Skip to main content

On the Expressive Power of IF-Logic with Classical Negation

  • Conference paper
Logic, Language, Information and Computation (WoLLIC 2011)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 6642))

Abstract

It is well-known that Independence Friendly (IF) logic is equivalent to existential second-order logic (\(\Sigma^1_1\)) and, therefore, is not closed under classical negation. The boolean closure of IF sentences, called Extended IF-logic, on the other hand, corresponds to a proper fragment of \(\Delta^1_2\). In this paper we consider IF-logic extended with Hodges’ flattening operator, which allows classical negation to occur also under the scope of IF quantifiers. We show that, nevertheless, the expressive power of this logic does not go beyond \(\Delta^1_2\). As part of the proof, we give a prenex normal form result and introduce a non-trivial syntactic fragment of full second-order logic that we show to be contained in \(\Delta^1_2\).

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

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Caicedo, X., Krynicki, M.: Quantifiers for reasoning with imperfect information and \(\Sigma^1_1\)-logic. Contemporary Mathematics 235, 17–31 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Enderton, H.: Finite partially ordered quantifiers. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 16, 393–397 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  3. Figueira, S., Gorín, D., Grimson, R.: On the formal semantics of IF-like logics. Journal of Computer and System Sciences 76(5), 333–346 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Hintikka, J.: The Principles of Mathematics Revisited. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  5. Hintikka, J., Sandu, G.: Game-theoretical semantics. In: van Benthem, J., ter Meulen, A. (eds.) Handbook of Logic and Language. The MIT press, Cambridge (1997)

    Google Scholar 

  6. Hodges, W.: Compositional semantics for a language of imperfect information. Logic Journal of the IGPL 5(4) (1997)

    Google Scholar 

  7. Janssen, T.M.V.: Independent choices and the interpretation of IF logic. Journal of Logic, Language and Information 11(3), 367–387 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Janssen, T.M.V., Dechesne, F.: Signalling in IF games: a tricky business. In: The Age of Alternative Logics, pp. 221–241. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  9. Kołodziejczyk, L.A.: The expressive power of Henkin quantifiers with dualization. Master’s thesis, Institute of Philosophy, Warsaw University, Poland (2002)

    Google Scholar 

  10. Mostowski, M.: Arithmetic with the Henkin quantifier and its generalizations. In: Gaillard, F., Richard, D. (eds.) Séminaire du Laboratoire Logique, Algorithmique et Informatique Clermontoise, vol. 2, pp. 1–25 (1991)

    Google Scholar 

  11. Väänänen, J.A.: On the semantics of informational independence. Logic Journal of the IGPL 10(3), 339–352 (2002)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Figueira, S., Gorín, D., Grimson, R. (2011). On the Expressive Power of IF-Logic with Classical Negation. In: Beklemishev, L.D., de Queiroz, R. (eds) Logic, Language, Information and Computation. WoLLIC 2011. Lecture Notes in Computer Science(), vol 6642. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20920-8_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-20920-8_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-20919-2

  • Online ISBN: 978-3-642-20920-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics