Geodesic
On Prenex Fragment of Provability Logic with Quantifiers on Proofs
Informatics and Automation, Mathematical logic and algebra, Tome 242 (2003), pp. 123-135

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We consider a fragment of provability logic with quantifiers on proofs that consists of formulas with no occurrences of quantifiers in the scope of the proof predicate. By definition, a logic ql is the set of formulas that are true in the standard model of arithmetic under every interpretation based on the standard Gödel proof predicate. We describe Kripke-style semantics for the logic ql and prove the corresponding completeness theorem. For the case of injective arithmetical interpretations, the decidability is proved. 
@article{TRSPY_2003_242_a9,
     author = {R. \`E. Yavorskii},
     title = {On {Prenex} {Fragment} of {Provability} {Logic} with {Quantifiers} on {Proofs}},
     journal = {Informatics and Automation},
     pages = {123--135},
     publisher = {mathdoc},
     volume = {242},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2003_242_a9/}
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R. È. Yavorskii. On Prenex Fragment of Provability Logic with Quantifiers on Proofs. Informatics and Automation, Mathematical logic and algebra, Tome 242 (2003), pp. 123-135. http://geodesic.mathdoc.fr/item/TRSPY_2003_242_a9/