Linked bibliography for the SEP article "Second-order and Higher-order Logic" by Jouko Väänänen

This is an automatically generated and experimental page

If everything goes well, this page should display the bibliography of the aforementioned article as it appears in the Stanford Encyclopedia of Philosophy, but with links added to PhilPapers records and Google Scholar for your convenience. Some bibliographies are not going to be represented correctly or fully up to date. In general, bibliographies of recent works are going to be much better linked than bibliographies of primary literature and older works. Entries with PhilPapers records have links on their titles. A green link indicates that the item is available online at least partially.

This experiment has been authorized by the editors of the Stanford Encyclopedia of Philosophy. The original article and bibliography can be found here.

  • Ajtai, M., 1979, “Isomorphism and Higher Order Equivalence”, Annals of Mathematical Logic, 16(3): 181–203. doi:10.1016/0003-4843(79)90001-9
  • –––, 1983, “\(\Sigma_1^1\)-Formulae on Finite Structures”, Annals of Pure and Applied Logic, 24(1): 1–48. doi:10.1016/0168-0072(83)90038-6
  • Asser, Günter, 1981, Einführung in Die Mathematische Logik: Teil Ill: Prädikatenlogik Höherer Stufe, Thun: Verlag Harri Deutsch.
  • Barwise, K. Jon, 1972a, “Absolute Logics and \(L_{\infty \omega }\)”, Annals of Mathematical Logic, 4(3): 309–340. doi:10.1016/0003-4843(72)90002-2
  • –––, 1972b, “The Hanf Number of Second Order Logic”, Journal of Symbolic Logic, 37(3): 588–594. doi:10.2307/2272748
  • Büchi, J. Richard, 1962, “On a Decision Method in Restricted Second Order Arithmetic”, in Logic, Methodology and Philosophy of Science: Proceedings of the 1960 International Congress, Ernest Nagel, Patrick Suppes, and Alfred Tarski (eds.), Stanford, CA: Stanford University Press, 1–11.
  • –––, 1973, “The Monadic Second Order Theory of \(\omega _{i}\)”, in Decidable Theories II: The Monadic Second Order Theory of All Countable Ordinals, by J. Richard Büchi and Dirk Siefkes, edited by G. H. Müller and D. Siefkes, (Lecture Notes in Mathematics 328), Berlin, Heidelberg: Springer Berlin Heidelberg, 1–127. doi:10.1007/BFb0082721
  • Buss, Samuel R. (ed.), 1998, Handbook of Proof Theory, (Studies in Logic and the Foundations of Mathematics 137), New York: Elsevier.
  • Button, Tim and Sean Walsh, 2016, “Structure and Categoricity: Determinacy of Reference and Truth Value in the Philosophy of Mathematics”, Philosophia Mathematica, 24(3): 283–307. doi:10.1093/philmat/nkw007
  • Church, Alonzo, 1956, Introduction to Mathematical Logic, volume 1, Revised and enlarged edition, (Princeton Mathematical Series 17), Princeton, NJ: Princeton University Press.
  • Cohen, Paul J., 1966, Set Theory and the Continuum Hypothesis, New York-Amsterdam: W. A. Benjamin.
  • Craig, William, 1965, “Satisfaction for \(n\)-th order languages defined in \(n\)-th order languages”, Journal of Symbolic Logic, 30: 13–25.
  • de la Cruz, Omar, 2002, “Finiteness and Choice”, Fundamenta Mathematicae, 173(1): 57–76. doi:10.4064/fm173-1-4
  • Durand, Arnaud, Ronald Fagin, and Bernd Loescher, 1998, “Spectra with Only Unary Function Symbols”, in Computer Science Logic, Mogens Nielsen and Wolfgang Thomas (eds.), (Lecture Notes in Computer Science 1414), Berlin, Heidelberg: Springer Berlin Heidelberg, 189–202. doi:10.1007/BFb0028015
  • Durand, Arnaud, Neil D. Jones, Johann A. Makowsky, and Malika More, 2012, “Fifty Years of the Spectrum Problem: Survey and New Results”, The Bulletin of Symbolic Logic, 18(4): 505–553. doi:10.2178/bsl.1804020
  • Elgot, Calvin C., 1961, “Decision Problems of Finite Automata Design and Related Arithmetics”, Transactions of the American Mathematical Society, 98(1): 21–21. doi:10.1090/S0002-9947-1961-0139530-9
  • Fagin, Ronald, 1974, “Generalized First-Order Spectra and Polynomial-Time Recognizable Sets”, in Complexity of Computation, Richard M. Karp (ed.), (SIAM-AMS Proceedings 7), Providence, RI: American Mathematical Society, 43–73.
