Course 5

Translating and Combining Logics

Hércules de Araujo Feitosa
CLE/ São Paulo State University – Unesp, Brazil
haf@fc.unesp.br

(with the assistance of Marcelo E. Coniglio)

Abstract

A general theory of translations, explicitly interested in the study of interrelations between logic systems in general, was introduced in da Silva, D’Ottaviano and Sette (1999). An extended definition for the concept of translation between logics, in order to single out what seems to be in fact the essential feature of a logical translation, was proposed: logics are characterized as pairs constituted by a set and a consequence operator, and translations between logics are defined as maps preserving consequence relations.

Feitosa and D’Ottaviano (2001) introduced the concept of conservative translation, having obtained conservative translations involving classical logic and several non-classical logics. Other stricter notions of translations, such as contextual translations, were introduced by Carnielli, Coniglio and D’Ottaviano (2009).

The use of (interesting) translations between logics constitutes a useful help for studying and analyzing interrelations between diverse logic systems, some intuitionistic logics, modal logics, Lukasiewicz and Post’s many-valued logics and several paraconsistent logics (D’Ottaviano and Feitosa 2019).

Translations between logical systems constitute an important tool in the theory of combination of logics. Besides leading to interesting and useful applications in situations in which it is necessary to work with different logics at the same time, combinations of logics are also of interest from the theoretical and philosophical perspective.

In areas such as knowledge representation (within artificial intelligence) and in formal specification and verification (within software engineering) well-behaved techniques for combining logic systems are very useful.

Several techniques for combining logics were introduced in the literature and these new tools permit us to combine propositional logics, non-truth-functional logics, first-order logics, modal logics and higher-order logics, among others (Carnielli et allia (2008)).

This tutorial gives an overview of several conceptual and technical aspects of translations between logics, on the one hand, and some important techniques for combining logics, on the other, with emphasis on fibring. The relevance of the notion of translation between logics for the theory of combining logics will be addressed. Several examples will be discussed.

References

Carnielli, W. A. and Coniglio, M. E. Combining Logics. In: The Stanford Encyclopedia of Philosophy, 2007. Ed.: E.N. Zalta.

Carnielli, W. A., Coniglio, M. E., Gabbay, D. , Gouveia, P. and Sernadas, C. Analysis and Synthesis of Logics. How to Cut and Paste Reasoning Systems. Applied Logic Series, vol. 35, Springer, 2008.

Carnielli, W., Coniglio, M. E., and D’Ottaviano, I.M. L. (2009) New Dimensions on traslations between logics. Logica Universalis, v.3, p. 1-19.

da Silva, J.J., D’ Ottaviano, I.M.L., and Sette, A.M. (1999) Translations between logics. In: Caicedo, X., and Montenegro, C.H. (Eds.) Models, algebras and proofs. New York: Marcel Dekker, p. 435-448.

D’Ottaviano, and I. M. L., Feitosa, H. A. (2019) Translations between Logics: a survey. The Proceedings of the 14th International Ludwig Wittgensteins Symposium, 2018. G.Mars, P. Weingartner, B. Ritter (eds.). Berlin: De Gruyter, p. 121-140.

Feitosa, H.A., and D’Ottaviano, I.M.L. (2001) Conservative translations. Annals of Pure and Applied Logic, Amsterdam, v. 108, p. 205-227.