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- W2983152153 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}))." @default.
- W2983152153 created "2019-11-22" @default.
- W2983152153 creator A5000014555 @default.
- W2983152153 date "2021-01-01" @default.
- W2983152153 modified "2023-09-25" @default.
- W2983152153 title "Hard Provability Logics" @default.
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- W2983152153 doi "https://doi.org/10.1007/978-3-030-53654-1_9" @default.
- W2983152153 hasPublicationYear "2021" @default.
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