  • –––, 1975, “Monadic Generalized Spectra”, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 21(1): 89–96. doi:10.1002/malq.19750210112
  • –––, 1993, “Finite-Model Theory—a Personal Perspective”, Theoretical Computer Science, 116(1): 3–31. doi:10.1016/0304-3975(93)90218-I
  • Fagin, Ronald, Larry J. Stockmeyer, and Moshe Y. Vardi, 1995, “On Monadic NP vs Monadic Co-NP”, Information and Computation, 120(1): 78–92. doi:10.1006/inco.1995.1100
  • Feferman, Solomon, 1999, “Does Mathematics Need New Axioms?”, The American Mathematical Monthly, 106(2): 99–111. doi:10.1080/00029890.1999.12005017
  • Feferman, Solomon and Georg Kreisel, 1966, “Persistent and Invariant Formulas Relative to Theories of Higher Order”, Bulletin of the American Mathematical Society, 72(3): 480–486. doi:10.1090/S0002-9904-1966-11507-0
  • Frege, Gottlob, 1879, Begriffsschrift: Eine Der Arithmetischen Nachgebildete Formelsprache Des Reinen Denkens, Halle.
  • –––, 1884, Die Grundlagen Der Arithmetik, Breslau: Verlage Wilhelm Koebner.
  • Garland, Stephen J., 1974, “Second-Order Cardinal Characterizability”, in Axiomatic Set Theory, Part 2, T. J. Jech (ed.), (Proceedings of Symposia in Pure Mathematics, 13.2), Providence, RI: American Math Society, 127–146. [Garland 1974 available online]
  • Gaßner, Christine, 1994, “The Axiom of Choice in Second-Order Predicate Logic”, Mathematical Logic Quarterly, 40(4): 533–546. doi:10.1002/malq.19940400410
  • Gödel, Kurt, 1931 [1986], “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I”, Monatshefte für Mathematik und Physik, 38(1): 173–198. Reprinted and translated as “On Formally Undecidable Propositions of Principia Mathematica and Related Systems” in Gödel 1986: 144–195. doi:10.1007/BF01700692
  • –––, 1936 [1986], “Über Die Länge von Beweisen”, Ergebnisse Eirtes Mathematischen Kolloquiums, 7: 23–24. Reprinted and translated as “On the Length of Proofs” in Gödel 1986: 396–399.
  • –––, 1939 [1990], “Consistency-Proof for the Generalized Continuum-Hypothesis”, Proceedings of the National Academy of Sciences, 25(4): 220–224. Reprinted in Gödel 1990: 28–32. doi:10.1073/pnas.25.4.220
  • –––, 1986, Collected Works, Volume 1: Publications 1929-1936, Solomon Feferman (ed.), New York: Oxford University Press.
  • –––, 1990, Collected Works, Volume 2: Publications 1938-1974, Solomon Feferman (ed.), New York: Oxford University Press.
  • Gurevich, Yuri, 1985, “Monadic Second-Order Theories”, in Model-Theoretic Logics, Jon Barwise and Sol Feferman (eds.) (Perspectives in Logic), New York: Springer-Verlag, 479–506. doi:10.1017/9781316717158.019
  • Gurevich, Yuri, Menachem Magidor, and Saharon Shelah, 1983, “The Monadic Theory of \(\omega_2\)”, Journal of Symbolic Logic, 48(2): 387–398. doi:10.2307/2273556
  • Hasenjaeger, Gisbert, 1961, “Unabhängigkeitsbeweise in Mengenlehre Und Stufenlogik Der Modelle.”, Jahresbericht Der Deutschen Mathematiker-Vereinigung, 63: 141–162.
  • Hellman, Geoffrey, 2001, “Three Varieties of Mathematical Structuralism†”, Philosophia Mathematica, 9(2): 184–211. doi:10.1093/philmat/9.2.184
  • Henkin, Leon, 1950, “Completeness in the Theory of Types”, Journal of Symbolic Logic, 15(2): 81–91. doi:10.2307/2266967
  • Herbrand, Jacques, 1930, Recherches sur la théorie de la démonstration, (Thèses de l’entre-deux-guerres 110), L’Université de Paris. [Herbrand 1930 available online]
  • Hilbert, David, 1900, “Über Den Zahlbegriff.”, Jahresbericht Der Deutschen Mathematiker-Vereinigung, 8: 180–183.
  • Hilbert, David and William Ackermann, 1928, Grundzüge Der Theoretischen Logik, (Grundlehren Der Mathematischen Wissenschaften in Einzeldarstellungen Mit Besonderer Berücksichtigung Der Anwendungsgebiete 27), Berlin: J. Springer.
  • –––, 1938, Grundzüge Der Theoretischen Logik, second edition, Berlin: Springer. doi:10.1007/978-3-662-41928-1
  • Hintikka, K. Jaakko J., 1955, “Reductions in the Theory of Types”, Acta Philosophica Fennica, 8: 57–115.
