Hostname: page-component-84b7d79bbc-c654p Total loading time: 0 Render date: 2024-07-30T05:21:04.893Z Has data issue: false hasContentIssue false
  • English
  • Français

The basic intuitionistic logic of proofs

Published online by Cambridge University Press:  12 March 2014

Sergei Artemov  and
Rosalie Iemhoff
Sergei Artemov
Affiliation:
Graduate Center, City University of New York, 365 Fifth Avenue, New York, NY 10016, U.S.A., E-mail: SArtemov@gc.cuny.edu
Rosalie Iemhoff
Affiliation:
Department of Philosophy, University Utrecht, Heidelberglaan 6–8, Utrecht, The Netherlands, E-mail: Rosalie.Iemhoff@phil.uu.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

The language of the basic logic of proofs extends the usual propositional language by forming sentences of the sort x is a proof of F for any sentence F. In this paper a complete axiomatization for the basic logic of proofs in Heyting Arithmetic HA was found.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 2 , June 2007 , pp. 439 - 451
Copyright
Copyright © Association for Symbolic Logic 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Alt, J. and Artemov, S., Reflexive lambda-calculus, Proceedings of the Dagstuhl-seminar on Proof Theory in Computer Science, Springer Lecture Notes in Computer Science, vol. 2183, 2001, pp. 22–37.CrossRefGoogle Scholar
[2]Artemov, S., Logic of proofs, Annals of Pure and Applied Logic, vol. 67 (1994), pp. 29–59.CrossRefGoogle Scholar
[3]Artemov, S., Operational modal logic, Technical Report MSI 95-29, Cornell University, 12 1995.Google Scholar
[4]Artemov, S., Explicit provability and constructive semantics, The Bulletin for Symbolic Logic, vol. 7 (2001), no. 1. pp. 1–36.CrossRefGoogle Scholar
[5]Artemov, S., Unified semantics for modality and λ-terms via proof polynomials, Algebras, diagrams and decisions in Language, Logic and Computation (Vermeulen, K. and Copestake, A., editors), CSLI Publications, Stanford University, 2002, pp. 89–119.Google Scholar
[6]Artemov, S., Kolmogorov and Gödel's approach to intuitionistic logic: current developments, Russian Mathematical Surveys, vol. 59 (2004), no. 2, pp. 203–229.CrossRefGoogle Scholar
[7]Artemov, S., Justified common knowledge, Theoretical Computer Science, vol. 357 (2006), pp. 4–22.CrossRefGoogle Scholar
[8]Artemov, S. and Beklemishev, L., Provability logic, Handbook of Philosophical Logic (Gabbay, D. and Guenthner, F., editors), vol. 13, Kluwer, Dordrecht, 2nd ed., 2004, pp. 229–403.Google Scholar
[9]Artemov, S. and Nogina, E., Introducing justification into epistemic logic, Journal of Logic and Computation, vol. 15 (2005), pp. 1059–1073.CrossRefGoogle Scholar
[10]Artemov, S. and Strassen, T., The basic logic of proofs, Lecture Notes in Computer Science, vol. 702, Springer, 1992.Google Scholar
[11]Bierman, G. and de Paiva, V., Intuitionistic necessity revisited, Proceedings of the Logic at Work Conference (Amsterdam), 12 1992, Second revision, June 1996 (http://theory.doc.ic.ac.uk/tfm/papers.html).Google Scholar
[12]de Jongh, D. and Japaridze, G., The logic of provability, Handbook of Proof Theory (Buss, S., editor), Studies in Logic and the Foundations of Mathematics, vol. 137, Elsevier, 1998, pp. 475–546.Google Scholar
[13]de Jongh, D. H. J., The maximality of the intuitionistic predicate calculus with respect to Heyting's Arithmetic, this Journal, vol. 36 (1970), p. 606.Google Scholar
[14]Fitting, M. C., The logic of proofs, semantically, Annals of Pure and Applied Logic, vol. 132 (2005), no. 1, pp. 1–25.CrossRefGoogle Scholar
[15]Ghilardi, S., Unification in intuitionistic logic, this Journal, vol. 64 (1999), no. 2, pp. 859–880.Google Scholar
[16]Ghilardi, S., Best solving modalequations, Annals of Pure and Applied Logic, vol. 102 (2000), pp. 183–198.CrossRefGoogle Scholar
[17]Gödel, K., Eine Interpretation des intuitionistischen Aussagenkalkuls, Ergebnisse eines mathematischen Kolloquiums, vol. 4 (1933), pp. 39–40.Google Scholar
[18]Gödel, K., Vortrag bei Zilsel (1938), Kurt Gödel collected works (Feferman, S., editor), vol. III, Oxford University Press, 1995, pp. 86–113.Google Scholar
[19]Iemhoff, R., On the admissible rules of intuitionistic propositional logic, this Journal, vol. 66 (2001), no. 1, pp. 281–294.Google Scholar
[20]Iemhoff, R., Provability logic and admissible rules, Ph.D. thesis, University of Amsterdam, 2001.Google Scholar
[21]Iemhoff, R., Preservativity logic (An analogue of interpretability logic for constructive theories), Mathematical Logic Quarterly, vol. 49 (2003), no. 3, pp. 1–21.CrossRefGoogle Scholar
[22]Iemhoff, R., Towards a proof system for admissibility, Computer Science Logic '03 (Baaz, M. and Makowsky, A., editors), Lecture Notes in Computer Science 2803, Springer, 2003, pp. 255–270.CrossRefGoogle Scholar
[23]Iemhoff, R., Intermediate logics and Visser's rules, Notre Dame Journal of Formal Logic, vol. 46 (2005), no. 1, pp. 65–81.CrossRefGoogle Scholar
[24]Kolmogoroff, A., Zur Deutung der intuitionistischen Logik, Mathematische Zeitschrift, vol. 35 (1932), pp. 58–65, In German. English translation in V. M. Tikhomirov, editor, Selected works of A. N. Kolmogorov. Volume I: Mathematics and Mechanics, pp. 151–158. Kluwer, Dordrecht, 1991.CrossRefGoogle Scholar
[25]Krupski, N., Typing in reflective combinatory logic, Annals of Pure and Applied Logic, vol. 141 (2006), pp. 243–256.CrossRefGoogle Scholar
[26]Martini, S. and Masini, A., A computational interpretation of modal proofs, Proof Theory of Modal Logics, (Workshop proceedings) (Wansing, , editor), Kluwer, 1994.Google Scholar
[27]Pfenning, F. and Wong, H. C., On a modal lambda-calculus for S4, Electronic Notes in Theoretical Computer Science, vol. 1, Elsevier, Amsterdam, 1995.Google Scholar
[28]Smorynski, C. A., Applications of Kripke models, Mathematical investigations of intuitionistic arithmetic and analysis (Troelstra, , editor), Springer Verlag, 1973, pp. 324–391.CrossRefGoogle Scholar
[29]Visser, A., Rules and arithmetics, Notre Dame Journal of Formal Logic, vol. 40 (1999), no. 1, pp. 116–140.CrossRefGoogle Scholar
[30]Visser, A., Substitutions of Σ-sentences: explorations between intuitionistic propositional logic and intuitionistic arithmetic, Annals of Pure and Applied Logic, vol. 114 (2002), no. 1–3, pp. 227–271.CrossRefGoogle Scholar