Logic and Philosophy of Time

Vol. 6 No. 1 (2025): 70 Years of Tense-Logic
Articles

Analytic Proof-Theory for Prior’s System Q: First Steps

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  • Torben Bräuner
Torben Bräuner
Roskilde Universitet
DOI: https://doi.org/10.54337/lpt.v6i1.9937

Published 13-01-2025

Keywords

  • Arthur Prior,
  • System Q,
  • tableau systems

How to Cite

Bräuner, T. (2025). Analytic Proof-Theory for Prior’s System Q: First Steps. Logic and Philosophy of Time, 6(1). https://doi.org/10.54337/lpt.v6i1.9937

Copyright (c) 2025 Logic and Philosophy of Time

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Abstract

Arthur Prior introduced the three-valued modal logic called the Q system. A few axiom systems for the Q system can be found in the literature, but no analytic proof-theory in the form of tableau-, sequent- or natural deduction systems. In the present paper we demonstrate how to turn a formal semantics for the Q system into a tableau system, whereby we provide a proof system that is suitable for actual reasoning.

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References

  1. Akama, S. and Y. Nagata (2007). Prior’s three-valued modal logic Q and its
  2. possible applications. Journal of Advanced Computational Intelligence and Intelligent Informatics 11, 105–110.
  3. Akama, S., Y. Nagata, and C. Yamada (2008). Three-valued temporal logic Qt
  4. and future contingents. Studia Logica 88, 215–231.
  5. Areces, C. and B. ten Cate (2007). Hybrid logics. In P. Blackburn, J. van Benthem, and F. Wolter (Eds.), Handbook of Modal Logic, pp. 821–868. Elsevier.
  6. Badie, F. (2023). On A.N. Prior’s logical system Q. In The History and Philosophy
  7. of Tense-logic, Volume 5 of Logic and Philosophy of Time. Aalborg University
  8. Press. 14 pages.
  9. Blackburn, P., T. Braüner, and J. Kofod (2020). Remarks on hybrid modal logic
  10. with propositional quantfiers. In The Metaphysics of Time: Themes from Prior,
  11. Volume 4 of Logic and Philosophy of Time, pp. 401–426. Aalborg University
  12. Press.
  13. Blackburn, P., T. Braüner, and J. Kofod (2023). An axiom system for basic hybrid logic with propositional quantifiers. In Logic, Language, Information, and
  14. Computation – 29th International Workshop, WoLLIC 2023, Proceedings, Volume
  15. of Lecture Notes in Computer Science, pp. 118–134. Springer-Verlag.
  16. Braüner, T. (2000). A cut-free Gentzen formulation of the modal logic S5. Logic
  17. Journal of the IGPL 8, 629–643.
  18. Braüner, T. (2011). Hybrid Logic and its Proof-Theory, Volume 37 of Applied Logic
  19. Series. Springer.
  20. Braüner, T. (2021). Hybrid logic. In E. Zalta (Ed.), The Stanford Encyclopedia
  21. of Philosophy. Stanford University. On-line encyclopedia article available at
  22. http://plato.stanford.edu/entries/logic-hybrid.
  23. Copeland, B. and A. Markoska-Cubrinovska (2023). Prior’s system Q and its
  24. extensions. In The History and Philosophy of Tense-logic, Volume 5 of Logic and
  25. Philosophy of Time. Aalborg Universitetsforlag. 24 pages.
  26. D’Agostino, M., D. Gabbay, R. Hähnle, and J. Posegga (Eds.) (1999). Handbook
  27. of Tableau Methods. Springer.
  28. Fine, K. and A. Prior (1977). Worlds, Times and Selves. Duckworth, London.
  29. Based on manuscripts by Prior with a preface and a postscript by K. Fine.
  30. Fitting, M. (1983). Proof Methods for Modal and Intuitionistic Logic. Reidel.
  31. Fitting, M. (2007). Modal proof theory. In P. Blackburn, J. van Benthem, and
  32. F. Wolter (Eds.), Handbook of Modal Logic, pp. 85–138. Elsevier.
  33. Gabbay, D. (1996). Labelled Deductive Systems. Oxford University Press.
  34. Greati, V., G. Greco, S. Marcelino, A. Palmigiano, and U. Rivieccio
  35. (2024). Generating proof systems for three-valued propositional logics.
  36. CoRR abs/2401.03274.
  37. Hintikka, J. (1955). Form and content in quantification theory. Acta Philosophica
  38. Fennica 8, 8–55.
  39. Priest, G. (2008). An Introduction to Non-Classical Logic, 2nd Edition. Cambridge
  40. Introductions to Philosophy. Cambridge University Press.
  41. Prior, A. (1957). Time and Modality. Clarendon/Oxford University Press.
  42. Segerberg, K. (1967). Some modal logics based on a three-valued logic. Theoria 33, 53–71.