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COMPLETE ADDITIVITY AND MODAL INCOMPLETENESS

  • WESLEY H. HOLLIDAY (a1) and TADEUSZ LITAK (a2)

Abstract

In this article, we tell a story about incompleteness in modal logic. The story weaves together an article of van Benthem (1979), “Syntactic aspects of modal incompleteness theorems,” and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, ${\cal V}$ -baos. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem’s article resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to ${\cal V}$ -baos, namely the provability logic $GLB$ (Japaridze, 1988; Boolos, 1993). We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is ${\cal V}$ -complete. After these results, we generalize the Blok Dichotomy (Blok, 1978) to degrees of ${\cal V}$ -incompleteness. In the end, we return to van Benthem’s theme of syntactic aspects of modal incompleteness.

Copyright

Corresponding author

*DEPARTMENT OF PHILOSOPHY AND GROUP IN LOGIC AND THE METHODOLOGY OF SCIENCE UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA 94720, USA E-mail: wesholliday@berkeley.edu
CHAIR FOR THEORETICAL COMPUTER SCIENCE (INFORMATIK 8) FAU ERLANGEN-NUREMBERG 91058 ERLANGEN, GERMANY E-mail: tadeusz.litak@fau.de

References

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Andréka, H., Gyenis, Z., & Németi, I. (2016). Ultraproducts of continuous posets. Algebra Universalis, 76(2), 231235.
Andréka, H., Németi, I., & Sain, I. (2001). Algebraic logic. In Gabbay, D. M. and Guenthner, F., editors. Handbook of Philosophical Logic, second edition. Dordrecht: Kluwer, pp. 133249.
Beklemishev, L. (2011). Ordinal completeness of bimodal provability logic GLB. In Bezhanishvili, N., Löbner, S., Schwabe, K., & Spada, L., editors. Logic, Language, and Computation: 8th International Tbilisi Symposium on Logic, Language, and Computation, TbiLLC 2009, Bakuriani, Georgia, September 21–25, 2009. Revised Selected Papers. Heidelberg: Springer, pp. 115.
Beklemishev, L., Bezhanishvili, G., & Icard, T. (2010). On topological models of GLP. In Schindler, R., editor. Ways of Proof Theory. Ontos Mathematical Logic, Vol. 2. Heusenstamm: Ontos Verlag, pp. 135155.
Beklemishev, L. & Gabelaia, D. (2014). Topological interpretations of provability logic. In Bezhanishvili, G., editor. Leo Esakia on Duality in Modal and Intuitionistic Logics. Outstanding Contributions to Logic, Vol. 4. Dordrecht: Springer, pp. 257290.
van Benthem, J. (1978). Two simple incomplete modal logics. Theoria, 44(1), 2537.
van Benthem, J. (1979). Syntactic aspects of modal incompleteness theorems. Theoria, 45(2), 6377.
van Benthem, J. (1983). Modal Logic and Classical Logic. Milan: Bibliopolis.
van Benthem, J. (2001). Correspondence theory. In Gabbay, D. and Guenthner, F., editors. Handbook of Philosophical Logic, second edition, Vol. 3. Dordrecht: Springer, pp. 325408.
Bezhanishvili, G. & Holliday, W. H. (2019). A semantic hierarchy for intuitionistic logic. Indagationes Mathematicae, 30(3), 403469.
Bezhanishvili, N. & Ghilardi, S. (2014). The bounded proof property via step algebras and step frames. Annals of Pure Applied Logic, 165(12), 18321863.
Bezhanishvili, N., & Kurz, A. (2007). Free modal algebras: A coalgebraic perspective. In Mossakowski, T., Montanari, U., and Haveraaen, M., editors. Algebra and Coalgebra in Computer Science. Lectures Notes in Computer Science, Vol. 4624. Berlin: Springer, pp. 143157.
Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal Logic. Cambridge Tracts in Theoretical Computer Science, Vol. 53. Cambridge: Cambridge University Press.
Blok, W. J. (1978). On the degree of incompleteness of modal logic and the covering relation in the lattice of modal logics. Technical Report 78-07, University of Amsterdam.
Blok, W. J. & Köhler, P. (1983). Algebraic semantics for quasi-classical modal logics. The Journal of Symbolic Logic, 48(4), 941964.
Blok, W. J. & Pigozzi, D. (1989). Algebraizable Logics. Memoirs of the American Mathematical Society, no. 396. Providence, RI: American Mathematical Society.
Boolos, G. (1993). The Logic of Provability. Cambridge: Cambridge University Press.
