Paper:

# Prior’s Three-Valued Modal Logic *Q* and its Possible Applications

## Seiki Akama^{*} and Yasunori Nagata^{**}

^{*}Dept. of Information Science, Teikyo Heisei University, 2289 Uruido, Ichihara, Chiba 290-0193, Japan

^{**}Dept. of Elect’l & Electronics Eng., University of the Ryukyus, 1 Senbaru, Nishihara, Okinawa 903-0213, Japan

*Ǫ*, Kripke semantics

*Q*as a “correct” modal logic from his philosophical motivations. Unfortunately, Prior’s

*Q*and many-valued modal logic have been neglected in the tradition of many-valued and modal logic. In this paper, we introduce a version of three-valued Kripke semantics for

*Q*, which aims to establish Prior’s ideas based on possible worlds. We investigate formal properties of

*Q*and prove the completeness theorem of

*Q*. We also compare our approach with others and suggest possible applications.

*Q*and its Possible Applications,”

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.11 No.1, pp. 105-110, 2007.

- [1] S. Akama, “A proof method for the six-valued logic for incomplete information,” Proc. of ISMVL’97, pp. 223-226, 1997.
- [2] S. Akama and Y. Nagata, “On Prior’s three-valued modal logic,” Proc. of ISMVL’2005, pp. 14-19, 2005.
- [3] R. A. Bull, “An axiomatization of Prior’s modal calculus Q,” Notre Dame Journal of Formal Logic, 5, pp. 211-214, 1964.
- [4] F. Correia, “Adequacy results from some Priorean modal logic,” Notre Dame Journal of Formal Logic, 40, pp. 236-249, 1999.
- [5] M. Fitting, “Many-valued modal logics,” Fundamenta Informatica, 15, pp. 235-254, 1992.
- [6] O. Garcia and M. Moussavi, “A six-valued logic for representing incomplete information,” Proc. of ISMVL’90, pp. 110-114, 1990.
- [7] G. Hughes and M. J. Cresswell, “An Introduction to Modal Logic,” Methuen, London, 1968.
- [8] A. N. Prior, “Time and Modality,” Clarendon Press, Oxford, 1957.
- [9] A. N. Prior, “Past, Present and Future,” Clarendon Press, Oxford, 1967.
- [10] A. N. Prior and K. Fine, “Worlds, Times and Selves,” Duckworth, London, 1977.
- [11] K. Segerberg, “Some modal logics based on a three-valued logic,” Theoria, 33, pp. 53-71, 1967.

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