Abstract
Let \(\mathsf{PL}(T,T')\) and \(\mathsf{PL}_{_{\Sigma _1}}(T,T')\) respectively indicate the provability logic and \(\Sigma _1\)-provability logic of \(T\) relative in \(T'\). In this paper we characterise the following relative provability logics: \(\mathsf{PL}_{_{\Sigma _1}}(\mathsf{HA},\mathbb {N})\), \(\mathsf{PL}_{_{\Sigma _1}}(\mathsf{HA},\mathsf{PA})\), \(\mathsf{PL}_{_{\Sigma _1}}(\mathsf{HA}^*,\mathbb {N})\), \(\mathsf{PL}_{_{\Sigma _1}}(\mathsf{HA}^*,\mathsf{PA})\), \(\mathsf{PL}(\mathsf{PA},\mathsf{HA})\), \(\mathsf{PL}_{_{\Sigma _1}}(\mathsf{PA},\mathsf{HA})\), \(\mathsf{PL}(\mathsf{PA}^*,\mathsf{HA})\), \(\mathsf{PL}_{_{\Sigma _1}}(\mathsf{PA}^*,\mathsf{HA})\), \(\mathsf{PL}(\mathsf{PA}^*,\mathsf{PA})\), \(\mathsf{PL}_{_{\Sigma _1}}(\mathsf{PA}^*,\mathsf{PA})\), \(\mathsf{PL}(\mathsf{PA}^*,\mathbb {N})\), \(\mathsf{PL}_{_{\Sigma _1}}(\mathsf{PA}^*,\mathbb {N})\) (see Table 9.3). It turns out that all of these provability logics are decidable. The notion of reduction for provability logics, first informally considered in (Ardeshir and Mojtahedi 2015). In this paper, we formalize a generalization of this notion (Definition 9.4.1) and provide several reductions of provability logics (see Diagram 9.5). The interesting fact is that \(\mathsf{PL}_{_{\Sigma _1}}(\mathsf{HA},\mathbb {N})\) is the hardest provability logic: the arithmetical completenesses of all provability logics listed above, as well as well-known provability logics like \(\mathsf{PL}(\mathsf{PA},\mathsf{PA})\), \(\mathsf{PL}(\mathsf{PA},\mathbb {N})\), \(\mathsf{PL}_{_{\Sigma _1}}(\mathsf{PA},\mathsf{PA})\), \(\mathsf{PL}_{_{\Sigma _1}}(\mathsf{PA},\mathbb {N})\) and \(\mathsf{PL}_{_{\Sigma _1}}(\mathsf{HA},\mathsf{HA})\), are all propositionally reducible to the arithmetical completeness of \(\mathsf{PL}_{_{\Sigma _1}}(\mathsf{HA},\mathbb {N})\).
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Acknowledgements
I would like to thank Mohammad Ardeshir for reading of the first draft of this paper and his very helpful comments, remarks and corrections. I am very grateful to the referee for her/his very helpful corrections, comments and suggestions.
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Mojtahedi, M. (2021). Hard Provability Logics. In: Mojtahedi, M., Rahman, S., Zarepour, M.S. (eds) Mathematics, Logic, and their Philosophies. Logic, Epistemology, and the Unity of Science, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-030-53654-1_9
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