  • Hyttinen, Tapani, Kaisa Kangas, and Jouko Vaananen, 2013, “On Second-Order Characterizability”, Logic Journal of IGPL, 21(5): 767–787. doi:10.1093/jigpal/jzs047
  • Jones, Neil D. and Alan L. Selman, 1974, “Turing Machines and the Spectra of First-Order Formulas”, Journal of Symbolic Logic, 39(1): 139–150. doi:10.2307/2272354
  • Krawczyk, A. and W. Marek, 1977, “On the Rules of Proof Generated by Hierarchies”, in Set Theory and Hierarchy Theory V, Alistair Lachlan, Marian Srebrny, and Andrzej Zarach (eds.) (Lecture Notes in Mathematics 619), Berlin: Springer Berlin Heidelberg, 227–239. doi:10.1007/BFb0067654
  • Kreisel, Georg, 1967, “Informal Rigour and Completeness Proofs”, in Problems in the Philosophy of Mathematics, Imre Lakatos (ed.), (Studies in Logic and the Foundations of Mathematics 47), Amsterdam: North-Holland, 138–186. doi:10.1016/S0049-237X(08)71525-8
  • Krynicki, Michał and Alistair H. Lachlan, 1979, “On the Semantics of the Henkin Quantifier”, Journal of Symbolic Logic, 44(2): 184–200. doi:10.2307/2273726
  • Kunen, Kenneth (ed.), 1980, Set Theory: An Introduction to Independence Proofs, (Studies in Logic and the Foundations of Mathematics 102), Amsterdam: North-Holland.
  • Lévy, Azriel, 1965, A Hierarchy of Formulas in Set Theory, (Memoirs of the American Mathematical Society 57), Providence, RI: American Mathematical Society.
  • Lindenbaum, Adolf and Andrzej Mostowski, 1938, “Über Die Unabhängigkeit Des Auswahlaxioms Und Einiger Seiner Folgerungen”, Sprawozdania z Posiedzeń Towarzystwa Naukowego Warszawskiego, Wydział III Nauk Matematyczno-Fizycznych (Comptes-Rendus Des Séances de La Société Des Sciences et Des Lettres de Varsovie,Classe III), 31: 27–32.
  • Löwenheim, Leopold, 1915, “Über Möglichkeiten im Relativkalkül”, Mathematische Annalen, 76(4): 447–470. doi:10.1007/BF01458217
  • Lyndon, Roger C., 1959, “An Interpolation Theorem in the Predicate Calculus.”, Pacific Journal of Mathematics, 9(1): 129–142.
  • Magidor, M., 1971, “On the Role of Supercompact and Extendible Cardinals in Logic”, Israel Journal of Mathematics, 10(2): 147–157. doi:10.1007/BF02771565
  • Makowsky, J.A., Saharon Shelah, and Jonathan Stavi, 1976, “δ-Logics and Generalized Quantifiers”, Annals of Mathematical Logic, 10(2): 155–192. doi:10.1016/0003-4843(76)90021-8
  • Manzano, María, 1996, Extensions of First Order Logic, (Cambridge Tracts in Theoretical Computer Science 19), Cambridge: Cambridge University Press.
  • Montague, Richard, 1963, “Reductions of Higher-Order Logic”, in The Theory of Models: Proceedings of the 1963 International Symposium at Berkeley, J.W. Addison, Leon Henkin, and Alfred Tarski (eds.), Amsterdam: North-Holland, 251–264. doi:10.1016/B978-0-7204-2233-7.50030-7
  • Moore, Gregory H., 1988, “The Emergence of First-Order Logic”, in History and Philosophy of Modern Mathematics, (Minnesota Studies in the Philosophy of Science 11), Minneapolis, MN: University of Minnesota Press, 95–135.
  • Mostowski, Andrzej, 1938, “O niezależności definicji skończoności w systemie logiki” (“On the independence of definitions of finiteness in a system of logic”), Dodatek do Rocznika Towarzystwa. Matematycznego, XI: 1–54; English translation in Mostowski 1979.
  • Mostowski, Andrzej, 1949, “An Undecidable Arithmetical Statement”, Fundamenta Mathematicae, 36(1): 143–164.
  • Mostowski, Andrzej, 1979, Foundational Studies. Selected works (Volume II), Studies in Logic and the Foundations of Mathematics: Volume 93, Amsterdam-New York: North-Holland Publishing Co.; PWN-Polish Scientific Publishers, Warsaw, 1979, edited by Kazimierz Kuratowski, Wiktor Marek, Leszek Pacholski, Helena Rasiowa, Czesław Ryll-Nardzewski and Paweł Zbierski.
