What to call the dual of "consequence"?
- 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
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.