Boolos, G. & Sambin, G. (1985). An incomplete systems of modal logic. Journal of Philosophical Logic, 14(4), 351358.
Buszkowski, W. (1986). Embedding Boolean structures into atomic Boolean algebras. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 32, 227228.
Buszkowski, W. (2004). A representation theorem for co-diagonalizable algebras. Reports on Mathematical Logic, 38, 1322.
ten Cate, B. & Litak, T. (2007). The importance of being discrete. Technical Report PP-2007-39, Institute for Logic, Language and Computation, University of Amsterdam.
Chagrov, A. V. (1990). Undecidable properties of extensions of a provability logic. II. Algebra and Logic, 29(5), 406413.
Chagrov, A. V. & Rybakov, M. N. (2003). How many variables does one need to prove PSPACE-hardness of modal logics? In Balbiani, P., Suzuki, N.-Y., Wolter, F., and Zakharyaschev, M., editors. Advances in Modal Logic, Vol. 4. London: King’s College Publications, pp. 7182.
Chagrov, A. V. & Zakharyaschev, M. (1993). The undecidability of the disjunction property of propositional logics and other related problems. The Journal of Symbolic Logic, 58(3), 9671002.
Chagrov, A. V. & Zakharyaschev, M. (1997). Modal Logic. Oxford Logic Guides. Oxford: Clarendon Press.
Chagrova, L. A. (1998). On the degree of neighbourhood incompleteness of normal modal logics. In Kracht, M., de Rijke, M., Wansing, H., and Zakharyaschev, M., editors. Advances in Modal Logic, Vol. 1. Stanford: CSLI Publications, pp. 6372.
Cirstea, C., Kurz, A., Pattinson, D., Schröder, L., & Venema, Y. (2011). Modal logics are coalgebraic. The Computer Journal, 54(1), 3141.
Conradie, W., Ghilardi, S., & Palmigiano, A. (2014). Unified correspondence. In Baltag, A. and Smets, S., editors. Johan van Benthem on Logic and Information Dynamics. Dordrecht: Springer, pp. 933975.
Conradie, W., Goranko, V., & Vakarelov, D. (2006). Algorithmic correspondence and completeness in modal logic. I. The core algorithm SQEMA. Logical Methods in Computer Science, 2(1), 126.
Coumans, D. C. S. & van Gool, S. J. (2013). On generalizing free algebras for a functor. Journal of Logic and Computation, 23(3), 645672.
Cresswell, M. J. (1984). An incomplete decidable modal logic. The Journal of Symbolic Logic, 49(2), 520527.
Czelakowski, J. (2001). Protoalgebraic Logics. Trends in Logic. Dordrecht: Springer.
Došen, K. (1989). Duality between modal algebras and neighborhood frames. Studia Logica, 48(2), 219234.
Dziobiak, W. (1978). A note on incompleteness of modal logics with respect to neighbourhood semantics. Bulletin of the Section of Logic, 7(4), 185190.
Fine, K. (1974). An incomplete logic containing S4. Theoria, 40(1), 2329.
Fine, K. (1975, 04). Normal forms in modal logic. Notre Dame Journal of Formal Logic, 16(2), 229237.
Font, J. M. (2006). Beyond Rasiowa’s algebraic approach to non-classical logic. Studia Logica, 82(2), 179209.
Font, J. M., Jansana, R., & Pigozzi, D. (2003). A survey of abstract algebraic logic. Studia Logica, 74(1), 1397.
Font, J. M., Jansana, R., & Pigozzi, D. (2009). Update to “A survey of abstract algebraic logic”. Studia Logica, 91(1), 125130.
Gargov, G. & Goranko, V. (1993). Modal logic with names. Journal of Philosophical Logic, 22(6), 607636.
Ghilardi, S. (1995). An algebraic theory of normal forms. Annals of Pure and Applied Logic, 71(3), 189245.
Ghilardi, S. & Meloni, G. (1997). Constructive canonicity in non-classical logic. Annals of Pure and Applied Logic, 86(1), 132.
Goldblatt, R. (2001). Persistence and atomic generation for varieties of Boolean algebras with operators. Studia Logica, 68(2), 155171.
Goldblatt, R. (2003). Mathematical modal logic: A view of its evolution. Journal of Applied Logic, 1(5–6), 309392.
Holliday, W. H. (2014). Partiality and adjointness in modal logic. In Goré, R., Kooi, B., and Kurucz, A., editors. Advances in Modal Logic, Vol. 10. London: College Publications, pp. 313332.
Holliday, W. H. (2015). Possibility frames and forcing for modal logic. UC Berkeley Working Paper in Logic and the Methodology of Science. February 2018 version. Available at: https://escholarship.org/uc/item/0tm6b30q.
