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Elementary Predicate Logic

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Handbook of Philosophical Logic

Part of the book series: Handbook of Philosophical Logic ((HALO,volume 1))

Abstract

Elementary (first-order) predicate logic is a child of many parents. At least three different groups of thinkers played their part in its conception, with three quite distinct motives. Maybe the mixture gave it hybrid strength. But whatever the reason, first-order logic is both the simplest, the most powerful and the most applicable branch of modern logic.

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Bibliography

  1. W. Ackermann. Solvable Cases of the Decision Problem. North-Holland, Amsterdam, 1962.

    Google Scholar 

  2. P. Aczel. Non-well-founded Sets. CSLI, Stanford CA, 1988.

    Google Scholar 

  3. J. E. J. Altham and N. W. Tennant. Sortal quantification. In E. L. Keenan, editor, Formal Semantics of Natural Language, pages 46–58. Cambridge University Press, 1975.

    Google Scholar 

  4. J. M. Anderson and H. W. Johnstone Jr. Natural Deduction: The Logical Basis of Axiom Systems. Wadsworth, Belmont, CA., 1962.

    Google Scholar 

  5. J. Ax and S. Kochen. Diophantine problems over local fields: I. American Journal of Mathematics, 87:605–630, 1965.

    Article  Google Scholar 

  6. J. Barwise. Abstract logics and L ∞ω .Annals Math Logic, 4:309–340, 1973.

    Article  Google Scholar 

  7. J. Barwise. Axioms for abstract model theory. Annals Math Logic, 7:221–265, 1974.

    Article  Google Scholar 

  8. J. Barwise. Admissible Sets and Structures. Springer, Berlin, 1975.

    Book  Google Scholar 

  9. J. Barwise and R. Cooper. Generalized quantifiers and natural langauges. Linguistics and Philosophy, 4:159–219, 1981.

    Article  Google Scholar 

  10. J. Barwise and J. Etchemendy. Tarski’s World 3.0. Cambridge University Press, 1991.

    Google Scholar 

  11. J. Barwise and J. Etchemendy. Hyperproof. CSLI, Stanford, 1994.

    Google Scholar 

  12. J. Barwise and L. Moss. Vicious Circles. CSLI, Stanford CA, 1996.

    Google Scholar 

  13. H. Behmann. Beiträge zur Algebra der Logik, insbesondere zum Entscheidungs-problem. Math Annalen, 86:163–229, 1922.

    Article  Google Scholar 

  14. J. L. Bell and M. Machover. A Course in Mathematical Logic. North-Holland, Amsterdam, 1977.

    Google Scholar 

  15. J. L. Bell and A. B. Slomson. Models and Ultraproducts. North-Holland, Amsterdam, 1969.

    Google Scholar 

  16. N. D. Belnap. Tonk, plonk and plink. Analysis, 22:130–134, 1962. Reprinted in [Strawson, 1967, pp. 132–137].

    Article  Google Scholar 

  17. P. Benacerraf and H. Putnam, editors. Philosophy of Mathematics: Selected Readings. Cambridge University Press, second edition, 1983.

    Google Scholar 

  18. P. Bernays. Review of Max Steck, ‘Ein unbekannter Brief von Gottlob Frege über Hilberts erste Vorlesung über die Grundlagen der Geometrie’. Journal of Symbolic Logic, 7:92 f., 1942.

    Article  Google Scholar 

  19. P. Bernays and A. A. Fraenkel. Axiomatic Set Theory. North-Holland, Amsterdam, 1958.

    Google Scholar 

  20. E. W. Beth. On Padoa’s method in the theory of definition. Koninklijke Nederlandse Akad. van Wetensch, 56 (ser. A, Math Sciences):330–339, 1953.

    Google Scholar 

  21. E. W. Beth. Semantic entailment and formal derivability. Mededelingen der Koninklijke Nederlandse Akad. van Wetensch, afd letterkunde 18, 1955. Reprinted in [Hintikka, 1969, pp. 9–41].

    Google Scholar 

  22. E. W. Beth. Formal Methods. Reidel, Dordrecht, 1962.

    Google Scholar 

  23. I. M. Bocheriski. A History of Formal Logic, translated by I. Thomas. Chelsea Publishing Co, New York, 1970.

    Google Scholar 

  24. B. Bolzano. Wissenschaftslehre. 1837. Edited and translated by R.George as Theory of Science, UCLA Press, Berkeley and Los Angeles, 1972.

    Google Scholar 

  25. G. Boole. The Mathematical Analysis of Logic. Macmillan, Barclay and Macmillan, Cambridge, 1847. Also pp. 45–124 of George Boole, Studies in Logic and Probability, Open Court, La Salle, IL, 1952.

    Google Scholar 

  26. G. Boole. An Investigation of the Laws of Thought. Walton and Maberley, London, 1854. Republished by Open Court, La Salle, IL, 1952.

    Google Scholar 

  27. G. S. Boolos and R. C. Jeffrey. Computability and Logic. Cambridge University Press, Cambridge, 1989.

    Google Scholar 

  28. G. Boolos. The Unprovability of Consistency: An Essay in Modal Logic. Cambridge University Press, 1979.

    Google Scholar 

  29. G. Boolos. The Logic of Provability. Cambridge University Press, 1993.

    Google Scholar 

  30. R. Carnap. Ein Gültigkeitskriterium für die Sätze der klassischen Mathematik. Monatshefte Math und Phys, 42:163–190, 1935.

