This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aiello, M. and van Benthem, J. (2002). A modal walk through space. Journal of Applied Non-Classical Logics, 12:319–363.
Artin, E. (1957). Geometric Algebra. Interscience, New York.
Balbiani, P. (1998). The modal multilogic of geometry. Journal of Applied Non-classical Logics, 8:259–281.
Balbiani, P., Fariñas del Cerro, L., Tinchev, T., and Vakarelov, D. (1997). Modal logics for incidence geometries. Journal of Logic and Computation, 7(1): 59–78.
Balbiani, P. and Goranko, V. (2002). Modal logics for parallelism, orthogonality, and affine geometries. Journal of Applied Non-Classical Logics, 12:365–397.
Basu, S. (1999). New results on quantifier elimination over real closed fields and applications to constraint databases. JACM, 46(4):537–555.
Basu, S., Pollack, R., and Roy, M.-F. (1996). On the combinatorial and algebraic complexity of quantifier elimination. Journal ACM, 43(6):1002–1045.
Batten, L. M. (1986). Combinatorics of Finite Geometries. CUP.
Behnke, H., Bachmann, F., Fladt, K., and Kunle, H., editors (1974). Fundamentals of Mathematics, Vol. II: Geometry. MIT Press, Cambridge, Massachusetts.
Bennett, M. K. (1995). Affine and Projective Geometry. J. Wiley, NY.
Beth, E. and Tarski, A. (1956). Equilaterality as the only primitive notion of Euclidean geometry. Indag. Math., 18:462–467.
Blackburn, P., de Rijke, M., and Venema, Y. (2001). Modal Logic. CUP.
Bledsoe, W.W. and Loveland, D.W., editors (1984). Contemporary Mathematics: Automated Theorem Proving - After 25 Years, Providence, RI. American Mathematical Society.
Blumenthal, L. (1961). A Modern View of Geometry. W.H. Freeman and Company, San Francisco.
Blumenthal, L. M. and Menger, K. (1970). Studies in Geometry. W. H. Freeman and Company, San Francisco.
Buchberger, B. (1985). Gröbner bases: an algorithmic method in polynomial ideal theory. In Bose, N., editor, Recent Trends in Multidimensional Systems Theory, pages 184–232. Reidel, Dordrecht.
Buchberger, B., Collins, G. E., and Kutzler, B. (1988). Algebraic methods for geometric reasoning. Annual Review of Computer Science.
Carnap, R. (1947). Meaning and Necessity. University of Chicago Press.
Caviness, B. F. and Johnson, J. R., editors (1998). Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer, New York.
Chang, C. C. and Keisler, H. J. (1973). Model Theory. North Holland Publishing Company, Amsterdam. 3rd ed. 1990.
Chou, S. C. (1984). Proving elementary geometry theorems using Wu’s algorithm. In Bledsoe and Loveland, 1984, pages 243–286.
Chou, S. C. (1987). A method for mechanical derivation of formulas in elementary geometry. Journal of Automated Reasoning, 3:291–299.
Chou, S. C. (1988). An introduction to Wu’s method for mechanical theorem proving in geometry. Journal of Automated Reasoning, 4:237–267.
Chou, S. C. (1990). Automated reasoning in geometries using the characteristic set method and Gröbner basis method. In Proceedings of the International Symposium on Symbolic and Algebraic Computation ISSAC’90, pages 255–260. ACM Press.
Chou, S. C. and Gao, X. S. (1990). Ritt-Wu’s decomposition algorithm and geometry theorem proving. In Stickel, M. E., editor, Proc. CADE-10, volume 449 of LNCS. Springer-Verlag.
Cohn, A. and Hazarika, S. (2001). Qualitative spatial representation and reasoning: an overview. Fundamenta Informaticae, 46:1–29.
Collins, G. E. (1975). Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. Lect. Notes Comput. Sci., 33:134–183.
Collins, G. E. (1998). Quantifier elimination by cylindrical algebraic decomposition—twenty years of progress. In Caviness, B. F. and Johnson, J. R., editors, Quantifier Elimination and Cylindrical Algebraic Decomposition, pages 8–23. Springer-Verlag, New York.
Coppel, W. A. (1998). Foundations of Convex Geometry. CUP.