Holliday, W. H. & Litak, T. (2018). One modal logic to rule them all? In Bezhanishvili, G., D’Agostino, G., Metcalfe, G., and Studer, T., editors. Advances in Modal Logic, Vol. 12. London: College Publications, pp. 367386.
Hopcroft, J. E., Motwani, R., & Ullman, J. D. (2003). Introduction to Automata Theory, Languages, and Computation (second edition). Boston: Addison-Wesley.
Humberstone, L. (2011). The Connectives. Cambridge, MA: MIT Press.
Jansana, R. (2006). Willem Blok’s contribution to abstract algebraic logic. Studia Logica, 83(1), 3148.
Japaridze, G. K. (1988). The polymodal logic of provability. In Smirnov, V. A. and Bezhanishvili, M. N., editors. Intensional Logics and the Logical Structure of Theories: Proceedings of the Fourth Soviet-Finnish Symposium on Logic, Telavi, May 1985. Tbilisi: Metsniereba, pp. 1648.
Jipsen, P. (1993). Discriminator varieties of Boolean algebras with residuated operators. In Algebraic Methods in Logic and in Computer Science. Banach Center Publications, Vol. 28. Warszawa: Institute of Mathematics, Polish Academy of Sciences, pp. 239252.
Jónsson, B. & Tarski, A. (1951). Boolean algebras with operators. Part I. American Journal of Mathematics, 73(4), 891939.
Jónsson, B. & Tarski, A. (1952). Boolean algebras with operators. Part II. American Journal of Mathematics, 74(1), 127162.
Kracht, M. (1999). Tools and Techniques in Modal Logic. Studies in Logic and the Foundations of Mathematics, Vol. 142. Amsterdam: Elsevier.
Kracht, M. & Wolter, F. (1991). Properties of independently axiomatizable bimodal logics. The Journal of Symbolic Logic, 56(4), 14691485.
Kracht, M. & Wolter, F. (1997). Simulation and transfer results in modal logic: A survey. Studia Logica, 59(2), 149177.
Kracht, M. & Wolter, F. (1999). Normal monomodal logics can simulate all others. The Journal of Symbolic Logic, 64(1), 99138.
Kurz, A. & Rosický, J. (2012). Strongly complete logics for coalgebras. Logical Methods in Computer Science, 8(3:14), 132.
Kuznetsov, A. V. (1975). On superintuitionistic logics. In Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1. Montreal, Quebec: Canadian Mathematical Congress, pp. 243249.
Lemmon, E. J. & Scott, D. (1977). The “Lemmon Notes”: An Introduction to Modal Logic, ed. Segerberg, K.. Number 11 in American Philosophical Quarterly Monograph Series. Oxford: Basil Blackwell.
Lewis, D. (1974). Intensional logics without iterative axioms. Journal of Philosophical Logic, 3(4), 457466.
Litak, T. (2004). Modal incompleteness revisited. Studia Logica, 76(3), 329342.
Litak, T. (2005a). An Algebraic Approach to Incompleteness in Modal Logic. Ph.D. Thesis, Japan Advanced Institute of Science and Technology.
Litak, T. (2005b). On notions of completeness weaker than Kripke completeness. In Schmidt, R., Pratt-Hartmann, I., Reynolds, M., and Wansing, H., editors. Advances in Modal Logic, Vol. 5. London: College Publications, pp. 149169.
Litak, T. (2006). Isomorphism via translation. In Governatori, G., Hodkinson, I. M., and Venema, Y., editors. Advances in Modal Logic, Vol. 6. London: College Publications, pp. 333351.
Litak, T. (2008). Stability of the Blok theorem. Algebra Universalis, 58(4), 385411.
Litak, T., Pattinson, D., Sano, K., & Schröder, L. (2012). Coalgebraic predicate logic. In Czumaj, A., Mehlhorn, K., Pitts, A., and Wattenhofer, R., editors. Automata, Languages, and Programming: 39th International Colloquium (ICALP). Lecture Notes in Computer Science, Vol. 7392. Heidelberg: Springer, pp. 299311.
Litak, T., Pattinson, D., Sano, K., & Schröder, L. (2018). Model theory and proof theory of coalgebraic predicate logic. Logical Methods in Computer Science, 14(1:22), 152.
Litak, T. & Wolter, F. (2005). All finitely axiomatizable tense logics of linear time flows are coNP-complete. Studia Logica, 81(2), 153165.
Łukasiewicz, J. & Tarski, A. (1930). Untersuchungen über den Aussagenkalkül. Comptes Rendus des séances de la Societé des Sciences et des Lettres de Varsovie, 23, 3050. English translation in Tarski 1956.
Makinson, D. (1971). Some embedding theorems for modal logic. Notre Dame Journal of Formal Logic, 12(2), 252254.
Moss, L. S. (2007). Finite models constructed from canonical formulas. Journal of Philosophical Logic, 36(6), 605640.