    Article  Google Scholar 

  31. R. Carnap. Meaning and Necessity. University of Chicago Press, second edition, 1956.

    Google Scholar 

  32. C. C. Chang and H. J. Keisler. Model Theory. North-Holland, Amsterdam, 1973.

    Google Scholar 

  33. C. Chastain. Reference and context. In K. Gunderson, editor, Minnesota Studies in the Philosophy of Science, VII, Language, Mind and Knowledge, pages 194–269. University of Minnesota Press, MI, 1975.

    Google Scholar 

  34. G. Cherlin. Model Theoretic Algebra: Selected Topics, volume 521 of Lecture Notes in Maths. Springer, Berlin, 1976.

    Google Scholar 

  35. A. Church. A note on the Entscheidungsproblem. Journal of Symbolic Logic, l:40f, 101f, 1936.

    Article  Google Scholar 

  36. A. Church. Introduction to Mathematical Logic, I. Princeton University Press, Princeton, NJ, 1956.

    Google Scholar 

  37. J. A. Coffa. The Semantic Tradition from Kant to Carnap: To the Vienna Station. Cambridge University Press, Cambridge, 1991.

    Book  Google Scholar 

  38. P. J. Cohen. Decision procedures for real and p-adic fields. Comm Pure Appl Math, 22:131–151, 1969.

    Article  Google Scholar 

  39. L. J. Cohen. Some remarks on Grice’s views about the logical particles of natural language. In Y. Bar-Hillel, editor, Pragmatics of Natural Languages, pages 60–68. Reidel, Dordrecht, 1971.

    Google Scholar 

  40. S. A. Cook. The complexity of theorem-proving procedures. In Proceedings of the Third Annual ACM Symposium on Theory of Computing, pages 151–158. ACM Press, NY, 1971.

    Chapter  Google Scholar 

  41. W. Craig. Linear reasoning. A new form of the Herbrand-Gentzen theorem. Journal of Symbolic Logic, 22:250–268, 1957.

    Article  Google Scholar 

  42. W. Craig. Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory. Journal of Symbolic Logic, 22:269–285, 1957.

    Article  Google Scholar 

  43. D. van Dalen. Logic and Structure. Springer, Berlin, 1980.

    Book  Google Scholar 

  44. R. Dedekind. Was sind und was sollen die Zahlen? Brunswick, 1888.

    Google Scholar 

  45. R. Dedekind. Letter to Keferstein, 1890. In J. Van Heijenoort, editor, From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931, pages 90–103. Harvard University Press, Cambridge, MA, 1967.

    Google Scholar 

  46. K. Dosen and P. Schroeder-Heister, editors. Sub-structural Logics. Oxford University Press, Oxford, 1993.

    Google Scholar 

  47. D. Dowty, R. Wall, and S. Peters. Introduction to Montague Semantics. Reidel, Dordrecht, 1981.

    Google Scholar 

  48. M. A. E. Dummett. Truth. Proc Aristotelian Soc, 59:141–162, 1958/59. Reprinted in [Strawson, 1967; pp. 49–68].

    Google Scholar 

  49. M. A. E. Dummett. Frege: Philosophy of Language. Duckworth, London, 1973.

    Google Scholar 

  50. M. A. E. Dummett. What is a theory of meaning? In Samuel Guttenplan, editor, Mind and Language. Clarendon Press, Oxford, 1975.

    Google Scholar 

  51. J. M. Dunn and N. D. Belnap. The substitution interpretation of the quantifiers. Nous, 2:177–185, 1968.

    Article  Google Scholar 

  52. H.-D. Ebbinghaus and J. Flum. Finite model theory. Springer, Berlin, 1995.

    Google Scholar 

  53. A. Ehrenfeucht. An application of games to the completeness problem for formalized theories. Fundamenta Math, 49:129–141, 1960.

    Google Scholar 

  54. H. B. Enderton. A Mathematical Introduction to Logic. Academic Press, New York, 1972.

    Google Scholar 

  55. J. Etchemendy. The Concept of Logical Consequence. Harvard University Press, Cambridge MA, 1990.

    Google Scholar 

  56. G. Evans. Pronouns. Linguistic Inquiry, 11:337–362, 1980.

    Google Scholar 

  57. S. Feferman. Lectures on proof theory. In Proc Summer School of Logic, Leeds 1967, Lecture Notes in Mathematics 70, pages 1–109. Springer, Berlin, 1968.

    Chapter  Google Scholar 

  58. S. Feferman. Persistent and invariant formulas for outer extensions. Compositio Math, 20:29–52, 1968.

    Google Scholar 

  59. S. Feferman. Set-theoretical foundations of category theory. In Reports of the Midwest Category Seminar III, Lecture Notes in Mathematics 106, pages 201–247. Springer, Berlin, 1969.

    Chapter  Google Scholar 

  60. S. Feferman. Applications of many-sorted interpolation theorems. In L. Henkin et al., editor, Proceedings of the Tarski Symposium, Proc Symposia in Pure Math. XXV, pages 205–223. American Mathematical Society, Providence, RI, 1974.

    Google Scholar 

  61. S. Feferman. Kurt Gödel: conviction and caution. Philosophia Naturalis, 21:546–562, 1984.

    Google Scholar 

  62. F. B. Fitch. Symbolic Logic. Ronald Press, New York, 1952.

    Google Scholar 

  63. J. Flum. First-order logic and its extensions. In ISILC Logic Conference, Lecture Notes in Mathematics 499, pages 248–307. Springer, Berlin, 1975.

    Google Scholar 

  64. A. Fraenkel. Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre. Math Annalen, 86:230–237, 1922.

    Article  Google Scholar 

  65. ss R. Fraïssé. Sur l’extension aux relations de quelques propriétés des ordres. Ann Sci École Norm Sup, 71:363–388, 1954.

    Google Scholar 

  66. ss R. Fraïssé. Sur quelques classifications des relations, basées sur des iso-morphismes restreints. Alger-Mathématiques, 2:16–60 and 273–295, 1955.

