Some Logics in the Vicinity of Interpretability Logics

Sergio Celani

doi:10.18778/0138-0680.2023.26

Authors

Sergio Celani Universidad Nacional del Centro and CONICET

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.

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 (1980).
Google Scholar DOI: https://doi.org/10.1017/CBO9780511621192

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

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, Neighborhood semantics for modal logic, An introduction, July, (2007), DOI: https://doi.org/10.1007/978-3-319-67149-9.
Google Scholar DOI: https://doi.org/10.1007/978-3-319-67149-9

R. Verbrugge, Generalized Veltman frames and models, Manuscript, Amsterdam, (1992).
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