Translating Graded Modalities into Predicate Logics

Translating Graded Modalities into Predicate Logics. Ohlbach, Jürgen, H., Schmidt, R. A., & Hustadt, U. In Proof Theory of Modal Logic, volume 2, of Applied Logic Series, pages 253--291. Kluwer, Dordrecht, The Netherlands, 1996.
abstract bibtex

In the logic of graded modalities it is possible to talk about sets of finite cardinality. Various calculi exist for graded modal logics and all generate vast amounts of case distinctions. In this paper we present an optimized translation from graded modal logic into many-sorted predicate logic. This translation has the advantage that in contrast to known approaches our calculus enables us to reason with cardinalities of sets symbolically. In many cases the length of proofs for theorems of this calculus is independent of the cardinalities. The translation is sound and complete.

@incollection{ OhlbachSchmidtHustadt96,
  author = {Ohlbach, Hans Jürgen and Schmidt, Renate A. and Hustadt, Ullrich},
  year = {1996},
  title = {Translating Graded Modalities into Predicate Logics},
  editor = {Wansing, H.},
  booktitle = {Proof Theory of Modal Logic},
  series = {Applied Logic Series},
  volume = {2},
  publisher = {Kluwer},
  address = {Dordrecht, The Netherlands},
  pages = {253--291},
  abstract = {In the logic of graded modalities it is possible to talk
  about sets of finite cardinality.  Various calculi exist for graded
  modal logics and all generate vast amounts of case distinctions.  In
  this paper we present an optimized translation from graded modal logic
  into many-sorted predicate logic.  This translation has the advantage
  that in contrast to known approaches our calculus enables us to reason
  with cardinalities of sets symbolically.  In many cases the length of
  proofs for theorems of this calculus is independent of the
  cardinalities. The translation is sound and complete.}
}

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