    Google Scholar 

  67. G. Frege. Begrifjsschrift. Halle, 1879. Translated in [Heijenoort, 1967, pp. 1–82].

    Google Scholar 

  68. G. Frege. Die Grundlagen der Arithmetik. Breslau, 1884. Translated by J. L. Austin, The Foundations of Arithmetic, 2nd edn., Blackwell, Oxford, 1953.

    Google Scholar 

  69. G. Frege. Funktion und Begriff. Jena, 1891. Also in [Frege, 1967, pp. 125–142] and translated in [Frege, 1952].

    Google Scholar 

  70. G. Frege. Grundgesetze der Arithmetik I. Jena, 1893. Partial translation with introduction by M. Furth, The Basic Laws of Arithmetic, University California Press, Berkeley, 1964.

    Google Scholar 

  71. G. Frege. Über die Grundlagen der Geometrie. Jahresbericht der Deutschen Mathematiker-Vereinigung, 15:293–309, 377–403 and 423–430, 1906. Translated in [Frege, 1971].

    Google Scholar 

  72. G. Frege. Anmerkungen zu: Philip E. B. Jourdain. The development of the theories of mathematical logic and the principles of mathematics, 1912. In [Frege, 1967, pp. 334–341].

    Google Scholar 

  73. G. Frege. Translations from the Philosophical Writings of Gottlob Frege. Blackwell, Oxford, 1952.

    Google Scholar 

  74. G. Frege. Kleine Schriften. Georg Olms Verlagsbuchhandlung, Hildesheim, 1967.

    Google Scholar 

  75. G. Frege. On the Foundations of Geometry and Formal Theories of Arithmetic. Yale University Press, New Haven, 1971. Translated with introduction by E. W. Kluge.

    Google Scholar 

  76. G. Frege and D. Hilbert. Correspondence leading to ‘On the foundations of geometry’, 1899–1900. In [Frege, 1967; pp. 407–418], translated in [Frege, 1971; pp. 6–21].

    Google Scholar 

  77. J. H. Gallier. Logic for Computer Science: foundations of Automatic Theorem Proving. Harper and Row, 1986.

    Google Scholar 

  78. R. O. Gandy. Set-theoretic functions for elementary syntax. In T. J. Jech, editor, Axiomatic Set Theory II, pages 103–126. American Mathematical Society, Providence, RI, 1974.

    Chapter  Google Scholar 

  79. M. R. Garey and D. S. Johnson. Computers and Intractability. W. H. Freeman, San Francisco, 1979.

    Google Scholar 

  80. G. Gentzen. Untersuchungen über das logische Schliessen. Math Zeitschrift, 39:176–210 and 405–431, 1934.

    Article  Google Scholar 

  81. J.-Y. Girard. Linear logic. Theoretical Computer Science, 50:1–102, 1987.

    Article  Google Scholar 

  82. J.-Y. Girard. Linear logic: its syntax and semantics. In J.-Y. Girard et al., editor, Advances in Linear — Logic, pages 1–42. Cambridge University Press, 1995.

    Chapter  Google Scholar 

  83. K. Gödel. Die Vollständigkeit der Axiome des logischen Funktionenkalküls. Monatshefte für Mathematik und Physik, 37:349–360, 1930. Translated in [Gödel, 1986, pp. 102–123] and [Heijenoort, 1967, pp. 582–591].

    Article  Google Scholar 

  84. K. Gödel. Eine Eigenschaft der Realisierungen des Aussagenkalküls. Ergebnisse Math Kolloq, 3:20–21, 1931. Translated in [Gödel, 1986, pp. 238–241].

    Google Scholar 

  85. K. Gödel. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik, 38:173–198, 1931. Translated in [Gödel, 1986, pp. 144–195] and [Heijenoort, 1967, pp. 596–616].

    Google Scholar 

  86. K. Gödel. What is Cantor’s continuum problem? American Mathematical Monthly, 54:515–525, 1947. Revised and expanded version in [Gödel, 1990, pp. 254–270].

    Article  Google Scholar 

  87. K. Gödel. Russell’s mathematical logic. In P. A. Schilpp, editor, The Philosophy of Bertrand Russell, pages pp. 123–153. Tudor Publ. Co, New York, 1951. Also in [Gödel, 1990, pp. 119–141].

    Google Scholar 

  88. K. Gödel. Collected Works. Volume I. Oxford University Press, New York, 1986. Edited by S. Feferman et al.

    Google Scholar 

  89. K. Gödel. Collected Works. Volume II. Oxford University Press, New York, 1990. Edited by S. Feferman et al.

    Google Scholar 

  90. R. Goldblatt. Axiomatizing the Logic of Computer Programming. Lecture Notes in Computer Science, 130, Springer, Berlin, 1982.

    Google Scholar 

  91. W. D. Goldfarb. Logic in the twenties: the nature of the quantifier. Journal of Symbolic Logic, 44:351–368, 1979.

    Article  Google Scholar 

  92. D. Goldson, S. Reeves and R. Bornat. A review of several programs for the teaching of logic, Computer Journal, 36:373–386, 1993.

    Article  Google Scholar 

  93. M. Gómez-Torrente. Tarski on logical consequence, Notre Dame Journal of Formal Logic, 37:125–151, 1996.

    Article  Google Scholar 

  94. H. P. Grice. Logic and conversation. In P. Cole et al., editor, Syntax and Semantics 3, Speech Acts, pp. 41–58. Academic Press, New York, 1975. Revised version in P. Grice, Studies in the Way of Words, Harvard University Press, Cambridge, MA, 1989, pp. 22–40.

