[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Forcing, Downward Löwenheim-Skolem and Omitting Types Theorems, Institutionally

  • Published:
Logica Universalis Aims and scope Submit manuscript

Abstract

In the context of proliferation of many logical systems in the area of mathematical logic and computer science, we present a generalization of forcing in institution-independent model theory which is used to prove two abstract results: Downward Löwenheim-Skolem Theorem (DLST) and Omitting Types Theorem (OTT). We instantiate these general results to many first-order logics, which are, roughly speaking, logics whose sentences can be constructed from atomic formulas by means of Boolean connectives and classical first-order quantifiers. These include first-order logic (FOL), logic of order-sorted algebras (OSA), preorder algebras (POA), as well as their infinitary variants \({{\bf FOL}_{\omega_1,\omega}}\) , \({{\bf OSA}_{\omega_1,\omega}}\) , \({{\bf POA}_{\omega_1,\omega}}\) . In addition to the first technique for proving OTT, we develop another one, in the spirit of institution-independent model theory, which consists of borrowing the Omitting Types Property (OTP) from a simpler institution across an institution comorphism. As a result we export the OTP from FOL to partial algebras (PA) and higher-order logic with Henkin semantics (HNK), and from the institution of \({{\bf FOL}_{\omega_1,\omega}}\) to \({{\bf PA}_{\omega_1,\omega}}\) and \({{\bf HNK}_{\omega_1,\omega}}\) . The second technique successfully extends the domain of application of OTT to non conventional logical systems for which the standard methods may fail.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adámek, J., Rosický, J.: Locally presentable and accessible categories. Number 189 in London Mathematical Society Lecture Notes. Cambridge University Press (1994)

  2. Ahmed T.S.: Omitting types for algebraizable extensions of first order logic. J. Appl. Non-Classic Logics. 15(4), 465–489 (2005)

    Article  MATH  Google Scholar 

  3. Ahmed T.S., Andréka H., Németi I.: Omitting types for finite variable fragments and complete representations of algebras. J. Symb. Log. 73(1), 65–89 (2008)

    Article  MATH  Google Scholar 

  4. Ahmed T.S., Samir B.: An omitting types theorem for first order logic with infinitary relation symbols. Math. Log. Q. 53(6), 564–570 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Astesiano E., Bidoit M., Kirchner H., Krieg-Brückner B., Mosses P.D., Sannella D., Tarlecki A.: CASL: the Common Algebraic Specification Language. Theor. Comput. Sci. 286(2), 153–196 (2002)

    Article  MATH  Google Scholar 

  6. Barwise J.: Notes on forcing and countable fragments. Mimeographed (1970)

  7. Chang, C.C., Keisler, H.J.: Model Theory. North-Holland, Amsterdam (1990)

  8. Church A.: A Formulation of the Simple Theory of Types. J. Symb. Log., 5(2), 56–68 (1940)

    Article  MathSciNet  Google Scholar 

  9. Codescu, M.: The model theory of higher order logic. Master’s thesis, Şcoala Normală Superioară Bucureşti (2007)

  10. Codescu M., Găină D.: Birkhoff completeness in Institutions. Logica Universalis 2(2), 277–309 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cohen, P.J.: The Independence of the Continuum Hypothesis. In: proceedings of the National academy of sciences of the United States of America, 50(6):1143–1148, December 1963

  12. Cohen, P.J.: The Independence of the Continuum Hypothesis, II. Proceedings of the National Academy of Sciences of the United States of America, 51(1):105–110, January 1964

  13. Diaconescu R.: Category-based constraint logic. Mathematical structures in computer science 10(3), 373–407 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Diaconescu R.: Institution-independent Ultraproducts. Fundamenta Informaticæ 55(3-4):321–348 (2003)

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Diaconescu R.: Elementary Diagrams in Institutions. J. Log. Comput. 14(5), 651–674 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Diaconescu R.: Herbrand Theorems in arbitrary institutions. Inf. Process. Lett. 90, 29–37 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Diaconescu R.: A categorical study on the finiteness of specifications. Inf. Process. Lett., 108(2), 75–80 (2008)

    Article  MathSciNet  Google Scholar 

  19. Diaconescu, R.: Institution-independent Model Theory. Studies in Universal Logic. Birkhäuser (2008)

  20. Diaconescu, R., Futatsugi, K.L.: CafeOBJ Report: The language, proof techniques, and methodologies for object-oriented algebraic specification, vol 6 of AMAST Series in Computing. World Scientific (1998)

