Skip to main content
Log in

Prehistoric Graph in Modal Derivations and Self-Referentiality

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

By terms-allowed-in-formulas capacity, Artemov’s Logic of Proofs LP Artemov includes self-referential formulas of the form t:ϕ(t) that play a crucial role in the realization of modal logic S4 in LP, and in the Brouwer–Heyting–Kolmogorov semantics of intuitionistic logic via LP. In an earlier work appeared in the Proceedings of CSR 2010 the author defined prehistoric loop in a sequent calculus of S4, and verified its necessity to self-referentiality in S4LP realization. In this extended version we generalize results there to T and K4, two modal logics smaller than S4 that yet call for self-referentiality in their realizations into corresponding justification logics JT and J4.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Here is the advantage of the !-ed notation for constants in JT, which makes the initial constant in a constant series a notational sub-constant of any constant in that series. Thus if c:A(!n c) belongs to \(\mathcal {CS}\), then so is c:A′(c) for some A′. Constant specification is also defined in some other works, like [10], where things get much more complicated when taking care of JT. We will also benefit from this notation in later sections.

  2. This is also an S4-theorem.

  3. In what follows, Γ,Δ,… , are multi-sets of formulas, and □Γ is an abbreviation of {□γ | γΓ}. For notational convenience, we may omit some “,” in sequents. Thus ΓΔ stands for Γ∪Δ, and Γα stans for Γ∪{α}, etc.

  4. In standard textbooks like [12], Γ,Δ are called side formulas. Since we will distinguish side formulas and weakening formulas in modal rules, and Γ,Δ here behave closely to weakening formulas in modal rules, we call them “weakening” formulas.

  5. Note that in (L→),(R→),(L□), there can be only one principal formula in the conclusion.

  6. Precisely speaking, the subproof with that (Cut) as its last rule.

  7. Precisely speaking, the remaining case is that one cut-formula is side in (R□), and the other is side in (R□) or principal. By Table 2, side formulas in (R□) are all negative, which implies that the positive cut-formula is principal. Since the negative cut-formula is from (R□), which determines the calculus (G3s), the positive cut-formula is principal in (R□).

  8. Note our special notation for constants in JT.

  9. This is just a matter of notation. There are no requirements on how we assign names to families (i.e., which family has number i as its name). Observations above show that ϵ can be taken to respect the order of families, thus we may assume we happened to assign it in the way that cooperates with ϵ.

References

  1. Artemov, S.N.: Explicit provability and constructive semantics. Bull. Symb. Log. 7(1), 1–36 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. Artemov, S.N.: The ontology of justifications in the logical setting. Technical Report 2011008, Ph.D. Program in Computer Science, The City University of New York (2011)

  3. Brezhnev, V.N.: On explicit counterparts of modal logics. Technical Report CFIS 2000-05, Cornell University (2000)

  4. Brezhnev, V.N., Kuznets, R.: Making knowledge explicit: how hard it is. Theor. Comput. Sci. 357(1–3), 23–34 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Brunnler, K., Goetschi, R., Kuznets, R.: A syntactic realization theorem for justification logics (2011) (draft)

  6. Fitting, M.: Proof Methods for Modal and Intuitionistic Logics. Reidel, Dordrecht (1983)

    Book  MATH  Google Scholar 

  7. Fitting, M.: The logic of proofs, semantically. Ann. Pure Appl. Log. 132(1), 1–25 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Fitting, M.: Realizations and LP. In: Artemov, S., Nerode, A. (eds.) LFCS 2007. LNCS, vol. 4514, pp. 212–223. Springer, Heidelberg (2007)

    Google Scholar 

  9. Kuznets, R.: Self-referentiality of justified knowledge. In: Hirsch, E.A., Razborov, A.A., Semenov, A., Slissenko, A. (eds.) CSR 2008. LNCS, vol. 5010, pp. 228–239. Springer, Heidelberg (2008)

    Google Scholar 

  10. Kuznets, R.: Self-referential justifications in epistemic logic. Theory Comput. Syst. 46(4), 636–661 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. Mkrtychev, A.: Models for the logic of proofs. In: Adian, S.I., Nerode, A. (eds.) LFCS ’97. LNCS, vol. 1234, pp. 266–275. Springer, Heidelberg (1997)

    Google Scholar 

  12. Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory, 2nd edn. Cambridge University Press, Cambridge (2000)

    Book  MATH  Google Scholar 

  13. Yu, J.: Prehistoric phenomena and self-referentiality. In: Ablayev, F., Mayr, E.W. (eds.) CSR 2010. LNCS, vol. 6072, pp. 384–396. Springer, Berlin (2010)

    Google Scholar 

Download references

Acknowledgements

The author is indebted to (in order of time of concern) Roman Kuznets, Fenrong Liu, anonymous referees of CSR 2010, Sergei Artemov, and an anonymous referee of this journal, who have offered detailed, helpful comments on earlier versions of this paper. The author would also like to thank the Program Committee of CSR 2010 (especially Ernst Mayr) and Springer-Verlag, who make it possible for this paper to appear after its conference version.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Junhua Yu.

Additional information

This research has been supported, in part, by NSF award 0830450 Justification Logic and Applications. This is an extended version of [13] published in the Proceeding of CSR 2010.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yu, J. Prehistoric Graph in Modal Derivations and Self-Referentiality. Theory Comput Syst 54, 190–210 (2014). https://doi.org/10.1007/s00224-013-9510-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-013-9510-z

Keywords

Navigation