    Google Scholar 

  95. J. Groenendijk and M. Stokhof. Dynamic predicate logic, Linguistics and Philosophy, 14:39–100, 1991.

    Article  Google Scholar 

  96. Y. Gurevich. Toward logic tailored for computational complexity. In M. M. Richter, et al., editors, Computation and Proof Theory, Lecture Notes in Mathematics 1104, pp. 175–216, Springer-Verlag, 1984.

    Chapter  Google Scholar 

  97. E. M. Hammer. Logic and Visual Information. CSLI and FoLLI, Stanford CA, 1995.

    Google Scholar 

  98. D. Harel. First-order Dynamic Logic. Lecture Notes in Computer Science, 68. Springer, Berlin, 1979.

    Book  Google Scholar 

  99. V. Harnik. Stability theory and set existence axioms. Journal of Symbolic Logic, 50:123–137, 1985.

    Article  Google Scholar 

  100. V. Harnik. Set existence axioms for general (not necessarily countable) stability theory. Annals of Pure and Applied Logic, 34:231–243, 1987.

    Article  Google Scholar 

  101. G. Hasenjaeger. Eine Bemerkung zu Henkins Beweis für die Vollständigkeit des Prädikatenkalküls der ersten Stufe. Journal of Symbolic Logic, 18:42–48, 1953.

    Article  Google Scholar 

  102. F. Hausdorff. Grundzüge der Mengenlehre. Veit, Leipzig, 1914.

    Google Scholar 

  103. J. van Heijenoort, editor. From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931. Harvard University Press, Cambridge, MA, 1967.

    Google Scholar 

  104. I. Heim. The Semantics of Definite and Indefinite Noun Phrases in English, Garland, New York, 1988.

    Google Scholar 

  105. L. Henkin. The completeness of the first-order functional calculus. Journal of Symbolic Logic, 14:159–166, 1949. Reprinted in [Hintikka, 1969].

    Article  Google Scholar 

  106. L. Henkin. Completeness in the theory of types. J. Symbolic Logic, 15:81–91, 1950. Reprinted in [Hintikka, 1969].

    Article  Google Scholar 

  107. L. Henkin. Some remarks on infinitely long formulas. In Infinitistic Methods: Proc. Symp. on Foundations of Mathematics, Warsaw, pages 167–183. Pergamon, London, 1961.

    Google Scholar 

  108. L. Henkin and A. Mostowski. Review of Mal’tsev [l94l]. Journal of Symbolic Logic, 24:55–57, 1959.

    Article  Google Scholar 

  109. J. Herbrand. Recherches sur la théorie de la démonstration. PhD thesis, University of Paris, 1930. Translated in [Herbrand, 1971, pp. 44–202].

    Google Scholar 

  110. J. Herbrand. Logical Writings. Harvard University Press, Cambridge, MA, 1971. Edited by W. D. Goldfarb.

    Book  Google Scholar 

  111. D. Hilbert. Grundlagen der Geometric Teubner, Leipzig, 1899.

    Google Scholar 

  112. D. Hilbert. Die logischen Grundlagen der Mathematik. Math Annalen, 88:151–165, 1923. Also in [Hilbert, 1970, pp. 178–195].

    Google Scholar 

  113. D. Hilbert. Über das Unendliche. Math Annalen, 95:161–190, 1926. Translated in [Heijenoort, 1967, pp. 367–392]; partial translation in [Benacerraf and Putnam, 1983, pp. 183–201].

    Article  Google Scholar 

  114. D. Hilbert. Die Grundlagen der Mathematik. Abhandlungen aus dem Math. Seminar der Hamburgischen Universität, 6:65–85, 1928. Translated in [Heijenoort, 1967, pp. 464–479].

    Google Scholar 

  115. D. Hilbert. Gesammelte Abhandlungen III: Analysis, Grundlagen der Mathematik, Physik, Verschiedenes. Springer, Berlin, 1970.

    Google Scholar 

  116. D. Hilbert and W. Ackermann. Grundzüge der theoretischen Logik. Springer, Berlin, 1928.

    Google Scholar 

  117. D. Hilbert and P. Bernays. Grundlagen der Mathematik I. Springer, Berlin, 1934.

    Google Scholar 

  118. D. Hilbert and P. Bernays. Grundlagen der Mathematik II. Springer, Berlin, 1939.

    Google Scholar 

  119. J. Hintikka. Distributive normal forms in the calculus of predicates. Acta Philosophica Fennica, 6, 1953.

    Google Scholar 

  120. J. Hintikka. Form and content in quantification theory. Acta Philosophica Fennica, 8:7–55, 1955.

    Google Scholar 

  121. J. Hintikka, editor. The Philosophy of Mathematics. Oxford University Press, 1969.

    Google Scholar 

  122. J. Hintikka. Logic, Language-games and Information. Oxford University Press, 1973.

    Google Scholar 

  123. J. Hintikka. The Principles of Mathematics Revisited, Cambridge University Press, Cambridge, 1996.

    Book  Google Scholar 

  124. W. Hodges. On order-types of models. Journal of Symbolic Logic, 37:69f, 1972.

    Google Scholar 

  125. W. Hodges. Logic. Penguin Books, Harmondsworth, Middx, 1977.

    Google Scholar 

  126. W. Hodges. Truth in a structure, Proceedings of Aristotelian Society, 86:135–151, 1985/6.

    Google Scholar 

  127. W. Hodges. Model Theory, Cambridge University Press, Cambridge, 1993.

    Book  Google Scholar 

  128. W. Hodges. Logical features of Horn clauses. In Handbook of Logic in Artificial Intelligence and Logic Programming, Volume 1: Logical Foundations, D. M. Gabbay, C. J. Hogger and J. A. Robinson, editors, pages 449–503. Clarendon Press, Oxford, 1993.