  21. Diaconescu R., Futatsugi K.: Logical foundations of CafeOBJ. Theor. Comput. Sci., 285(2), 289–318 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Diaconescu, R., Futatsugi, K., Ogata, K.: CafeOBJ: Logical Foundations and Methodologies. Computers and Artificial Intelligence 22(3) (2003)

  23. Goguen J., Burstall R.: Institutions: Abstract model theory for specification and programming. J. Assoc. Comput Mach 39(1), 95–146 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  24. Goguen J.A., Diaconescu R.: An Oxford Survey of Order Sorted Algebra. Mathematical structures in computer science 4(3), 363–392 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  25. Goguen J.A., Meseguer J.: Order-Sorted Algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theor. Comput. Sci. 105(2), 217–273 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. Goguen J.A., Rosu G.: Institution Morphisms. Formal Asp. Comput. 13(3-5), 274–307 (2002)

    Article  MATH  Google Scholar 

  27. Găină D., Petria M.: Completeness by forcing. J. Log. Comput. 20(6), 1165–1186 (2010)

    Article  MATH  Google Scholar 

  28. Găină D., Popescu A.: An Institution-independent Generalization of Tarski’s Elementary Chain Theorem. J. Logic. Comput. 16(6), 713–735 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Găină D., Popescu A.: An Institution-Independent Proof of Robinson Consistency Theorem. Studia Logica 85(1), 41–73 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Henkin L.: Completeness in the theory of types. J. Symb. Log. 15(2), 81–91 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  31. Henkin L.: A Generalization of the Concept of ω-consistency. J. Symb. Log. 19(3), 183–196 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  32. Keisler, H.: Forcing and the omitting types theorem. In: Morley, M., (ed), studies in model theory, MAA studies in mathematics, vol. 8, pp. 96–133 (1973)

  33. Mac Lane, S.: Categories for the Working Mathematician. Springer, 2nd edn (1998)

  34. Meseguer, J.: General logics. In Logic Colloquium 87, pp. 275–329. North Holland (1989)

  35. Meseguer J., Goguen J.A.: Order-sorted algebra solves the constructor-selector, multiple representation, and coercion problems. Inf. Comput. 103(1), 114–158 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  36. Möller, B., Tarlecki, A., Wirsing, M.: Algebraic specifications of reachable higher-order algebras. In: Sannella, D., Tarlecki, A. (eds.) ADT, Lecture notes in computer science, vol: 332 pp. 154–169. Springer (1987)

  37. Mossakowski T.: Relating casl with other specification languages: the institution level. Theor. Comput. Sci. 286(2), 367–475 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  38. Orey S.: On ω-consistency and related properties. J. Symb. Log. 21(3), 246–252 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  39. Petria, M.: An Institutional Version of Gödel’s Completeness Theorem. In: Mossakowski, T., Montanari, U., Haveraaen, M., (eds.), CALCO Lecture Notes in Computer Science, vol. 4624 pages 409–424. Springer (2007)

  40. Petria M., Diaconescu R.: Abstract Beth definability in institutions. J. Symb. Logic. 71(3), 1002–1028 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  41. Robinson A.: Forcing in model theory. Symposia Mathematica, 5, 69–82 (1971)

    Google Scholar 

  42. Tarlecki A.: On the existence of free models in abstract algebraic institutuons. Theor. Comput. Sci. 37, 269–304 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  43. Tarlecki, A.: Bits and pieces of the theory of institutions. In Pitt, D., Abramsky, S., Poigné, A., Rydeheard, D., (eds.), Proceedings of the summer workshop on category theory and computer programming, Lecture notes in computer science, vol. 240, pp. 334–360. Springer (1986)

  44. Tarlecki A.: Quasi-varieties in abstract algebraic institutions. J. Comput. Syst. Sci. 33(3), 333–360 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  45. Tarlecki, A.: Moving between logical systems. In recent trends in data type specification, pp. 478–502. Springer (1998)

Download references

Author information

Authors and Affiliations

  1. Research Center for Software Verification, Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi, Ishikawa, 923-1292, Japan

    Daniel Găină

Authors
  1. Daniel Găină
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Găină.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Găină, D. Forcing, Downward Löwenheim-Skolem and Omitting Types Theorems, Institutionally. Log. Univers. 8, 469–498 (2014). https://doi.org/10.1007/s11787-013-0090-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11787-013-0090-0

Mathematics Subject Classification (2010)

Keywords

Search

Navigation