org/0000-0003-2542-4128

DOI:

https://doi.org/10.18778/0138-0680.2023.26

Keywords:

interpretability logic, Kripke frames, neighbourhood frames, Veltman semantics

Abstract

In this paper we shall define semantically some families of propositional modal logics related to the interpretability logic \(\mathbf{IL}\). We will introduce the logics \(\mathbf{BIL}\) and \(\mathbf{BIL}^{+}\) in the propositional language with a modal operator \(\square\) and a binary operator \(\Rightarrow\) such that \(\mathbf{BIL}\subseteq\mathbf{BIL}^{+}\subseteq\mathbf{IL}\). The logic \(\mathbf{BIL}\) is generated by the relational structures \(\left<X,R,N\right>\), called basic frames, where \(\left<X,R\right>\) is a Kripke frame and \(\left<X,N\right>\) is a neighborhood frame. We will prove that the logic \(\mathbf{BIL}^{+}\) 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 \(\mathbf{BIL}^{+}\) and its extensions. We shall also study some axiomatic extensions of \(\mathsf{\mathbf{BIL}}\) 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.

References

P. Blackburn, M. de Rijke, Y. Venema, Modal Logic, no. 53 in Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, Cambridge (2001), DOI: https://doi.org/10.1017/CBO9781107050884
Google Scholar DOI: https://doi.org/10.1017/CBO9781107050884

G. Boolos, The logic of provability, Cambridge University Press, Cambridge (1995).
Google Scholar DOI: https://doi.org/10.1017/CBO9780511625183

S. A. Celani, Properties of saturation in monotonic neighbourhood models and some applications, Studia Logica, vol. 103(4) (2015), pp. 733–755, DOI: https://doi.org/10.1007/s11225-014-9590-z
Google Scholar DOI: https://doi.org/10.1007/s11225-014-9590-z

B. F. Chellas, Modal logic: an introduction, Cambridge University Press, Cambridge (1980).
Google Scholar DOI: https://doi.org/10.1017/CBO9780511621192

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: https://doi.org/10.1007/978-1-4613-0609-2_3
Google Scholar DOI: https://doi.org/10.1007/978-1-4613-0609-2_3

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

E. Pacuit, Introduction and Motivation, [in:] Neighborhood semantics for modal logic, Springer, Cham (2017), pp. 1–38, DOI: https://doi.org/10.1007/978-3-319-67149-9
Google Scholar DOI: https://doi.org/10.1007/978-3-319-67149-9_1

R. Verbrugge, Generalized Veltman frames and models (1992), manuscript.
Google Scholar

A. Visser, Interpretability logic, [in:] P. P. Petkov (ed.), Mathematical logic, Springer US, Boston, MA (1990), pp. 175–209, DOI: https://doi.org/10.1007/978-1-4613-0609-2_13
Google Scholar DOI: https://doi.org/10.1007/978-1-4613-0609-2_13

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