An Institution-Independent Proof of the Robinson Consistency Theorem


We prove an institutional version of A. Robinson’s Consistency Theorem. This result is then appliedto the institution of many-sorted first-order predicate logic and to two of its variations, infinitary and partial, obtaining very general syntactic criteria sufficient for a signature square in order to satisfy the Robinson consistency and Craig interpolation properties.

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  1. [1]

    Aiguier, M., and F. Barbier, ‘On a generalization of some results of model theory: Robinson, Craig and Beth theorems’. Submitted. (Ask authors for current version at aiguier,

  2. [2]

    Andr' emeti, and I. Sain, ‘Craig property of a logic and decomposability eka, H., I. N' of theories’, in P. Dekker and M. Stokhof (eds.), Proceedings ofthe Ninth Amsterdam Colloquium, vol. 3229 of Lecture Notes in Computer Science, 1993, pp. 87–93.

  3. [3]

    Barwise, J., and J. Feferman, Model-Theoretic Logics, Springer, 1985.

  4. [4]

    Bergstra, J., J. Heering, and P.Klint, ‘Module algebra’, Journal of the Association for Computing Machinery, 37(2):335–372, 1990. An Institution-Independent Proof of the Robinson Consistency Theorem

    Google Scholar 

  5. [5]

    Bicarregui, D. G. J., T. Dimitrakos and T. Maibaum, ‘Interpolation in practical formal development’, Logic Journal of the IGPL, 9(1):231–243, 2001.

    Article  Google Scholar 

  6. [6]

    Borzyszkowski, T., ‘Generalized interpolation in CASL’, Inf. Process. Lett., 76(1–2):19–24, 2000.

    Article  Google Scholar 

  7. [7]

    Borzyszkowski, T., ‘Logical systems for structured specifications’, Theor. Comput. Sci., 286(2):197–245, 2002.

    Article  Google Scholar 

  8. [8]

    Burmeister, P., A Model Theoretic Oriented Appraoch to Partial Algebras, Akademie-Verlag Berlin, 1986.

    Google Scholar 

  9. [9]

    Burstall, R., and J. Goguen, ‘Semantics of Clear’, Unpublished notes handed out at the 1978 Symposium on Algebra and Applications. Stefan Banach Center, Warsaw, Poland, 1978.

  10. [10]

    Cengarle, M. V., Formal Specifications with High-Order Parametrization, PhD thesis, Institute for Informatics, Ludwig-Maximilians University, Munich, 1994.

  11. [11]

    Chang, C. C., and H. J. Keisler, Model Theory, North Holland, Amsterdam, 1973.

    Google Scholar 

  12. [12]

    CoFI task group on semantics, CASL — The Common Algebraic Specification Language, Semantics,, July 1999.

  13. [13]

    Craig, W., ‘Linear reasoning. A new form of the Herbrand-Gentzen Theorem’, Journal of Symbolic Logic, 22:250–268, 1957.

    Article  Google Scholar 

  14. [14]

    Diaconescu, R., Institution-independent Model Theory, Book draft. (Ask author for current version at

  15. [15]

    Diaconescu, R., ‘Institution-independent ultraproducts’, Fundamenta Informaticae, 55(3–4):321–348, 2003.

    Google Scholar 

  16. [16]

    Diaconescu, R., ‘Elementarydiagrams in institutions’. Journal of Logic and Computation, 14(5):651–674, 2004.

    Article  Google Scholar 

  17. [17]

    Diaconescu, R., ‘Herbrand theorems in arbitrary institutions’, Inf. Process. Lett., 90:29–37, 2004.

    Article  Google Scholar 

  18. [18]

    Diaconescu, R., ‘An institution-independent proof of Craig interpolation theorem’, Studia Logica, 77(1):59–79, 2004.

    Article  Google Scholar 

  19. [19]

    Diaconescu, R., ‘Interpolation in Grothendieck institutions’, Theor. Comput. Sci., 311:439–461, 2004.

    Article  Google Scholar 

  20. [20]

    Diaconescu, R., J. Goguen, and P. Stefaneas, ‘Logical support for modularization’, in G. Huet and G. Plotkin (eds.), Logical Environments, Cambridge, 1993, pp. 83–130.

    Google Scholar 

  21. [21]

    Dimitrakos, T., and T. Maibaum, ‘On a generalised modularization theorem’, Inf. Process. Lett., 74(2):65–71, 2000.

    Article  Google Scholar 

  22. [22]

    Feferman, S., ‘Lectures on proof theory’, in M. Lob (ed.), Proceedings of the Summer School in Logic, Lecture Notes in Mathematics, vol. 70, pp. 1–107, 1968.

  23. [23]

    G.a, D., and A. Popescu, ‘An institution-independent generalization of Tarski's ain. elementary chain theorem’, Journal of Logic and Computation, To appear.

  24. [24]

    Goguen, J., and R. Burstall, ‘Institutions: Abstract model theory for specification and programming’, Journal of the Association for Computing Machinery, 39(1):95–146, January 1992.

    Google Scholar 

  25. [25]

    Keisler, H. J., Model Theory for Infinitary Logic, North-Holland, 1971.

  26. [26]

    Mac Lane, S., Categories for the Working Mathematician, Springer, 1971.

  27. [27]

    Madar' asz, J. X., ‘Interpolation and amalgamation; Pushing the limits. Part ii’, Studia Logica, 62(1):1–19, 1999.

    Article  Google Scholar 

  28. [28]

    Makkai, M., and R. Par' e, Accessible Categories: The Foundations of Categorical Model Theory, Providence, 1989.

  29. [29]

    Makkai, M., and G. Reyes, ‘First order categorical logic’, Lecture Notes in Mathematics, (611), 1977.

