Skip to main content

Conservative Translations Revisited

Journal of Philosophical Logic (2022)Cite this article

Abstract

We provide sufficient conditions for the existence of a conservative translation from a consequence system to another one. We analyze the problem in many settings, namely when the consequence systems are generated by a deductive calculus or by a logic system including both proof-theoretic and model-theoretic components. We also discuss reflection of several metaproperties with the objective of showing that conservative translations provide an alternative to proving such properties from scratch. We discuss soundness and completeness, disjunction property and metatheorem of deduction among others. We provide several illustrations of conservative translations.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Barwise, J. (1974). Axioms for abstract model theory. Annals for Mathematical Logic, 7, 221–265.

    Article  Google Scholar 

  2. Blok, W.J., & Pigozzi, D. (1989). Algebraizable Logics, Memoirs of the American Mathematical Society (Vol. 396). AMS.

  3. Brown, D.J., Suszko, R., & Bloom, S.L. (1973). Abstract logics. Dissertationes Mathematicae, 102, 52.

    Google Scholar 

  4. Bueno-Soler, J., & Carnielli, W.A. (2005). Possible-translations algebraization for paraconsistent logics. Bulletin of the Section of Logic, 34(2), 77–92.

    Google Scholar 

  5. Carnielli, W.A. (1990). Many-valued logics and plausible reasoning. In Proceedings of the twentieth international symposium on multiple-valued logic (pp. 328–335).

  6. Carnielli, W.A., & Coniglio, M.E. (2016). Paraconsistent logic: consistency. Contradiction and negation. Springer.

  7. Carnielli, W.A., Coniglio, M.E., & D’Ottaviano, I.L. (2009). New dimensions on translations between logics. Logica Universalis, 3(1), 1–18.

    Article  Google Scholar 

  8. Diaconescu, R. (2008). institution-independent model theory. Studies in Universal Logic, Birkhäuser.

  9. D’Ottaviano, I.M.L., & Feitosa, H.A. (2000). Paraconsistent logics and translations. Synthese, 125(1-2), 77–95.

    Google Scholar 

  10. Feitosa, H.A., & D’Ottaviano, I.M.L. (2001). Conservative translations. Annals of Pure and Applied Logic, 108(1-3), 205–227.

    Article  Google Scholar 

  11. Fiadeiro, J.L., & Sernadas, A. (1987). Structuring theories on consequence. In ADT 1987: recent trends in data type specification, Vol. 332 of lecture notes in computer science (pp. 44–72). Springer.

  12. Glivenko, V. (1929). Sur quelques points de la logique de M. Brouwer. Bulletins de la Classe des Sciences, 15(5), 183–188.

    Google Scholar 

  13. Gödel, K. (1986). Collected works (vol. I). Oxford University Press.

  14. Goguen, J.A., & Burstall, R.M. (1984). Introducing institutions. In Logics of programs, vol. 164 of lecture notes in computer science (pp. 221–256). Springer.

  15. Goguen, J.A., & Burstall, R.M. (1992). Institutions: abstract model theory for specification and programming. Journal of the Association for Computing Machinery, 39(1), 95–146.

    Article  Google Scholar 

  16. Jeřábek, E. (2012). The ubiquity of conservative translations. The Review of Symbolic Logic, 5(4), 666–678.

    Article  Google Scholar 

  17. Kamide, N., & Wansing, H. (2012). Proof theory of Nelson’s paraconsistent logic: a uniform perspective. Theoretical Computer Science, 415, 1–38.

    Article  Google Scholar 

  18. Kolmogorov, A. (1925). On the principle “tertium non datur”. Mathematicheskii Sbornik, 32, 646–667. English translation, On the principle of excluded middle, in From Frege to Gödel: A Source Book on Mathematical Logic. J. van Heijenoort 1967, pp 414–437.

    Google Scholar 

  19. Kubyshkina, E. (2021). Conservative translations of four-valued logics in modal logic. Synthese, 198(suppl. 22), S5555–S5571.

    Article  Google Scholar 

  20. Marcos, J. (2008). Possible-translations semantics for some weak classically-based paraconsistent logics. Journal of Applied Non-Classical Logics, 18(1), 7–28.