    Google Scholar 

  129. W. Hodges. A Shorter Model Theory, Cambridge University Press, Cambridge, 1997.

    Google Scholar 

  130. W. Hodges. Compositional semantics for a language of imperfect information, Logic Journal of the IGPL, 5:539–563, 1997.

    Article  Google Scholar 

  131. E. V. Huntington. The Continuum and Other Types of Serial Order, with an Introduction to Cantor’s Transfinite Numbers. Harvard University Press, Cambridge, MA, 1904.

    Google Scholar 

  132. R. C. Jeffrey. Formal Logic: its Scope and Limits. McGraw-Hill, New York, 1967.

    Google Scholar 

  133. P. N. Johnson-Laird and R. M. J. Byrne. Deduction. Lawrence Erlbaum Associates, Hove, 1991.

    Google Scholar 

  134. P. T. Johnstone. Topos Theory. Academic Press, London, 1977.

    Google Scholar 

  135. D. Kalish and R. Montague. Logic: Techniques of Formal Reasoning. Harcourt, Brace and World, New York, 1964.

    Google Scholar 

  136. L. Kalmar. Über die Axiomatisierbarkeit des Aussagenkalküls. Acta Scient. Math. Szeged, 7:222–243, 1934/5.

    Google Scholar 

  137. H. Kamp. Formal properties of ‘Now’. Theoria, 37:227–273, 1971.

    Article  Google Scholar 

  138. H. Kamp. A theory of truth and semantic representation. In J. A. G. Groenendijk et al., editor, Formal Methods in the Study of Language, pages 277–322. Math Centrum, Amsterdam, 1981.

    Google Scholar 

  139. H. Kamp and U. Reyle. From Discourse to Logic, Kluwer, Dordrecht, 1993.

    Google Scholar 

  140. D. Kaplan. What is Russell’s theory of descriptions? In Proceedings of Internat Colloquium on Logic, Physical Reality and History, Denver, 1966, pages 227–244. Plenum, New York, 1966. Reprinted in [Pears, 1972, pp. 227–244].

    Google Scholar 

  141. C. Karp. Finite quantifier equivalence. In J. Addison et al., editor, The Theory of Models. North-Holland, Amsterdam, 1965.

    Google Scholar 

  142. R. Kempson, editor. Bulletin of the IGPL, volume 3 numbers 2, 3: Special Issue on Deduction and Language, 1995.

    Google Scholar 

  143. S.C. Kleene. Recursive predicates and quantifiers. Trans Amer Math Soc, 53:41–73, 1943.

    Article  Google Scholar 

  144. S. C. Kleene. Introduction to Metamathematics. North-Holland, Amsterdam, 1952.

    Google Scholar 

  145. V. Klenk. Intended models and the Löwenheim-Skolem theorem. J. Philos. Logic, 5:475–489, 1976.

    Article  Google Scholar 

  146. W. Kneale. The province of logic. In H. D. Lewis, editor, Contemporary British Philosophy, 3rd Series, pages 237–261. George Allen and Unwin, London, 1956.

    Google Scholar 

  147. R. Kowalski. Logic for problem solving, North-Holland, New York, 1979.

    Google Scholar 

  148. G. Kreisel. Informal rigour and completeness proofs. In Lakatos, editor, Problems in the Philosophy of Mathematics, pages 138–157. North-Holland, Amsterdam, 1967. Partially reprinted in [Hintikka, 1969, pp. 78–94].

    Google Scholar 

  149. G. Kreisel and J. L. Krivine. Elements of Mathematical Logic (Model Theory). North-Holland, Amsterdam, 1967.

    Google Scholar 

  150. S. Kripke. Is there a problem about substitutional quantification? In G. Evans and J. McDowell, editors, Truth and Meaning: Essays in Semantics, pages 325–419. Clarendon Press, Oxford, 1976.

    Google Scholar 

  151. L. Kronecker. Grundzüge einer arithmetischen Theorie der algebraischen Grössen. Crelle’s Journal, 92:1–122, 1882.

    Google Scholar 

  152. G. Lakoff. Linguistics and natural logic. In D. Davidson and G. Harman, editors, Semantics of Natural Languages, pages 545–665. Reidei, Dordrecht, 1972.

    Chapter  Google Scholar 

  153. C. H. Langford. Some theorems on deducibility. Annals of Math, 28:16–40, 1927.

    Article  Google Scholar 

  154. A. C. Leisenring. Mathematical Logic and Hilbert’s e-symbol. Gordon and Breach, New York, 1969.

    Google Scholar 

  155. E. J. Lemmon. Beginning Logic. Nelson, London, 1965.

    Google Scholar 

  156. A. Levy. A hierarchy of formulas in set theory. Memoirs of the American Mathematical Society, 57, 1965.

    Google Scholar 

  157. A. Levy. Basic Set Theory. Springer, New York, 1979.

    Book  Google Scholar 

  158. m, 1969] P. Lindström. On extensions of elementary logic. Theoria, 35:1–11, 1969.

    Article  Google Scholar 

  159. P. Lorenzen. Ein dialogisches Konstruktivitätskriterium. In Infìnitistic Methods, Proc of a Symp on Foundations of Mathematics, Warsaw, pages 193–200, Pergamon, London, 1961.

    Google Scholar 

  160. P. Lorenzen. Metamathematik. Bibliographisches Institut, Mannheim, 1962.

    Google Scholar 

  161. P. Lorenzen and O. Schwemmer. Konstruktive Logic, Ethik und Wissenschaftstheorie. Bibliographisches Institut, Mannheim, 1975.

