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.
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Barwise, J. (1974). Axioms for abstract model theory. Annals for Mathematical Logic, 7, 221–265.
Blok, W.J., & Pigozzi, D. (1989). Algebraizable Logics, Memoirs of the American Mathematical Society (Vol. 396). AMS.
Brown, D.J., Suszko, R., & Bloom, S.L. (1973). Abstract logics. Dissertationes Mathematicae, 102, 52.
Bueno-Soler, J., & Carnielli, W.A. (2005). Possible-translations algebraization for paraconsistent logics. Bulletin of the Section of Logic, 34(2), 77–92.
Carnielli, W.A. (1990). Many-valued logics and plausible reasoning. In Proceedings of the twentieth international symposium on multiple-valued logic (pp. 328–335).
Carnielli, W.A., & Coniglio, M.E. (2016). Paraconsistent logic: consistency. Contradiction and negation. Springer.
Carnielli, W.A., Coniglio, M.E., & D’Ottaviano, I.L. (2009). New dimensions on translations between logics. Logica Universalis, 3(1), 1–18.
Diaconescu, R. (2008). institution-independent model theory. Studies in Universal Logic, Birkhäuser.
D’Ottaviano, I.M.L., & Feitosa, H.A. (2000). Paraconsistent logics and translations. Synthese, 125(1-2), 77–95.
Feitosa, H.A., & D’Ottaviano, I.M.L. (2001). Conservative translations. Annals of Pure and Applied Logic, 108(1-3), 205–227.
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.
Glivenko, V. (1929). Sur quelques points de la logique de M. Brouwer. Bulletins de la Classe des Sciences, 15(5), 183–188.
Gödel, K. (1986). Collected works (vol. I). Oxford University Press.
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.
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.
Jeřábek, E. (2012). The ubiquity of conservative translations. The Review of Symbolic Logic, 5(4), 666–678.
Kamide, N., & Wansing, H. (2012). Proof theory of Nelson’s paraconsistent logic: a uniform perspective. Theoretical Computer Science, 415, 1–38.
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.
Kubyshkina, E. (2021). Conservative translations of four-valued logics in modal logic. Synthese, 198(suppl. 22), S5555–S5571.
Marcos, J. (2008). Possible-translations semantics for some weak classically-based paraconsistent logics. Journal of Applied Non-Classical Logics, 18(1), 7–28.
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.
Meseguer, J. (1989). General logics. In Logic Colloquium’87, Vol. 129 of Stud. Logic Found. Math. (pp. 275–329). North-Holland.
Mints, G. (2000). A short introduction to intuitionistic logic. Kluwer.
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.
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.
Rybakov, V. (1997). Admissibility of logical inference rules. North-Holland.
Schurz, G. (2021). Why classical logic is privileged: justification of logics based on translatability. Synthese, 199(5-6), 13067–13094.
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.
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.
Voutsadakis, G. (2005). Categorical abstract algebraic logic: models of π-institutions. Notre Dame Journal of Formal Logic, 46(4), 439–460.
Sernadas, A., Sernadas, C., & Rasga, J. (2012). On meet-combination of logics. Journal of Logic and Computation, 22(6), 1453–1470.
Wójcicki, R. (1988). Theory of logical calculi. Kluwer.
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.
Carnielli, W. A., Coniglio, M. E., Gabbay, D., Gouveia, P., & Sernadas, C. (2008). Analysis and synthesis of logics. Springer.
Sernadas, C., Rasga, J., & Carnielli, W. A. (2002). Modulated fibring and the collapsing problem. The Journal of Symbolic Logic, 67(4), 1541–1569.
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.
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.
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Ramos, J., Rasga, J. & Sernadas, C. Conservative Translations Revisited. J Philos Logic (2022). https://doi.org/10.1007/s10992-022-09691-3
- Consequence system
- Conservative translation
- Reflection of metatheorems by conservative translation