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On the characterization of fibred logics, with applications to conservativity and finite-valuedness

Marcelino, S. ; Caleiro, C.

Journal of Logic and Computation Vol. 27, Nº 7, pp. 2063 - 2088, August, 2016.

ISSN (print): 0955-792X
ISSN (online): 1465-363X

Scimago Journal Ranking: 0,42 (in 2016)

Digital Object Identifier: 10.1093/logcom/exw023

Abstract
Fibring is a general mechanism for combining logics that provides valuable insight on designing and understanding complex logical systems. To date, most research on fibring has focused on its model and proof-theoretic aspects, and on transference results for relevant metalogical properties. But we are still far from understanding in full the way mixed reasoning emerges from the logics being combined, which is preventing us from having a fully satisfactory semantics for fibred logics and, consequently, limiting the usability of the general results obtained. In previous work, assuming no shared connectives, we have presented an effective characterization of mixed reasoning in terms of the component logics, taking only variables as hypotheses. Despite these restrictions, the result immediately proved to have very interesting applications. In this article, we extend our previous characterization of mixed reasoning for disjoint fibring to arbitrary non-mixed hypotheses. While still not completely satisfactory, as the characterization still cannot cover reasoning from mixed hypotheses, and even less fibred logics with shared connectives, the result again proves to be extremely useful. We illustrate its power by exploring two meaningful applications. To start with, we provide the first full characterization of conservativity for logics obtained by disjoint fibring, extending the partial results of Schechter (2011). Then, we take a semantic detour and use our characterization of mixed reasoning to show that (disjoint) fibring does not preserve finite (N)valuedness.