    Article  Google Scholar 

  21. McKinsey, J.C.C., & Tarski, A. (1948). Some theorems about the sentential calculi of Lewis and Heyting. The Journal of Symbolic Logic, 13, 1–15.

    Article  Google Scholar 

  22. Meseguer, J. (1989). General logics. In Logic Colloquium’87, Vol. 129 of Stud. Logic Found. Math. (pp. 275–329). North-Holland.

  23. Mints, G. (2000). A short introduction to intuitionistic logic. Kluwer.

  24. Mints, G. (2012). The Gödel-Tarski translations of intuitionistic propositional formulas. In E. Edem, J. Lee, Y. Lierler, & D. Pearce (Eds.) Correct reasoning (pp. 487–491). Springer.

  25. Prawitz, D., & Malmnäs, P.-E. (1968). A survey of some connections between classical, intuitionistic and minimal logic. In Contributions to mathematical logic colloquium (pp. 215–229). North-Holland.

  26. Rybakov, V. (1997). Admissibility of logical inference rules. North-Holland.

  27. Schurz, G. (2021). Why classical logic is privileged: justification of logics based on translatability. Synthese, 199(5-6), 13067–13094.

    Article  Google Scholar 

  28. Schurz, G. (2022). Meaning-preserving translations of non-classical logics into classical logic: between pluralism and monism. Journal of Philosophical Logic, 51(1), 27–55.

    Article  Google Scholar 

  29. Rasga, J., Sernadas, C., & Carnielli, W. A. (2021). Reduction techniques for proving decidability in logics and their meet-combination. The Bulletin of Symbolic Logic, 27(1), 39–66.

    Article  Google Scholar 

  30. Voutsadakis, G. (2005). Categorical abstract algebraic logic: models of π-institutions. Notre Dame Journal of Formal Logic, 46(4), 439–460.

    Article  Google Scholar 

  31. Sernadas, A., Sernadas, C., & Rasga, J. (2012). On meet-combination of logics. Journal of Logic and Computation, 22(6), 1453–1470.

    Article  Google Scholar 

  32. Wójcicki, R. (1988). Theory of logical calculi. Kluwer.

  33. Cruz-Filipe, L., Sernadas, A., & Sernadas, C. (2008). Heterogeneous fibring of deductive systems via abstract proof systems. Logic Journal of the IGPL, 16(2), 121–153.

    Article  Google Scholar 

  34. Carnielli, W. A., Coniglio, M. E., Gabbay, D., Gouveia, P., & Sernadas, C. (2008). Analysis and synthesis of logics. Springer.

  35. Sernadas, C., Rasga, J., & Carnielli, W. A. (2002). Modulated fibring and the collapsing problem. The Journal of Symbolic Logic, 67(4), 1541–1569.

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to acknowledge the support of Instituto de Telecomunicações Research Unit ref. UIDB/50008/2020 funded by Fundação para a Ciência e a Tecnologia (FCT) and the Department of Mathematics of Instituto Superior Técnico, Universidade de Lisboa. The authors acknowledge the suggestions and comments of the two anonymous reviewers that helped to shape the final version of the paper.

Funding

The authors would like to acknowledge the support of Instituto de Telecomunicações, Research Unit ref. UIDB/50008/2020 funded by Fundação para a Ciência e a Tecnologia (FCT) and the Department of Mathematics of Instituto Superior Técnico, Universidade de Lisboa.

Author information

Authors and Affiliations

  1. Department of Mathematics, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal

    J. Ramos, J. Rasga & C. Sernadas

  2. Instituto de Telecomunicações, Aveiro, Portugal

    J. Ramos, J. Rasga & C. Sernadas

Authors
  1. J. Ramos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Rasga
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. Sernadas
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors whose names appear on the submission

1) made substantial contributions to the conception or design of the work;

2) drafted the work or revised it critically for important intellectual content;

3) approved the version to be published; and

4) agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.

Corresponding author

Correspondence to C. Sernadas.

Ethics declarations

Competing interests

The authors did not have and do not have any relationship with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ramos, J., Rasga, J. & Sernadas, C. Conservative Translations Revisited. J Philos Logic (2022). https://doi.org/10.1007/s10992-022-09691-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10992-022-09691-3

Keywords

  • Consequence system
  • Conservative translation
  • Reflection of metatheorems by conservative translation