Skip to main content
×
×
Home

THE UBIQUITY OF CONSERVATIVE TRANSLATIONS

  • EMIL JEŘÁBEK (a1)
Abstract

We study the notion of conservative translation between logics introduced by (Feitosa & D’Ottaviano2001). We show that classical propositional logic (CPC) is universal in the sense that every finitary consequence relation over a countable set of formulas can be conservatively translated into CPC. The translation is computable if the consequence relation is decidable. More generally, we show that one can take instead of CPC a broad class of logics (extensions of a certain fragment of full Lambek calculus FL) including most nonclassical logics studied in the literature, hence in a sense, (almost) any two reasonable deductive systems can be conservatively translated into each other. We also provide some counterexamples, in particular the paraconsistent logic LP is not universal.

Copyright
Corresponding author
*INSTITUTE OF MATHEMATICS AS CR, ŽITNÁ 25, 115 67 PRAHA 1, CZECH REPUBLIC, E-mail: jerabek@math.cas.cz, URL: http://math.cas.cz/~jerabek
References
Hide All
Carnielli, W. A., Coniglio, M. E., & D’Ottaviano, I. M. L. (2009). New dimensions on translations between logics. Logica Universalis, 3(1), 118.
Czelakowski, J. (2001). Protoalgebraic Logic, Vol. 10 of Trends in Logic. Dordecht: Kluwer.
da Silva, J. J., D’Ottaviano, I. M. L., & Sette, A. M. (1999). Translations between logics. In Caicedo, X., and Montenegro, C., editors. Models, Algebras, and Proofs, Vol. 203 of Lecture Notes in Pure and Applied Mathematics. New York: Marcel Dekker, pp. 435448.
D’Ottaviano, I. M. L., & Feitosa, H. A. (1999). Many-valued logics and translations. Journal of Applied Non-classical Logics, 9(1), 121140.
D’Ottaviano, I. M. L., & Feitosa, H. A. (2000). Paraconsistent logics and translations. Synthese, 125(1–2), 7795.
D’Ottaviano, I. M. L., & Feitosa, H. A. (2006). Translating from Lukasiewicz’s logics into classical logic: Is it possible? Poznan Studies in the Philosophy of the Sciences and the Humanities, 91(1), 157168.
D’Ottaviano, I. M. L., & Feitosa, H. A. (2007). Deductive systems and translations. In Béziau, J.-Y., and Costa-Leite, A., editors. Perspectives on Universal Logic. Monza: Polimetrica, pp. 125157.
Feitosa, H. A., & D’Ottaviano, I. M. L. (2001). Conservative translations. Annals of Pure and Applied Logic, 108, 205227.
Galatos, N., Jipsen, P., Kowalski, T., & Ono, H. (2007). Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Vol. 151 of Studies in Logic and the Foundations of Mathematics. Amsterdam: Elsevier.
Lau, D. (2006). Function Algebras on Finite Sets: A Basic Course on Many-Valued Logic and Clone Theory. New York: Springer.
Mossakowski, T., Diaconescu, R., & Tarlecki, A. (2009). What is a logic translation? Logica Universalis, 3(1), 95124.
Post, E. L. (1941). The Two-Valued Iterative Systems of Mathematical Logic. Number 5 in Annals of Mathematics Studies. Princeton, NJ: Princeton University Press.
Shoesmith, D. J., & Smiley, T. J. (1978). Multiple-Conclusion Logic. Cambridge University Press.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Review of Symbolic Logic
  • ISSN: 1755-0203
  • EISSN: 1755-0211
  • URL: /core/journals/review-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed