Some Logics in the Vicinity of Interpretability Logics
DOI:
https://doi.org/10.18778/0138-0680.2023.26Keywords:
interpretability logic, Kripke frames, neighbourhood frames, Veltman semanticsAbstract
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
Downloads
Published
How to Cite
Issue
Section
License
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.