% Open session                 %
% To be presented in LUATCS'99 %
% December 1999                %

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\title{{\bf Possible-Translations Semantics:} \\ a special way of combining logics}
\author{Jo\~ao Marcos \\ IFCH -- Unicamp, Brazil \\ {\small\tt <vegetal@cle.unicamp.br>}}
\date{}

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\maketitle

The task of finding, for non-classical logics, semantics which are
both strongly adequate and intuitively acceptable is often a hard
one. Possible-translations semantics were devised in order to help
solving this predicament \cite{Car90}. Here we shall understand
logics simply as sets (of formulas) endowed with a consequence
relation, and translations as morphisms between logics, i.e. maps
preserving their consequence relations.  The idea of {\sl
possible-translations semantics} is to base an interpretation to a
given logic \(\cal{L}\) on the combination of an appropriate set
of translations of the formulas of \(\cal{L}\) into a class of
logics with known semantics.

Some examples will be considered.  We will begin by presenting
some paraconsistent fragments of classical logic which, despite
not being characterizable by finite matrices, can be characterized
by a suitable combination of three-valued logics \cite{Mar99}. In
this process, decidability of the considered logics is attained as
a byproduct: to evaluate a given formula we just have to evaluate
all of its possible translations using the three-valued matrices.
Next we will see how possible-translations semantics provide a
straightforward way of talking about a deductive limit to a given
hierarchy of logics \cite{CM9?}. Finally, we will show how
many-valued logics themselves could be seen as suitable
combinations of copies of classical logic, giving rise to the
concept of {\sl society semantics} \cite{CLM99}.

One of the most general approaches to the combination of logics
comes from Gabbay's {\sl fibring} \cite{Gab99}.  Examples of
fibred logics usually deal with the construction of previously
unknown rich and complex logics by way of fibring simpler ones. We
call this {\sl splicing logics}.  The examples of
possible-translations semantics mentioned above clearly go the
other way around, showing how a complex logic could be analyzed
into its ingredients. We call this {\sl splitting logics}.  Of
course, both possible-translations semantics and fibring can go
either direction, splitting and splicing logics.  We will mention
some open problems in this study and give some hints on a
categorial approach to these questions, by way of sheaves
\cite{CC99}.


\begin{thebibliography}{mm}

\bibitem{Car90}
W. Carnielli, Many-valued logics and plausible reasoning. In:{\it
Proceedings of the {\tt XX}\/ International Congress on
Many-Valued Logics}, IEEE Computer Society, (1990) pp.328--35.

\bibitem{CC99}
W. Carnielli and M. E. Coniglio, A categorial approach to the
combination of logics.  To appear in {\it Manuscrito}, v.22, n.2,
(1999), Campinas, CLE. 17p.

\bibitem{CLM99}
W. Carnielli and M. Lima-Marques, Society semantics and
multiple-valued logics. In:{\it Advances in Contemporary Logic and
Computer Science:} Proceedings of the {\tt XI} Brazilian
Conference on Mathematical Logic, May 1996, Salvador, Bahia,
Brazil. Contemporary Mathematics, v.235, (1999) American
Mathematical Society, pp.33--52.

\bibitem{CM9?}
W. Carnielli and J. Marcos, Limits for paraconsistent calculi.
Submitted to publication, 13p.

\bibitem{Gab99}
D. Gabbay, {\it Fibring Logics}.  Clarendon Press, Oxford
University Press, 1999. 476p.

\bibitem{Mar99}
J. Marcos, {\it Possible-Translations Semantics} (in Portuguese).
Campinas, August 1999. {\tt xxviii} $+$ 240p. Thesis -- IFCH,
Unicamp. \\ see \small{\tt
http://www.unicamp.br/cle/recent\_students.htm}

\end{thebibliography}


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