Studia Logica

, Volume 69, Issue 1, pp 97–131

# Ground and Free-Variable Tableaux for Variants of Quantified Modal Logics

• Marta Cialdea Mayer
• Serenella Cerrito
Article

## Abstract

In this paper we study proof procedures for some variants of first-order modal logics, where domains may be either cumulative or freely varying and terms may be either rigid or non-rigid, local or non-local. We define both ground and free variable tableau methods, parametric with respect to the variants of the considered logics. The treatment of each variant is equally simple and is based on the annotation of functional symbols by natural numbers, conveying some semantical information on the worlds where they are meant to be interpreted.

This paper is an extended version of a previous work where full proofs were not included. Proofs are in some points rather tricky and may help in understanding the reasons for some details in basic definitions.

Quantified Modal Logics Tableaux

## References

1. [1]
Abadi, M., and Z. MANNA: 1986, 'Modal Theorem Proving' In: Proc. of the 8th Int. Conf. on Automated Deduction, Berlin, pp. 172-189.Google Scholar
2. [2]
Artosi, A., P. Benassi, G. Governatori, and A. Rotolo: 1998, 'Shakespearian modal logic: A labelled treatment of modal identity' in: M. Kracht, M. de Rijke, H. Wansing, and M. Zakharyaschev (eds.), Advances in Modal Logic, Stanford: CSLI Publications, pp. 1-21.Google Scholar
3. [3]
Auffray, Y., and P. Enjalbert: 1992, 'Modal Theorem Proving: an equational viewpoint' Journal of Logic and Computation 2, 247-297.Google Scholar
4. [4]
Baaz, M., and C. Fermüller: 1995, 'Non-elementary speedups between different versions of tableaux' in: Proc. of Tableaux 95, pp. 217-230.Google Scholar
5. [5]
Basin, D., M. Matthews, and L. Viganò: 1998, 'Labelled Modal Logics: Quanti-fiers' Journal of Logic Language and information 7(3), 237-263.Google Scholar
6. [6]
Beckert, B., R. Hähnle, and P. H. Schmitt: 1993, 'The even more liberalized δ-rule in free variable semantic tableaux' in: Proc. of the 3rd Kurt G¨odel Colloquium KGC'93, pp. 108-119.Google Scholar
7. [7]
Belnap, N.: 1982, 'Display Logic' J. of Philosophical Logic 11, 375-417.Google Scholar
8. [8]
Cialdea, M.: 1991, 'Resolution for some first order modal systems' Theoretical Computer Science 85, 213-229.Google Scholar
9. [9]
Cialdea, M. and L. Fariñas del Cerro: 1986, 'A Modal Herbrand's Property' Z. Math. Logik Grundlag. Math. 32, 523-539.Google Scholar
10. [10]
Cialdea Mayer, M., and S. Cerrito: 2000, 'Variants of First-Order Modal Logics' in: R. Dyckhoff (ed.), Proc. of Tableaux 2000.Google Scholar
11. [11]
Fitting, M.: 1983, Proof Methods for Modal and Intuitionistic Logics, Reidel Publishing Company.Google Scholar
12. [12]
Fitting, M.: 1988, 'First-Order Modal Tableaux' Journal of Automated Reasoning 4, 191-213.Google Scholar
13. [13]
Fitting, M.: 1999, 'On quantified modal logic' Fundamenta informaticæ 39, 105-121.Google Scholar
14. [14]
Garson, J.W.: 1984, 'Quantification in modal logic' in: D. Gabbay and F. Guenthner (eds.): Handbook of Philosophical Logic, Vol. II, D. Reidel Publ. Co., pp. 249-307.Google Scholar
15. [15]
Goré, R.: 1999, 'Tableau methods for modal and temporal logics' in: M. D'Agostino, G. Gabbay, R. Hähnle, and J. Posegga (eds.): Handbook of tableau methods, Kluwer.Google Scholar
16. [16]
Hähnle, R. and P. H. Schmitt: 1994, 'The liberalized δ-rule in Free variable semantic tableaux' Journal of Automated Reasoning 13, 211-222.Google Scholar
17. [17]
Jackson, P. and H. Reichgelt: 1987, 'A general proof method for first-order modal logic' in: Proc. of the 10th Joint Conf. on Artificial intelligence (IJCAI '87), pp. 942-944.Google Scholar
18. [18]
Konolige, K.: 1986, 'Resolution and quantified epistemic logics' in: J.H. Siekmann (ed.): Proc. of the 8th int. Conf. on Automated Deduction (CADE 86), pp. 199-208.Google Scholar
19. [19]
Mints, G.: 1997, 'Indexed systems of sequents and cut-elimination' J. of Philosophical Logic 26, 671-696.Google Scholar
20. [20]
Ohlbach, H. J.: 1991, 'Semantics Based Translation Methods for Modal Logics' Journal of Logic and Computation 1(5), 691-746.Google Scholar
21. [21]
Wallen, L.A.: 1990, Automated Deduction in Nonclassical Logics: Efficient Matrix Proof Methods for Modal and intuitionistic Logics, MIT Press.Google Scholar
22. [22]
Wansing, H.: 1962, 'Predicate logics on display' Studia Logica 62(1), 49-75.Google Scholar