## Notre Dame Journal of Formal Logic

### Modal Consequence Relations Extending $\mathbf{S4.3}$: An Application of Projective Unification

#### Abstract

We characterize all finitary consequence relations over $\mathbf{S4.3}$, both syntactically, by exhibiting so-called (admissible) passive rules that extend the given logic, and semantically, by providing suitable strongly adequate classes of algebras. This is achieved by applying an earlier result stating that a modal logic $L$ extending $\mathbf{S4}$ has projective unification if and only if $L$ contains $\mathbf{S4.3}$. In particular, we show that these consequence relations enjoy the strong finite model property, and are finitely based. In this way, we extend the known results by Bull and Fine, from logics, to consequence relations. We also show that the lattice of consequence relations over $\mathbf{S4.3}$ (the lattice of quasivarieties of $\mathbf{S4.3}$-algebras) is countable and distributive and it forms a Heyting algebra.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 57, Number 4 (2016), 523-549.

Dates
Accepted: 11 November 2013
First available in Project Euclid: 19 July 2016

https://projecteuclid.org/euclid.ndjfl/1468952203

Digital Object Identifier
doi:10.1215/00294527-3636512

Mathematical Reviews number (MathSciNet)
MR3565536

Zentralblatt MATH identifier
06663939

#### Citation

Dzik, Wojciech; Wojtylak, Piotr. Modal Consequence Relations Extending $\mathbf{S4.3}$ : An Application of Projective Unification. Notre Dame J. Formal Logic 57 (2016), no. 4, 523--549. doi:10.1215/00294527-3636512. https://projecteuclid.org/euclid.ndjfl/1468952203

#### References

• [1] Baader, F., and S. Ghilardi, “Unification in modal and description logics,” Logic Journal of the IGPL, vol. 19 (2011), pp. 705–30.
• [2] Blackburn, P., M. de Rijke, and Y. Venema, Modal Logic, vol. 53 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, Cambridge, 2001.
• [3] Bloom, S. L., “Some theorems on structural consequence operations,” Studia Logica, vol. 34 (1975), pp. 1–9.
• [4] Bull, R. A., “That all normal extensions of S$4.3$ have the finite model property,” Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 314–44.
• [5] Bull, R. A., and K. Segerberg, “Basic modal logic,” pp. 1–88 in Handbook of Philosophical Logic, Vol. II, edited by D. Gabbay and F. Guenthner, vol. 165 of Synthese Library, Reidel, Dordrecht, 1979.
• [6] Chagrov, A., and M. Zakharyaschev, Modal Logic, vol. 35 of Oxford Logic Guides, Oxford University Press, Oxford, 1997.
• [7] Dzik, W., “Unitary unification of S5 logic and its extensions,” pp. 19–26 in Applications of Algebra, VI (Zakopane-Jaszczurówka, 2002), vol. 32 of Bulletin of the Section of Logic, 2003.
• [8] Dzik, W., “Splittings of lattices of theories and unification types,” Contributions to General Algebra, vol. 17 (2006), pp. 73–84.
• [9] Dzik, W., Unification Types in Logic, vol. 2554 of Scientific Publications of the University of Silesia, Silesian University Press, Katowice, 2007.
• [10] Dzik, W., and P. Wojtylak, “Projective unification in modal logic,” Logic Journal of the IGPL, vol. 20 (2012), pp. 121–53.
• [11] Fine, K., “The logics containing $\mathrm{S}4.3$,” Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 17 (1971), pp. 371–76.
• [12] Gencer, C., and D. de Jongh, “Unifiability in extensions of K4,” Logic Journal of the IGPL, vol. 17 (2009), pp. 159–72.
• [13] Ghilardi, S., “Best solving modal equations,” Annals of Pure and Applied Logic, vol. 102 (2000), pp. 183–98.
• [14] Jeřabek, E., “Admissible rules in modal logic,” Journal of Logic and Computation, vol. 15 (2005), pp. 411–31.
• [15] Kagan, J., and R. Quackenbush, “Monadic algebras,” Reports on Mathematical Logic, vol. 7 (1976), pp. 53–61.
• [16] Kracht, M., Tools and Techniques in Modal Logic, vol. 142 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1999.
• [17] Kracht, M., “Modal consequence relations,” of Handbook of Modal Logic, edited by P. Blackburn, J. van Benthem, and F. Wolter, vol. 3 of Studies in Logic and Practical Reasoning, Elsevier, Amsterdam, 2006.
• [18] Łoś, J., and R. Suszko, “Remarks on sentential logics,” Indagationes Mathematicae, vol. 20 (1958), pp. 177–83.
• [19] Pogorzelski, W. A., “Structural completeness of the propositional calculus,” Bulletin de l’Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 19 (1971), pp. 349–51.
• [20] Pogorzelski, W. A., and P. Wojtylak, Completeness Theory for Propositional Logics, Studies in Universal Logic, Birkhäuser, Basel, 2008.
• [21] Rautenberg, W., “Splitting lattices of logics,” Archive for Mathematical Logic, vol. 20 (1980), pp. 155–59.
• [22] Rybakov, V. V., “Admissible rules for logics containing $\mathrm{S4.3}$,” Siberian Mathematical Journal, vol. 25 (1984), pp. 141–45.
• [23] Rybakov, V. V., “Hereditarily structurally complete modal logics,” Journal of Symbolic Logic, vol. 60 (1995), pp. 266–88.
• [24] Rybakov, V. V., Admissibility of Logical Inference Rules, vol. 136 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1997.
• [25] Scroggs, S. J., “Extensions of the Lewis system $\mathrm{S}5$,” Journal of Symbolic Logic, vol. 16 (1951), pp. 112–20.
• [26] Wójcicki, R., Theory of Logical Calculi: Basic Theory of Consequence Operations, vol. 199 of Synthese Library, Kluwer Academic, Dordrecht, 1988.
• [27] Wojtylak, P., “Matrix representations for structural strengthenings of a propositional logic,” Studia Logica, vol. 38 (1979), pp. 263–66.
• [28] Wroński, A., “Transparent unification problem,” pp. 105–7 in First German-Polish Workshop on Logic & Logical Philosophy (Bachotek, 1995), vol. 29 of Reports on Mathematical Logic, Jagiellonian Press, Krakow, 1995.
• [29] Zakharyaschev, M., F. Wolter, and A. Chagrov, “Advanced modal logic,” pp. 83–266 in Handbook of Philosophical Logic, Vol. 3, edited by D. Gabbay and F. Guenthner, Kluwer Academic, Dordrecht, 2001.