    Google Scholar 

  162. L. Löwenheim. Über Möglichkeiten im Relativkallkül. Math Annalen, 76:447–470, 1915. Translated in [Heijenoort, 1967, pp. 228–251].

    Article  Google Scholar 

  163. J. Lukasiewicz and A. Tarski. Untersuchungen über den Aussagenkalkül. Comptes Rendus des séances de la Société des Sciences et des Lettres de Varsovie, 23 cl. iii:30–50, 1930. Translated in [Tarski, 1983, pp. 38–59].

    Google Scholar 

  164. A. I. Mal’tsev. Untersuchungen aus dem Gebiete der Mathematischen Logik. Mat Sbornik, 1:323–336, 1936. Translated in [Mal’tsev, 1971, pp. 1–14].

    Google Scholar 

  165. A. I. Mal’tsev. On a general method for obtaining local theorems in group theory (Russian). Ivanov Gos. Ped. Inst. Uc. Zap. Fiz.-Mat. Fak., 1:3–9, 1941. Translated in [Mal’tsev, 1971, pp. 15–21].

    Google Scholar 

  166. tsev, 1971] A. I. Mal’cev. The Metamathematics of Algebraic Systems; Collected Papers 1936–1967. North-Holland, Amsterdam, 1971. Translated and edited by B. F. Wells III.

    Google Scholar 

  167. K. I. Manktelow and D. E. Over. Inference and Understanding, Routledge, London, 1990.

    Google Scholar 

  168. B. Mates. Elementary Logic. Oxford University Press, New York, 1965.

    Google Scholar 

  169. Members of the Johns Hopkins University, Boston. Studies in Logic. Little, Brown and Co, 1883.

    Google Scholar 

  170. E. Mendelson. Introduction to Mathematical Logic, Third edition. Van Nostrand, Princeton, NJ, 1964.

    Google Scholar 

  171. O. H. Mitchell. On a new algebra of logic. In Members of the Johns Hopkins University, Boston, Studies in Logic, pages 72–106. Little, Brown and Co, 1883.

    Chapter  Google Scholar 

  172. R. Montague. English as a formal language. In B. Visentini et al., editor, Linguaggi nella Società e nella Tecnica. Milan, 1970. Also in [Montague, 1974, pp. 188–221].

    Google Scholar 

  173. R. Montague. The proper treatment of quantification in ordinary English. In J. Hintikka et al., editor, Approaches to Natural Language. Reidel, Dordrecht, 1973. Also in [Montague, 1974, pp. 247–270].

    Google Scholar 

  174. R. H. Thomason, editor. Formal Philosophy, Selected Papers of Richard Montague, Yale University Press, New Haven, 1974.

    Google Scholar 

  175. R. Montague and R. L. Vaught. Natural models of set theory. Fundamenta Math, 47:219–242, 1959.

    Google Scholar 

  176. G. H. Moore. Beyond first-order logic: the historical interplay between mathematical logic and axiomatic set theory. History and Philosophy of Logic, 1:95–137, 1980.

    Article  Google Scholar 

  177. G. V. Morrill. Type Logical Grammar: Categorial Logic of Signs. Kluwer, Dordrecht, 1994.

    Book  Google Scholar 

  178. R. E. Nisbett, G. T. Fong, D. R Lehman and P. W. Cheng. Teaching reasoning. Science, 238:625–631, 1987.

    Article  Google Scholar 

  179. P. Padawitz. Computing in Horn Clause Theories. Springer, Berlin, 1988.

    Book  Google Scholar 

  180. B. Partee. Bound variables and other anaphors. In D. Waltz, editor, Tinlap-2, Theoretical Issues in Natural Language Processing, pages 248–280. Association for Computing Machinery, New York, 1978.

    Google Scholar 

  181. G. Peano. Arithmetices Principia, Nova Methodo Exposita. Turin, 1889. Translation in [Heijenoort, 1967, pp. 85–97].

    Google Scholar 

  182. D. F. Pears, editor. Bertrand Russell. A Collection of Critical Essays. Anchor Books, Doubleday, New York, 1972.

    Google Scholar 

  183. C. S. Peirce. A theory of probable inference. Note B. The logic of relatives. In Boston Members of the Johns Hopkins University, editor, Studies in Logic. Little, Brown and Co, 1883. Reprinted in [Peirce, 1933, Vol III, pp. 195–209].

    Google Scholar 

  184. C. S. Peirce. On the algebra of logic. Amer. J. Math., 7:180–202, 1885. Reprinted in [Peirce, 1933, Vol. III, pp. 210–238].

    Google Scholar 

  185. C. S. Peirce. The simplest mathematics. In C. Hartshorne et al., editor, Collected Papers of Charles Sanders Peirce, volume IV, pages 189–262. Harvard University Press, Cambridge, MA, 1902.

    Google Scholar 

  186. C. S. Peirce. In C. Hartshorne et al., editor, Collected Papers of Charles Sanders Peirce. Harvard University Press, Cambridge, MA, 1933.

    Google Scholar 

  187. J. Perry. Frege on demonstratives. Philosophical Review, 86:474–497, 1977. Reprinted in P. Yourgram, editor, Demonstratives, pages 50–70, Oxford University Press, New York, 1990.

    Article  Google Scholar 

  188. K. R. Popper. Logic without assumptions. Proc. Aristot. Soc, pages 251–292, 1946/47.

    Google Scholar 

  189. E. Post. Introduction to a general theory of elementary propositions. American Journal of Mathematics, 43:163–185, 1921. Reprinted in [Heijenoort, 1967, pp. 264–283].

