On the Concept of a Notational Variant

  • Alexander W. KocurekEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10455)


In the study of modal and nonclassical logics, translations have frequently been employed as a way of measuring the inferential capabilities of a logic. It is sometimes claimed that two logics are “notational variants” if they are translationally equivalent. However, we will show that this cannot be quite right, since first-order logic and propositional logic are translationally equivalent. Others have claimed that for two logics to be notational variants, they must at least be compositionally intertranslatable. The definition of compositionality these accounts use, however, is too strong, as the standard translation from modal logic to first-order logic is not compositional in this sense. In light of this, we will explore a weaker version of this notion that we will call schematicity and show that there is no schematic translation either from first-order logic to propositional logic or from intuitionistic logic to classical logic.


Translation Notational variant Lindenbaum-Tarski algebras Compositionality Schematicity 


  1. 1.
    Andréka, H., István, N., van Benthem, J.F.A.K.: Modal languages and bounded fragments of predicate logic. J. Philos. Logic 27(3), 217–274 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001)CrossRefzbMATHGoogle Scholar
  3. 3.
    Caleiro, C., Gonçalves, R.: Equipollent logical systems. In: Beziau, J.Y. (ed.) Logica Universalis, pp. 97–109. Springer, Heidelberg (2007). doi: 10.1007/978-3-7643-8354-1_6 CrossRefGoogle Scholar
  4. 4.
    Carnielli, W.A., Coniglio, M.E., D’Ottaviano, I.M.L.: New dimensions on translations between logics. Logica Universalis 3, 1–18 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Epstein, R.L.: The semantic foundations of logic. In: Epstein, R.L. (ed.) The Semantic Foundations of Logic Volume 1: Propositional Logics. Springer, Dordrecht (1990). doi: 10.1007/978-94-009-0525-2_11 CrossRefGoogle Scholar
  6. 6.
    French, R.: Translational embeddings in modal logic. Ph.D. thesis (2010)Google Scholar
  7. 7.
    Gödel, K.: Zur Intuitionistischen Arithmetik und Zahlentheorie. Ergebnisse eines mathematischen Kolloquiums 4, 34–38 (1933). Reprinted in Gödel 1986, pp. 286–295zbMATHGoogle Scholar
  8. 8.
    Gödel, K.: Collected Works. Oxford University Press, Oxford (1986)zbMATHGoogle Scholar
  9. 9.
    Kracht, M., Wolter, F.: Normal monomodal logics can simulate all others. J. Symb. Logic 64(01), 99–138 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jeřábek, E.: The ubiquity of conservative translations. Rev. Symb. Logic 5, 666–678 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mossakowski, T., Diaconescu, R., Tarlecki, A.: What is a logic translation? Logica Universalis 3, 95–124 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Pelletier, F.J., Urquhart, A.: Synonymous logics. J. Philos. Logic 32, 259–285 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Straßburger, L.: What is a logic, and what is a proof? In: Beziau, J.Y. (ed.) Logica Universalis, pp. 135–152. Springer, Basel (2007)CrossRefGoogle Scholar
  14. 14.
    Thomason, S.K.: Reduction of tense logic to modal logic. I. J. Symb. Logic 39(3), 549–551 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Thomason, S.K.: Reduction of tense logic to modal logic II. Theoria 41(3), 154–169 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wójcicki, R.: Theory of Logical Calculi: Basic Theory of Consequence Operators. Springer, Dordrecht (1988). doi: 10.1007/978-94-015-6942-2 CrossRefzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.University of California, BerkeleyBerkeleyUSA

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