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On the Universality of Atomic and Molecular Logics via Protologics

Abstract

After observing that the truth conditions of connectives of non–classical logics are generally defined in terms of formulas of first–order logic, we introduce ‘protologics’, a class of logics whose connectives are defined by arbitrary first–order formulas. Then, we introduce atomic and molecular logics, which are two subclasses of protologics that generalize our gaggle logics and which behave particularly well from a theoretical point of view. We also study and introduce a notion of equi-expressivity between two logics based on different classes of models. We prove that, according to that notion, every pure predicate logic with \(k\ge 0\) variables and constants is as expressive as a predicate atomic logic, some sort of atomic logic. Then, we prove that the class of protologics is equally expressive as the class of molecular logics. That formally supports our claim that atomic and molecular logics are somehow ‘universal’. Finally, we identify a subclass of molecular logics that we call predicate molecular logics and which constitutes its representative core: every molecular logic is as expressive as a predicate molecular logic.

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Notes

  1. I thank Peter Arndt for checking and proving that result.

  2. The definition of molecular connectives and molecular logics of [6] is less general than in the present article, in the sense that it is not possible there to have the same argument at different places in the definition of a molecular connective. This said, it is nevertheless possible to define an appropriate notion of \({\textsf {C}}\)-bisimulation for the more general definition of molecular logics of the present article.

  3. [5] differs very slightly from [4]. It essentially corrects minor mistakes and typos and proves that the rule of associativity is derivable in \(\textsf {GGL}_{{\textsf {C}}}\).

References

  1. Allwein, G., Dunn, M.J.: Kripke models for linear logic. J. Symbol. Logic 58(2), 514–545 (1993)

  2. Arndt, P., Freire, R.A., Luciano, O.O., Mariano, H.L.: A global glance on categories in logic. Logica Universalis 1(1), 3–39 (2007)

    MathSciNet  Article  Google Scholar 

  3. Aucher, G.: Displaying Updates in Logic. J. Logic Comput. 26(6), 1865–1912 (2016)

    MathSciNet  Article  Google Scholar 

  4. Aucher, G.: Towards universal logic: gaggle logics. J. Applied Logics. 7(6), 875–945 (2020)

  5. Aucher, G.: Selected Topics from Contemporary Logics, chapter Towards Universal Logic: Gaggle Logics, pages 5–73. Landscapes in Logic. College Publications, October (2021)

  6. Aucher, G.: A van Benthem theorem for atomic and molecular logics. In Proceedings of Non-classical logics. Theory and applications. (NCL’22), EPTCS, (2022)

  7. Barwise, J.: Axioms for abstract model theory. Annal. Math. Logic 7(2), 221–265 (1974)

    MathSciNet  Article  Google Scholar 

  8. Béziau, J-Y.: Logica universalis, chapter from consequence operator to universal logic: a survey of general abstract logic. Birkhäuser Basel, (2007)

  9. Béziau, J.-Y.: Editorial: Introduction to the universal logic corner. J. Log. Comput. 19(6), 1111 (2009)

    Article  Google Scholar 

  10. Bimbo, K., Dunn, M.J.: Generalized galois logics: relational semantics of nonclassical logical calculi. number 188. center for the study of language and information, (2008)

  11. Chang, C.C., Keisler, H.J.: Model theory. studies in logic and the foundations of mathematics. Elsevier, Amsterdam (1998)

    Google Scholar 

  12. Dunn, M.J.: Gaggle theory: an abstraction of galois connections and residuation, with applications to negation, implication, and various logical operators. European workshop on logics in artificial intelligence. Springer, Berlin Heidelberg (1990)

  13. Dunn, M.J.: Philosophy of language and logic, of philosophical perspectives, chapter perp and star two treatments of negation. Ridgeview Publishing Company, USA (1993)

  14. Dunn, M.J., Hardegree, G.M.: Algebraic methods in philosophical logic in oxford logic guides. Clarendon Press, Oxford (2001)

  15. Dunn, M.J., Zhou, C.: Negation in the context of gaggle theory. Studia Logica 80(2–3), 235–264 (2005)

  16. Enderton, H.: An introduction to mathematical logic. Academic Press, USA (2001)

    MATH  Google Scholar 

  17. Gabbay, D. (ed.): What is a logical system?. Studies in logic and computation. Oxford University Press, USA (1994)

    Google Scholar 

  18. Gabbay, D.: Labelled deductive systems. Oxford University Press, USA (1996)

    MATH  Google Scholar 

  19. Garcia-Matos, M., Väänänen, J.: Abstract model theory as a framework for universal logic. In Jean-Yves Béziau, editor, Logica Universalis, pages 19–33, Basel, (2007). Birkhäuser Basel