Coxeter, H. (1969). Introduction to Geometry. John Wiley & Sons, NY.
Davenport, James H. and Heintz, Joos (1988). Real quantifier elimination is doubly exponential. J. Symb. Comput., 5(1/2):29–35.
de Rijke, M. (1992). The modal logic of inequality. Journal of Symbolic Logic, 57:566–584.
de Rijke, M. (1995). The logic of Peirce algebras. Journal of Logic, Language and Information, 4:227–250.
Demri, S. (1996). A simple tableau system for the logic of elsewhere. In Miglioli, P., Moscato, U., Mundici, D., and Ornaghi, M., editors, Theorem Proving with Analytic Tableaux and Related Methods, volume 1071 of LNAI. Springer.
Doets, K. (1996). Basic Model Theory. CSLI Publications, Stanford.
Dolzmann, A., Sturm, A., and Weispfenning, V. (1998). A new approach for automatic theorem proving in real geometry. Journal of Automated Reasoning, 21:357–380.
Enderton, H. (1972). A Mathematical Introduction to Logic. Harcourt Academic Press, New York. 2nd ed. 2001.
Esser, M. (1951). Self-dual postulates for n-dimensional geometry. Duke Math. Journal, 18:475–479.
Esser, M. (1973). Self-dual axioms for many-dimensional projective geometry. Trans. Amer. Math. Soc., 177:221–236.
Eves, H. (1972). A Survey of Geometry. Allyn and Bacon Inc., Boston.
Gabbay, D. (1981). An irreflexivity lemma with applications to axiomatizations of conditions in tense frames. In Monnich, U., editor, Aspects of Philosophical Logic, pages 67–89. Reidel, Dordrecht.
Gemignani, M. (1971). Axiomatic Geometry. Addison-Wesley Publ. Co.
Goldblatt, R. (1987). Orthogonality and Space-Time Geometry. Springer-Verlag.
Goranko, V. (1990). Modal definability in enriched languages. Notre Dame Journal of Formal Logic, 31:81–105.
Goranko, V. and Passy, S. (1992). Using the universal modality: gains and questions. Journal of Logic and Computation, 2:5–30.
Hartshorne, R. (1967). Foundations of Projective Geometry. W. A. Benjamin Inc., New York.
Heintz, J., Roy, M., and Solerno, P. (1990). Sur la complexité du principe de tarski-seidenberg. Bull. Soc. Math. France, 118:101–126.
Henkin, L., Suppes, P., and Tarski, A., editors (1959). The Axiomatic Method, with Special Reference to Geometry and Physics. North-Holland Publishing Company, Amsterdam.
Heyting, A. (1963). Axiomatic Projective Geometry. P. Noordhoff (Groningen) and North-Holland Publishing Company (Amsterdam).
Hilbert, D. (1950). Foundations of Geometry. La Salle, Illinois.
Hodges, W. (1993). Model Theory. Cambridge University Press.
Hughes, D. R. and Piper, F. C. (1973). Projective Planes. Graduate Texts in Mathematics no. 6. Springer-Verlag, New York.
Hughes, G. and Cresswell, M. (1996). A New Introduction to Modal Logic. Routledge.
Kapur, D. (1986). Geometry theorem proving using Hilbert’s Nullstellensatz. In Proc. of SYMSAC’86, pages 202–208, Waterloo.
Karzel, H., Sörensen, K., and Windelberg, D. (1973). Einführung in die Geometrie. Studia mathematica/Mathematische Lehrbücher, Taschenbuch 1. Uni-Taschenbücher, No. 184. Vandenhoeck & Ruprecht, Göttingen.
Kordos, M. (1982). Bisorted projective geometry. Bull. Acad. Polon. Sci. Sér. Sci. Math., 30(no. 9–10):429–432 (1983).
Kramer, R. (1993). The undefinability of intersection from perpendicularity in the three-dimensional Euclidean geometry of lines. Geometriae Dedicata, 46:207–210.
Lemon, O. and Pratt, I. (1998). On the incompleteness of modal logics of space: advancing complete modal logics of place. In Kracht, M., de Rijke, M., Wansing, H., and Zakharyaschev, M., editors, Advances in Modal Logic: Volume 1, pages 115–132. CSLI Publications.
