In this paper we consider, with a categorial approach, previous ideas and previous work of Walter Carnielli (learned by the authors directly from Walter’s talks at CLE-Unicamp) concerning possible translation semantics and remote algebrization. We show that if logic remotely protoalgebrizable—i.e. it has a conservative translation into a protoalgebraizable logic—then it is has a theorem or it is “non-implosive” (i.e. any formula that is a consequence of every non-empty set is a theorem). A similar result is proved for the existence of conservative translations into weakly-equivalential logics, a concept here introduced, and into algebraizable logics.

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Possible-Translation Semantics from a Categorical Point of View

  • Peter Arndt,
  • Hugo Luiz Mariano,
  • Darllan Conceição Pinto

摘要

In this paper we consider, with a categorial approach, previous ideas and previous work of Walter Carnielli (learned by the authors directly from Walter’s talks at CLE-Unicamp) concerning possible translation semantics and remote algebrization. We show that if logic remotely protoalgebrizable—i.e. it has a conservative translation into a protoalgebraizable logic—then it is has a theorem or it is “non-implosive” (i.e. any formula that is a consequence of every non-empty set is a theorem). A similar result is proved for the existence of conservative translations into weakly-equivalential logics, a concept here introduced, and into algebraizable logics.