  • Parsons, Charles, 1990, “The Structuralist View of Mathematical Objects”, Synthese, 84(3): 303–346. doi:10.1007/BF00485186
  • Quine, W. V. O., 1970, Philosophy of Logic, Cambridge, MA: Harvard University Press.
  • Rabin, Michael O., 1968, “Decidability of Second-Order Theories and Automata on Infinite Trees”, Bulletin of the American Mathematical Society, 74(5): 1025–1030. doi:10.1090/S0002-9904-1968-12122-6
  • Rabin, Michael O. and Dana Scott, 1959, “Finite Automata and Their Decision Problems”, IBM Journal of Research and Development, 3(2): 114–125. doi:10.1147/rd.32.0114
  • Resnik, Michael D., 1988, “Second-Order Logic Still Wild”, The Journal of Philosophy, 85(2): 75–87. doi:10.2307/2026993
  • Schmidt, Arnold, 1951, “Die Zulässigkeit der Behandlung mehrsortiger Theorien mittels der üblichen einsortigen Prädikatenlogik”, Mathematische Annalen, 123(1): 187–200. doi:10.1007/BF02054948
  • Scott, Dana, 1961, “Measurable Cardinals and Constructible Sets”, Bulletin de l' Académie Polonaise des Sciences, Série des sciences mathématiques, astronomiques et physiques, 9: 521–524.
  • Shapiro, Stewart, 1991, Foundations without Foundationalism: A Case for Second-Order Logic, New York: Oxford University Press. doi:10.1093/0198250290.001.0001
  • Shelah, Saharon, 1973, “There Are Just Four Second-Order Quantifiers”, Israel Journal of Mathematics, 15(3): 282–300. doi:10.1007/BF02787572
  • –––, 2004, “Spectra of Monadic Second Order Sentences”, Scientiae Mathematicae Japonicae, (Special issue on set theory and algebraic model theory), 59(2): 351–355. [Shelah 2004 available online]
  • Simpson, Stephen G., 1999 [2009], Subsystems of Second Order Arithmetic, second edition, (Perspectives in Logic), Cambridge: Cambridge University Press. first edition in 1999, New York: Springer. doi:10.1017/CBO9780511581007
  • Tarski, Alfred, 1933 [1956], Pojęcie Prawdy w Językach Nauk Dedukcyjnych (Le concept de vérité dans les langages formalisés), (Prace Towarzystwa Naukowego Warszawskiego, Wydział III Nauk Matematyczno-fizycznych/Travaux de la Société des Sciences et des Lettres de Varsovie, Classe III Sciences Mathématiques et Physiques, 34), Warsaw. German translation see Tarski 1936. English translation as “The Concept of Truth in Formalized Languages” in Tarski 1956: chapter 8: 152–278.
  • –––, 1936 [1956], “Der Wahrheitsbegriff in Den Formalisierten Sprachen”, Studia Philosophica, 1: 261–405. Translated as “The Concept of Truth in Formalized Languages” in Tarski 1956: chapter 8.
  • –––, 1956, Logic, Semantics, Metamathematics; Papers from 1923 to 1938, J. H. Woodger (trans.), Oxford: Clarendon Press.
  • Tharp, Leslie H., 1973, “The Characterization of Monadic Logic”, Journal of Symbolic Logic, 38(3): 481–488. doi:10.2307/2273046
  • Väänänen, Jouko, 1979, “Abstract Logic and Set Theory. I. Definability”, in Logic Colloquium ’78: Proceedings of the Colloquium Held in Mons, Maurice Boffa, Dirk van Dalen, and Kenneth Mcaloon (eds.), (Studies in Logic and the Foundations of Mathematics 97), Amsterdam: Elsevier, 391–421. doi:10.1016/S0049-237X(08)71637-9
  • –––, 1980, “Boolean Valued Models and Generalized Quantifiers”, Annals of Mathematical Logic, 18(3): 193–225. doi:10.1016/0003-4843(80)90005-4
  • –––, 2012, “Second Order Logic or Set Theory?”, The Bulletin of Symbolic Logic, 18(1): 91–121. doi:10.2178/bsl/1327328440
  • Väänänen, Jouko and Tong Wang, 2015, “Internal Categoricity in Arithmetic and Set Theory”, Notre Dame Journal of Formal Logic, 56(1): 121–134. doi:10.1215/00294527-2835038
  • Walmsley, James, 2002, “Categoricity and Indefinite Extensibility”, Proceedings of the Aristotelian Society, 102(1): 239–257. doi:10.1111/j.0066-7372.2003.00052.x
  • Wang, Hao, 1952, “Logic of Many-Sorted Theories”, Journal of Symbolic Logic, 17(2): 105–116. doi:10.2307/2266241
  • Zermelo, Ernst, 1930, “Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre”, Fundamenta Mathematicæ, 16: 29–47.

Generated Wed Nov 25 05:12:18 2020