Geodesic
Provability logic with operations over proofs
Fundamentalʹnaâ i prikladnaâ matematika, Tome 3 (1997) no. 4, pp. 1173-1197.

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We present a natural axiomatization for propositional logic with modal operator for formal provability (Solovay, [5]) and labeled modalities for individual proofs with operations over them (Artemov, [2]). For this purpose the language is extended by two new operations. The obtained system $\mathcal{MLP}$ naturally includes both Solovay's provability logic GL and Artemov's operational modal logic $\mathcal{LP}$. All finite extensions of the basic system $\mathcal{MLP}_{0}$ are proved to be decidable and arithmetically complete. It is shown that $\mathcal{LP}$ realizes all operations over proofs admitting description in the modal propositional language. 
@article{FPM_1997_3_4_a17,
     author = {T. L. Sidon},
     title = {Provability logic with operations over proofs},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {1173--1197},
     publisher = {mathdoc},
     volume = {3},
     number = {4},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_1997_3_4_a17/}
}
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T. L. Sidon. Provability logic with operations over proofs. Fundamentalʹnaâ i prikladnaâ matematika, Tome 3 (1997) no. 4, pp. 1173-1197. http://geodesic.mathdoc.fr/item/FPM_1997_3_4_a17/