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What to call the dual of "consequence"?

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  • Vaughan Pratt
    When a formula p is in the theory TM(S) of the class M(S) of models of a set S of formulas, we call p a *consequence* of S. The dual relationship obtains when
    Message 1 of 1 , Aug 2, 2006
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      When a formula p is in the theory TM(S) of the class M(S) of models of a
      set S of formulas, we call p a *consequence* of S. The dual
      relationship obtains when a structure m is in MT(C), the models of the
      theory of a class C of structures. Is there a word for this
      relationship between m and C? And if not can you supply a suitable one?

      When I posed this question to another group in 1990, Sol Feferman and
      Robit Parikh both had vague recollections that Tarski may have proposed
      a suitable term in the 1950s.

      The readership of the list then proposed various names, including
      "homologue" by Bill Rounds which seemed very good at the time. Thus one
      would define a Boolean algebra as any equational homologue of the
      2-element Boolean algebra. One could also speak of elementary
      homologues of the field of complex numbers, and so on for other logical
      frameworks admitting a suitable Galois connection between models and
      theories.

      While one might worry about the proximity to homology, the risk there
      seemed less than that of calling a Boolean algebra of sets under union
      and intersection and complement a field of sets, as proposed by Birkhoff.

      Vaughan Pratt
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