  20. Goguen, J.A., Burstall, R.M.: Institutions: Abstract model theory for specification and programming. J. ACM (JACM) 39(1), 95–146 (1992)

    MathSciNet  Article  Google Scholar 

  21. Goldblatt, R.: Semantic analysis of orthologic. J. Phil. Logic 3, 19–35 (1974)

    MathSciNet  Article  Google Scholar 

  22. Goré, R.: Substructural logics on display. Logic J. IGPL 6(3), 451–504 (1998)

    MathSciNet  Article  Google Scholar 

  23. Grishin, V.: On a generalization of the Ajdukiewicz-Lambek system. In: Mikhailov, A.I. (ed.) Studies in nonclassical logics and formal systems, pp. 315–334. Nauka, Moscow (1983)

    Google Scholar 

  24. Janin D., Lenzi G.: Relating levels of the mu-calculus hierarchy and levels of the monadic hierachy. In: LICS, pages 347–356, Boston, United States, IEEE computer society (2001)

  25. Kripke, S.A.: Semantical analysis of modal logic, i: normal propositional calculi. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 8, 113–116 (1963)

    MathSciNet  Article  Google Scholar 

  26. Kripke, S.A.: Formal Systems and Recursive Functions, chapter semantical analysis of intuitionistic logic, I, pages 92–130. North Holland, Amsterdam, (1965)

  27. Kuhn, S.T.: Quantifiers as modal operators. Studia Logica 39(2), 145–158 (1980)

    MathSciNet  Article  Google Scholar 

  28. Lambek, J.: The mathematics of sentence structure. Am. Math. Monthly 65, 154–170 (1958)

    MathSciNet  Article  Google Scholar 

  29. Marx, M., Venema, Y.: Multi-dimensional Modal Logic, volume 4 of Applied logic series. Kluwer (1997)

  30. Meseguer, J.: General logics. In: Ebbinghaus, H.-D., Fernandez-Prida, J., Garrido, M., Lascar, D., Rodriquez Artalejo, M. (eds.) Logic Colloquium’87, of studies in logic and the foundations of mathematics. Elsevier, Amsterdam (1989)

    Google Scholar 

  31. Moortgat, M.: Symmetries in natural language syntax and semantics: the lambek-grishin calculus in logic, language information and computation. Springer, Cham (2007)

    MATH  Google Scholar 

  32. Moss, L.S.: Applied logic: a manifesto. mathematical problems from applied logic I. Springer, Cham (2006)

    MATH  Google Scholar 

  33. Mossakowski, T., Diaconescu, R., Tarlecki, A.: What is a logic translation? Logica Universalis 3(1), 95–124 (2009)

    MathSciNet  Article  Google Scholar 

  34. Mossakowski, T., Goguen, J., Diaconescu, R., Tarlecki, A.: What is a logic? In: Beziau, J.-Y. (ed.) Logica Universalis, pp. 111–133. Basel, Birkhäuser (2007)

    Chapter  Google Scholar 

  35. Olkhovikov, G.K.: On expressive power of basic modal intuitionistic logic as a fragment of classical FOL. J. Appl. Log. 21, 57–90 (2017)

    MathSciNet  Article  Google Scholar 

  36. Priest, G.: An introduction to non-classical logic. Cambridge University Press, USA (2011)

    MATH  Google Scholar 

  37. Prior, A.: Past, present and future. Clarendon Press, Oxford (1967)

    Book  Google Scholar 

  38. Restall, G.: An introduction to substructural logics. Routledge, UK (2000)

  39. Rotman, J.J.: An introduction to the theory of groups. graduate texts in mathematics. Springer, New York (1995)

    Book  Google Scholar 

  40. Sambin, G., Battilotti, G., Faggian, C.: Basic logic: reflection, symmetry, visibility. J. Symbol. Logic 65(03), 979–1013 (2000)

    MathSciNet  Article  Google Scholar 

  41. Väänänen, J.: Second-order and higher-order logic. In: Edward, N., Zalta, editor, The stanford encyclopedia of philosophy. metaphysics research lab, Stanford University, Fall 2021 edition, (2021)

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Correspondence to Guillaume Aucher.

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This paper is the winner of the Louis Couturat Logic Prize 2021 (France)

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Aucher, G. On the Universality of Atomic and Molecular Logics via Protologics. Log. Univers. 16, 285–322 (2022). https://doi.org/10.1007/s11787-022-00298-5

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Keywords

  • Universal logic
  • Expressivity
  • First-order logics
  • Non-classical logics

Mathematics Subject Classification

  • 03
  • 03B
  • 03C