Lenz, H. (1954). Zur Begründung der analytischen Geometrie. S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 1954, pages 17–72 (1955).
Lenz, H. (1989). Zur Begründung der affinen Geometrie des Raumes. Mitt. Math. Ges. Hamburg, 11(6):763–775.
Lenz, H. (1992). Konvexität in Anordnungsräumen. Abh. Math. Sem. Univ. Hamburg, 62:255–285.
Lombard, M. and Vesley, R. (1998). A common axiom set for classical and intuitionistic plane geometry. Annals of Pure and Applied Logic, 95: 229–255.
Marx, M. (1996). Dynamic arrow logic. In Marx, M., Pólos, L., and Masuch, M., editors, Arrow Logic and Multi-Modal Logic, pages 109–123. CSLI Publications.
Menger, K. (1948). Independent self-dual postulates in projective geometry. Rep. Math. Colloquium (2), 8:81–87.
Menger, K. (1950). The projective space. Duke Math. Journal, 17:1–14.
Menghini, M. (1991). On configurational propositions. Pure Math. Appl. Ser. A., 2(1–2):87–126.
Meserve, B. (1983). Fundamental Concepts of Geometry. Dover Publ., New York, second edition.
Mihalek, R. (1972). Projective Geometry and Algebraic Structures. Academic Press, New York.
Pambuccian, V. (1989). Simple axiom systems for Euclidean geometry. Mathematical Chronicle, 18:63–74.
Pambuccian, V. (1995). Ternary operations as primitive notions for constructive plane geometry VI. Math. Logic Quarterly, 41:384–394.
Pambuccian, V. (2001a). Constructive axiomatizations of plane absolute, Euclidean and hyperbolic geometry. Mathematical Logic Quarterly, 47: 129–136.
Pambuccian, V. (2001b). Fragments of Euclidean and hyperbolic geometry. Scientiae Mathematicae Japonicae, 53(2):361–400.
Pambuccian, V. (2003). Sphere tangency as single primitive notion for hyperbolic and Euclidean geometry. Forum Mathematicum, 15:943–947.
Pambuccian, V. (2004). Axiomatizations of Euclidean geometry in terms of points, equilateral triangles or squares, and incidence. Indagationes Mathematicae, 15(3):413–417.
Pambuccian, V. (2006). Axiomatizations of hyperbolic and absolute geometries. In Prékopa, A. and Molnár, E., editors, Non-Euclidean Geometries: János Bolyai Memorial Volume, pages 119–153. Springer Verlag, New York.
Pasch, M. (1882). Vorlesungen über neuere Geometrie. B.G.Teubner, Leipzig.
Pieri, M. (1908). La geometria elementare istituita sulle nozioni “punto” e “sfera”. Memorie di Matematica e di Fisica della Societa Italiana delle Scienze, 15:345–450.
Renegar, J. (1992). On the computational complexity and geometry of the first-order theory of the reals. J. Symb. Comput., 13(3):255–352.
Schwabhäuser, W. and Szczerba, L. (1975). Relations on lines as primitive notions for Euclidean geometry. Fund. Math., LXXXII:347–355.
Schwabhäuser, W., Szmielew, W., and Tarski, A. (1983). Metamathematische Methoden in der Geometrie. Springer-Verlag, Berlin.
Scott, D. (1956). A symmetric primitive of Euclidean geometry. Indag. Math., 18:456–461.
Scott, D. (1959). Dimension in elementary Euclidean geometry. In Henkin et al., 1959, pages 53–67.
Segerberg, K. (1981). A note on the logic of elsewhere. Theoria, 47:183–187.
Seidenberg, A. (1954). A new decision method for elementary algebra. Ann. Math., 60:365–374.
Spaan, E. (1993). Complexity of Modal Logics. PhD thesis, University of Amsterdam.
Stebletsova, V. (2000). Algebras, Relations, Geometries. PhD thesis, Zeno Institute of Philosophy, Univ. of Utrecht.
Stockmeyer, L. J. (1977). The polynomial-time hierarchy. Theoretical Computer Science, 3(1):1–22.
Szczerba, L. (1972). Weak general affine geometry. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 20:753–761.
