Obviously a logic, such as classical propositional logic (or
the modal logic S5), is not a structure, but a class of equivalent or
identical structures, since it can be presented in many different ways
(choice of primitive connectives, consequence relation vs tautologies,
etc.) .
The problem is to find a good definition of equivalence or
identity between logics which is reasonable and applies to a wide range
of particular cases.
The standard definition, "to have a common expansion by
definition up to isomorphism", does not apply straightforwardly, as
pointed by Béziau et al. (2001). Proposals such that one of
Pollard (1998) or Pelletier and Urquhart (2003) apply only to some
special situations.
So this fundamental problem is still open and we hope that
someone will present a satisfactory solution in Montreux.
References
*J.-Y. Béziau, R.P. de Freitas, J.P. Viana. What is Classical
Propositional Logic? (A Study in Universal Logic), Logica Studies
7, 2001.
*F.J. Pelletier & A. Urquhart. Synonymous
Logics , Journal of Philosophical Logic, 32, 2003, pp. 259-285.
*Stephen Pollard. Homeomorphism
and the Equivalence of Logical Systems , Notre Dame Journal of
Formal Logic, 39, 1998, pp.422-435.