    Article  Google Scholar 

  190. D. Prawitz. Natural Deduction: a Proof-theoretical Study. Almqvist and Wiksell, Stockholm, 1965.

    Google Scholar 

  191. D. Prawitz. Proofs and the meaning and the completeness of the logical constants. In J. Hintikka, I. Niiniluoto, and E. Saarinen, editors, Essays on Mathematical and Philosophical Logic, pages 25–40. Reidel, Dordrecht, 1979.

    Chapter  Google Scholar 

  192. A. N. Prior. The runabout inference ticket. Analysis, 21:38–39, 1960. Reprinted in [Strawson, 1967, pp. 129–131].

    Article  Google Scholar 

  193. A. N. Prior. Formal Logic. Oxford University Press, 1962.

    Google Scholar 

  194. H. Putnam. Models and reality. Journal of Symbolic Logic, 45:464–482, 1980. Reprinted in [Benacerraf, 1983; pp. 421–444].

    Article  Google Scholar 

  195. W. V. Quine. Mathematical Logic. Harvard University Press, Cambridge, MA, 1940. Revised edition 1951.

    Google Scholar 

  196. W. V. Quine. Methods of Logic. Holt, New York, 1950.

    Google Scholar 

  197. W. V. Quine. Philosophy of Logic. Prentice-Hall, Englewood Cliffs, NJ, 1970.

    Google Scholar 

  198. R. Quirk and S. Greenbaum. A University Grammar of English. Longman, London, 1973.

    Google Scholar 

  199. H. Rasiowa and R. Sikorski. A proof of the completeness theorem of Gödel. Fundamenta Math, 37:193–200, 1950.

    Google Scholar 

  200. H. Rasiowa and R. Sikorski. The Mathematics of Metamathematics. Monografie Matematyczne, Polska Akad. Nauk, 1963.

    Google Scholar 

  201. S. Reeves and M. Clarke. Logic for Computer Science. Addison-Wesley, 1990.

    Google Scholar 

  202. L. J. Rips. The Psychology of Proof. MIT Press, Cambridge Mass., 1994.

    Google Scholar 

  203. A. Robinson. The metaphysics of the calculus. In Lakatos, editor, Problems in the Philosophy of Mathematics, pages 28–40. North-Holland, Amsterdam, 1967. Reprinted in [Hintikka, 1969, pp. 153–163], and in Selected papers of Abraham Robinson, Vol. 2, edited by H. J. Keisler et al., pp. 537–555. Yale University Press, New Haven, 1979.

    Chapter  Google Scholar 

  204. B. Russell. On denoting. Mind, 14:479–493, 1905. Reprinted in [Russell, 1956].

    Article  Google Scholar 

  205. B. Russell. In R. C. Marsh, editor, Logic and Knowledge, Essays 1901–1950. George Allen and Unwin, London, 1956.

    Google Scholar 

  206. G. E. Sacks. Saturated Model Theory. Benjamin, Reading, MA, 1972.

    Google Scholar 

  207. H. A. Schmidt. Über deduktive Theorien mit mehreren Sorten von Grunddingen. Math Annalen, 115:485–506, 1938.

    Article  Google Scholar 

  208. der, 1895] E. Schröder. Vorlesungen über die Algebra der Logik, volume 3. Leipzig, 1895.

    Google Scholar 

  209. tte, 1956] K. Schütte. Ein System des verknüpfenden Schliessens. Arch. Math. Logik Grundlagenforschung, 2:55–67, 1956.

    Article  Google Scholar 

  210. tte, 1977] K. Schütte. Proof Theory. Springer, Berlin, 1977. Translated by J. N. Crossley.

    Book  Google Scholar 

  211. W. R. Scott. Group Theory. Prentice-Hall, Englewood Cliffs, NJ, 1964.

    Google Scholar 

  212. D. J. Shoesmith and T. J. Smiley. Multiple-Conclusion Logic. Cambridge University Press, 1978.

    Book  Google Scholar 

  213. T. Skolem. Untersuchungen über die Axiome des Klassenkalküls und über Produktations- und Summationsprobleme, welche gewisse von Aussagen betreffen. Videnskapsselskapets Skrifter, I. Matem.-naturv. klasse, no 3, 1919. Reprinted in [Skolem, 1970, pp. 67–101].

    Google Scholar 

  214. T. Skolem. Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen. Videnskapsselskapets Skrifter, I. Matem.-Naturv. Klasse 4, 1920. Reprinted in [Skolem, 1970, pp. 103–136]; partial translation in [Heijenoort, 1967, pp. 252–263].

    Google Scholar 

  215. T. Skolem. Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre. Matematikerkongressen i Helsingfors den 4–7 Juli 1922, 1922. Reprinted in [Skolem, 1970, pp. 137–152]; translation in [Heijenoort, 1967, pp. 290–301].

    Google Scholar 

  216. T. Skolem. Begründung der elementaren Arithmetik durch die rekurrierende Denkweise ohne Anwendung scheinbarer Veränderlichen mit unendlichem Ausdehnungsbereich. Videnskapsselskapets Skrifter I, Matem.-naturv. Klasse 6, 1923. Translation in [Heijenoort, 1967, pp. 303–333].

    Google Scholar 

  217. T. Skolem. Über die mathematische Logik. Norsk. Mat. Tidsk., 10:125–142, 1928. Reprinted in [Skolem, 1970, pp. 189–206]; translation in [Heijenoort, 1967, pp. 513–524].

    Google Scholar 

  218. T. Skolem. Über einige Grundlagenfragen der Mathematik. Skr. Norsk. Akad. Oslo I Mat.-Natur Kl 4, pages 1–49, 1929. Reprinted in [Skolem, 1970, pp. 227–273].

    Google Scholar 

  219. T. Skolem. Über die Nichtcharakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen. Fundamenta Math, 23:150–161, 1934. Reprinted in [Skolem, 1970, pp. 355–366].