Pitts, A. M. (2013). Nominal Sets: Names and Symmetry in Computer Science. Cambridge Tracts in Theoretical Computer Science, Vol. 57. Cambridge: Cambridge University Press.
Pitts, A. M. (2016). Nominal techniques. ACM SIGLOG News, 3(1), 5772.
Rabin, M. O. (1969). Decidability of second-order theories and automata on infinite trees. Transactions of the American Mathematical Society, 141, 135.
Rasiowa, H. (1974). An Algebraic Approach to Non-classical Logics. Amsterdam: North-Holland.
Rautenberg, W., Wolter, F., & Zakharyaschev, M. (2006). Willem Blok and modal logic. Studia Logica, 83(1–3), 1530. Special issue in memory of Willem Johannes Blok.
Rice, H. G. (1953). Classes of recursively enumerable sets and their decision problems. Transactions of the American Mathematical Society, 74(2), 358366.
Schröder, L. (2008). Expressivity of coalgebraic modal logic: The limits and beyond. Theoretical Computer Science, 390(2–3), 230247. Special issue on Foundations of Software Science and Computational Structures.
Schröder, L. & Pattinson, D. (2010). Rank-1 modal logics are coalgebraic. Journal of Logic and Computation, 20(5), 11131147.
Segerberg, K. (1971). An Essay in Classical Modal Logic. Filosofiska Studier, Vol. 13. Uppsala, Sweden: University of Uppsala.
Shapirovsky, I. (2008). PSPACE-decidability of Japaridze’s polymodal logic. In Areces, C. and Goldblatt, R., editors. Advances in Modal Logic, Vol. 7. London: College Publications, pp. 289304.
Simpson, S. G. (2009). Subsystems of Second Order Arithmetic. Perspectives in Logic. New York: Association for Symbolic Logic, Cambridge University Press.
Solovay, R. M. (1976). Provability interpretations of modal logic. Israel Journal of Mathematics, 25(3), 287304.
Surendonk, T. J. (2001). Canonicity for intensional logics with even axioms. The Journal of Symbolic Logic, 66(3), 11411156.
Suzuki, T. (2010, 009). Canonicity results of substructural and lattice-based logics. The Review of Symbolic Logic, 4(1), 142.
Tarski, A. (1956). Logic, Semantics, Metamathematics: Papers from 1923 to 1938. Oxford: Clarendon Press. Translated by J. H. Woodger.
Thomason, S. K. (1972). Semantic analysis of tense logics. The Journal of Symbolic Logic, 37(1), 150158.
Thomason, S. K. (1974a). An incompleteness theorem in modal logic. Theoria, 40(1), 3034.
Thomason, S. K. (1974b). Reduction of tense logic to modal logic. I. The Journal of Symbolic Logic, 39(3), 549551.
Thomason, S. K. (1974c). Reduction of tense logic to modal logic II. Theoria, 40(3), 154169.
Thomason, S. K. (1975a). Categories of frames for modal logic. The Journal of Symbolic Logic, 40(3), 439442.
Thomason, S. K. (1975b). Reduction of second-order logic to modal logic. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 21(1), 107114.
Thomason, S. K. (1982). Undecidability of the completeness problem of modal logic. Banach Center Publications, 9(1), 341345.
Venema, Y. (2003). Atomless varieties. The Journal of Symbolic Logic, 68(2), 607614.
Venema, Y. (2007). Algebras and coalgebras. In Blackburn, P., van Benthem, J., and Wolter, F., editors. Handbook of Modal Logic. Amsterdam: Elsevier, pp. 331426.
Švejdar, V. (2003). The decision problem of provability logic with only one atom. Archive for Mathematical Logic, 42(8), 763768.
Wolter, F. (1993). Lattices of Modal Logics. Ph.D. Thesis, Fachbereich Mathematik, Freien Universität Berlin.
Wolter, F. (1996a). Properties of tense logics. Mathematical Logic Quarterly, 42(1), 481500.
Wolter, F. (1996b). Tense logic without tense operators. Mathematical Logic Quarterly, 42(1), 145171.
Wolter, F. & Zakharyaschev, M. (2006). Modal decision problems. In Blackburn, P., van Benthem, J., and Wolter, F., editors. Handbook of Modal Logic. Amsterdam: Elsevier, pp. 427489.
Zakharyaschev, M., Wolter, F., & Chagrov, A. V. (2001). Advanced modal logic. In Gabbay, D. M. and Guenthner, F., editors. Handbook of Philosophical Logic, (second edition), Vol. 3. Dordrecht: Springer, pp. 83266.

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COMPLETE ADDITIVITY AND MODAL INCOMPLETENESS

  • WESLEY H. HOLLIDAY (a1) and TADEUSZ LITAK (a2)

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