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Mohammad Ardeshir, S. Mojtaba Mojtahedi, Reduction of provability logics to Σ1-provability logics, Logic Journal of the IGPL, Volume 23, Issue 5, October 2015, Pages 842–847, https://doi.org/10.1093/jigpal/jzv031
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Abstract
We show that the provability logic of PA, GL and the truth provability logic, i.e. the provability logic of PA relative to the standard model , GLS are reducible to their -provability logics, GLV and GLSV, respectively, by only propositional substitutions.
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