Complete infinitary type logics

Studia Logica 63 (1):85-119 (1999)
For each regular cardinal , we set up three systems of infinitary type logic, in which the length of the types and the length of the typed syntactical constructs are . For a fixed , these three versions are, in the order of increasing strength: the local system (), the global system g() (the difference concerns the conditions on eigenvariables) and the -system () (which has anti-selection terms or Hilbertian -terms, and no conditions on eigenvariables). A full cut elimination theorem is proved for the local systems, and about the -systems we prove that they admit cut-free proofs for sequents in the -free language common to the local and global systems. These two results follow from semantic completeness proofs. Thus every sequent provable in a global system has a cut-free proof in the corresponding -systems. It is, however, an open question whether the global systems in themselves admit cut elimination.
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