  30. [30]

    McMillan, K. L., ‘Interpolation and sat-based model checking’, in Computed Aided Verification 2003, vol. 2725 of Lecture Notes in Computer Science, Springer-Verlag, 2003, pp. 1–13.

  31. [31]

    McMillan, K. L., ‘An interpolating theorem prover’, in TACAS, vol. 2988, 2004, pp. 16–30.

    Google Scholar 

  32. [32]

    Meseguer, J., ‘General logics’. in H.-D. Ebbinghaus et al. (eds.), Proceedings, Logic Colloquium 1987, North-Holland, 1989, pp. 275–329.

  33. [33]

    Monk, J. D., Mathematical Logic, Springer-Verlag, 1976.

  34. [34]

    Mossakowski, T., J. Goguen, R. Diaconescu, and A. Tarlecki, ‘What is a logic?’, in J.-Y. Beziau (ed.), Logica Universalis, Birkhauser, 2005, pp. 113–133.

    Google Scholar 

  35. [35]

    Mundici, D., ‘Compactness + Craig interpolation = Robinson consistency in any logic’, Manuscript, University of Florence, 1979.

  36. [36]

    Mundici, D., ‘Robinson consistency theorem in soft model theory’, Atti Accad. Nat. Lincei. Redingoti, 67:383–386, 1979.

    Google Scholar 

  37. [37]

    Mundici, D., ‘Robinson's consistency theorem in soft model theory’, Trans. of the AMS, 263:231–241, 1981.

    Article  Google Scholar 

  38. [38]

    Nelson, G., and D. C. Oppen, ‘Simplification by cooperating decision procedures’, ACM Trans. Program. Lang. Syst., 1(2):245–257, 1979.

    Article  Google Scholar 

  39. [39]

    Oppen, D. C., ‘Complexity, convexity and combinations of theories’, Theoretical Computer Science, 12:291–302, 1980.

    Article  Google Scholar 

  40. [40]

    Petria, M., and R. Diaconescu, ‘Abstract Beth definability institutionally’, Submitted. (Ask authors for current version at

  41. [41]

    Reichel, H., Structural Induction on Partial Algebras, Akademie-Verlag Berlin, 1984.

    Google Scholar 

  42. [42]

    Robinson, A., ‘A result on consistency and its application to the theory of definition’, vol. 59 of Konikl. Ned. Akad. Wetenschap. (Amsterdam)Proc A, 1956, pp. 47–58.

  43. [43]

    Roşu, G., and J. Goguen, ‘On equational Craig interpolation’, Journal of Universal Computer Science, 6(1):194–200, 2000.

    Google Scholar 

  44. [44]

    Rodenburg, P.H., ‘A simple algebraic proof of the equational interpolation theorem’, Algebra Universalis, 28:48–51, 1991.

    Article  Google Scholar 

  45. [45]

    Sain, I., ‘Beth's and Craig's properties via epimorphisms and amalgamation in algebraic logic’, in Algebraic Logic and Universal Algebra in Computer Science, 1988, pp. 209–225.

  46. [46]

    Salibra, A., and G. Scollo, ‘A soft stairway to institutions’, in M. Bidoit and C. Choppy (eds.), Recent Trends in Data Type Specification, vol. 655 of Lecture Notes in Computer Science, Springer-Verlag, 1992, pp. 310–329.

  47. [47]

    Salibra, A., and G. Scollo, ‘Interpolation and compactness in categories of pre- institutions’, Math. Struct. in Comp. Science, 6:261–286, 1996. An Institution-Independent Proof of the Robinson Consistency Theorem

    Article  Google Scholar 

  48. [48]

    Sannella, D., and A. Tarlecki, ‘Specifications in an arbitrary institution’, Information and Control, 76:165–210, 1988.

    Google Scholar 

  49. [49]

    Tarlecki, A., ‘Bits and pieces of the theory of institutions’, in D. Pitt, S. Abramsky, A. Poign' e, and D. Rydeheard (eds.), Proceedings, Summer Workshop on Category Theory and Computer Programming, Lecture Notes in Computer Science, vol. 240, Springer, 1986, pp. 334–360.

  50. [50]

    Tarlecki, A., ‘On the existence of free models in abstract algebraic institutions’, Theoretical Computer Science, 37:269–304, 1986.

    Article  Google Scholar 

  51. [51]

    Tarlecki, A., ‘Quasi-varieties in abstract algebraic institutions’, Journal of Computer and System Sciences, 33(3):333–360, 1986.

    Article  Google Scholar 

  52. [52]

    Tinelli, C., and C. Zarba, ‘Combining decision procedures for sorted theories’, in J.J. Alferes and J. A.Leite (eds.), Logics in Artificial Intelligence, vol. 3229 of Lecture Notes in Computer Science, Springer, 2004, pp. 641–653.

  53. [53]

    Weiss, W. A. R., and C. D'Mello, ‘Fundamentals of model theory’, Lecture Notes, University of Toronto, 1997.

  54. [54]

    Wirsing, M., ‘Structured specifications: Syntax, semantics and proof calculus’, in F. Bauer, W. Bauer, and H. Schwichtenberg (eds.), Logic and Algebra of Specification, vol. 94 of NATO ASI Series F: Computer and System Sciences, Springer-Verlag, 1991, pp. 411–442.

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Correspondence to Daniel Gâinâ.

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Presented by Robert Goldblatt

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Gâinâ, D., Popescu, A. An Institution-Independent Proof of the Robinson Consistency Theorem. Stud Logica 85, 41–73 (2007).

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  • institution
  • Robinson consistency
  • Craig interpolation
  • elementary diagram
  • many-sorted first-order logic