The Basic Logic of Proofs. Artemov, S. & Strassen, T. Technical Report iam–92–018, Institut für Informatik und angewandte Mathematik, 1992.
The Basic Logic of Proofs [link]Paper  abstract   bibtex   
Propositional Provability Logic was axiomatized in~[Sol76]. This logic describes the behaviour of the arithmetical operator ``$y$~is provable''. The aim of the current paper is to provide propositional axiomatizations of the predicate ``$x$~is a proof of~$y$'' by means of modal logic, with the intention of meeting some of the needs of computer science.
@TechReport{as92,
  Author         = {Sergei Artemov and Tyko Strassen},
  Title          = {{The Basic Logic of Proofs}},
  Institution    = {Institut f{\"u}r Informatik und angewandte Mathematik},
  Number         = {iam--92--018},
  abstract       = "Propositional Provability Logic was axiomatized
                   in~[Sol76]. This logic describes the behaviour of the
                   arithmetical operator ``$y$~is provable''. The aim of
                   the current paper is to provide propositional
                   axiomatizations of the predicate ``$x$~is a proof
                   of~$y$'' by means of modal logic, with the intention of
                   meeting some of the needs of computer science.",
  url            = {http://www.iam.unibe.ch/publikationen/techreports/1992/iam-92-018},
  year           = 1992
}

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