    Google Scholar 

  220. T. Skolem. A critical remark on foundational research. Kongelige Norsk. Vidensk. Forhand. Trondheim, 28:100–105, 1955. Reprinted in [Skolem, 1970, pp. 581–586].

    Google Scholar 

  221. T. Skolem. In Selected Works in Logic. J. E. Fenstad, editor, Universitetsforlaget, Oslo, 1970.

    Google Scholar 

  222. R. Smullyan. First-Order Logic. Springer, Berlin, 1968.

    Book  Google Scholar 

  223. J. D. Sneed. The Logical Structure of Mathematical Physics. Reidel, Dordrecht, 1971.

    Book  Google Scholar 

  224. ller, 1976] W. Stegmüller. The Structure and Dynamics of Theories. Springer, New York, 1976.

    Google Scholar 

  225. M. Steiner. Mathematical Knowledge. Cornell University Press, Ithaca, 1975.

    Google Scholar 

  226. K. Stenning, R. Cox and J. Oberlander. Contrasting the cognitive effects of graphical and sentential logic teaching: reasoning, representation and individual differences, Language and Cognitive Processes, 10:333–354, 1995.

    Article  Google Scholar 

  227. L. Stevenson. Frege’s two definitions of quantification. Philos. Quarterly, 23:207–223, 1973.

    Article  Google Scholar 

  228. P. F. Strawson, editor. Philosophical Logic. Oxford University Press, 1967.

    Google Scholar 

  229. P. Suppes. Introduction to Logic. Van Nostrand, Princeton, NJ, 1957.

    Google Scholar 

  230. P. Suppes. Axiomatic Set Theory. Dover, NY, 1972.

    Google Scholar 

  231. A. Tarski. Der Wahrheitsbegriff in den formalisierten Sprachen, based on a paper in Ruch. Filozoficzny xii (1930/1), 1935. Translated in [Tarski, 1983, pp. 152–278].

    Google Scholar 

  232. A. Tarski. O pojciu wynikania logicznego. Przeglad Filozoficzny, 39:58–68, 1936. Translated as ‘On the concept of logical consequence’ in [Tarski, 1983, pp. 409–420].

    Google Scholar 

  233. A. Tarski. Contributions to the theory of models I, II. Indag. Math, 16:572–588, 1954.

    Google Scholar 

  234. A. Tarski. Logic, Semantics, Metamathematics, Papers from 1923 to 1938. Hackett, Indianapolis, 1983. Translated by J. H. Woodger with analytical index by J. Corcoran.

    Google Scholar 

  235. A. Tarski and S. Givant. A Formalization of Set Theory without Variables. American Mathematical Society, Providence RI, 1987.

    Google Scholar 

  236. A. Tarski and R. L. Vaught. Arithmetical extensions of relational systems. Compositio Math, 13:81–102, 1956.

    Google Scholar 

  237. A. Tarski, A. Mostowski, and R. M. Robinson. Undecidable Theories. North-Holland, Amsterdam, 1953.

    Google Scholar 

  238. N. W. Tennant. Natural Logic. Edinburgh University Press, 1978.

    Google Scholar 

  239. R. H. Thomason. Symbolic Logic, An Introduction. Macmillan, London, 1970.

    Google Scholar 

  240. R. L. Vaught. Model theory before 1945. In L. Henkin et al, editor, Proceedings of the Tarski Symposium, pages 153–172. AMS, Providence, RI, 1974.

    Chapter  Google Scholar 

  241. J. Von Neumann. Eine Axiomatisierung der Mengenlehre. J. für die Reine und Angew Math, 154:219–240, 1925. Translated in [Heijenoort, 1967, pp. 393–413].

    Google Scholar 

  242. H. Wang. Logic of many-sorted theories. Journal of Symbolic Logic, 17:105–116, 1952.

    Article  Google Scholar 

  243. H. Wang. A survey of Skolem’s work in logic, 1970. In [Skolem, 1970, pp. 17–52].

    Google Scholar 

  244. H. Wang. From Mathematics to Philosophy. Routledge and Kegan Paul, NY, 1974.

    Google Scholar 

  245. P. C. Wason. Reasoning. In New Horizons in Psychology, B. Foss, ed., pages 135–151. Penguin, Harmondsworth, 1966.

    Google Scholar 

  246. A. N. Whitehead and B. Russell. Principia Mathematica I. Cambridge University Press, 1910. Up to to 56*, reprinted 1962.

    Google Scholar 

  247. J. E. Wiredu. Deducibility and inferability. Mind, 82:31–55, 1973.

    Article  Google Scholar 

  248. L. Wittgenstein. Tractatus Logico-Philosophicus. Annalen der Naturphilosophie, 1910. Reprinted with translation by D. F. Pears and B. F. McGuin-ness, Routledge and Kegan Paul, London, 1961.

    Google Scholar 

  249. E. Zermelo. Untersuchungen über die Grundlagen der Mengenlehre I. Math Annalen, 65:261–281, 1908. Translated in [Heijenoort, 1967, pp. 199–215].

    Article  Google Scholar 

  250. J. Zucker. The correspondence between cut-elimination and normalisation. Annals of Math Logic, 7:1–156, 1974.

    Article  Google Scholar 

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  1. Wilfrid Hodges
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  1. King’s College, London, UK

    D. M. Gabbay

  2. Centrum für Informations- und Sprachverarbeitung, Ludwig-Maximilians-Universität München, Germany

    F. Guenthner

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Hodges, W. (2001). Elementary Predicate Logic. In: Gabbay, D.M., Guenthner, F. (eds) Handbook of Philosophical Logic. Handbook of Philosophical Logic, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9833-0_1

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