Szczerba, L. and Tarski, A. (1965). Metamathematical properties of some affine geometries. In Bar-Hillel, Y., editor, Proceedings of the 1964 International Congress for Logic, Methodology and Philosophy of Science, Studies in Logic and the Foundations of Mathematics, Amsterdam. North-Holland Publishing Company.
Szczerba, L. and Tarski, A. (1979). Metamathematical discussion of some affine geometries. Fund. Math., 104:155–192.
Szmielew, W. (1959). Some metamathematical problems concerning elementary hyperbolic geometry. In Henkin et al., 1959, pages 30–52.
Szmielew, W. (1983). From Affine to Euclidean Geometry: An Axiomatic Approach. D. Reidel and PWN-Warsaw.
Tarski, A. (1949a). On essential undecidability. Journal of Symbolic Logic, 14:75–76.
Tarski, A. (1949b). Undecidability of the theories of lattices and projective geometries. Journal of Symbolic Logic, 14:77–78.
Tarski, A. (1951). A decision method for elementary algebra and geometry. Technical report, UCLA. Prepared for publ. by J. McKinsey.
Tarski, A. (1956). A general theorem concerning primitive notions of Euclidean geometry. Indag. Math., 18:468–474.
Tarski, A. (1959). What is elementary geometry? In Henkin et al., 1959, pages 16–29.
Tarski, A. (1967). The completeness of elementary algebra and geometry. Technical report, Institut Blaise Pascal, Paris.
Tarski, A. and Givant, S. (1999). Tarski’s system of geometry. Bull. of Symb. Logic, 5(2):175–214.
Tarski, A. and Mostowski, A. (1949). Undecidability in the arithmetic of integers and in the theory of rings. Journal of Symbolic Logic, 14:76.
Tarski, A., Mostowski, A., and Robinson, R. (1953). Undecidable theories. North-Holland.
van Benthem, J. (1983). The Logic of Time. Kluwer.
van Benthem, J. (1984). Correspondence theory. In Gabbay, D. and Guenthner, F., editors, Handbook of Philosophical Logic: Volume II, pages 167–247. Reidel.
van Benthem, J. (1994). A note on dynamic arrow logics. In van Eijck, J. and Visser, A., editors, Logic and Information Flow, pages 15–29. MIT Press.
van Benthem, J. (1996). Complexity of contents versus complexity of wrappings. In Marx, M., Masuch, M., and Pólos, editors, Arrow Logic and Multimodal Logic, pages 203–219. CSLI Publications, Stanford.
van Benthem, J. (1999). Temporal patterns and modal structure. Logic Journal of IGPL, 7:7–26.
Veblen, O. (1904). A system of axioms for geometry. Transactions of the American Mathematical Society, 5:343–384.
Veblen, O. (1914). The foundations of geometry. In Young, J., editor, Monographs on topics of modern mathematics, relevant to the elementary field, pages 1–51. Longsman, Green, and Company, New York.
Venema, Y. (1993). Derivation rules as anti-axioms in modal logic. Journal of Symbolic Logic, 58:1003–1034.
Venema, Y. (1999). Points, lines and diamonds: a two-sorted modal logic for projective planes. Journal of Logic and Computation, 9:601–621.
von Plato, J. (1995). The axioms of constructive geometry. Annals of Pure and Applied Logic, 76:169–200.
von Wright, G. (1979). A modal logic of place. In Sosa, E., editor, The Philosophy of Nicholas Rescher, pages 65–73. Reidel.
Wu, W. (1984). Some recent advances in mechanical theorem proving in geometries. In Bledsoe and Loveland, 1984, pages 235–242.
Wu, W. (1986). Basic principles of mechanical theorem proving in geometries. Journal of Automated Reasoning, 2(4):221–252.
Ziegler, M. (1982). Einige unentscheidbare Körpertheorien. Enseign. Math. (2), 28(3–4):269–280.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer
About this chapter
Cite this chapter
Balbiani, P., Goranko, V., Kellerman, R., Vakarelov, D. (2007). Logical Theories for Fragments of Elementary Geometry. In: Aiello, M., Pratt-Hartmann, I., Van Benthem, J. (eds) Handbook of Spatial Logics. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5587-4_7
Download citation
DOI: https://doi.org/10.1007/978-1-4020-5587-4_7
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-5586-7
Online ISBN: 978-1-4020-5587-4
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)