Abstract
What is nowadays the central part of any introduction to logic, and indeed to some the logical theory par excellence, used to be a modest fragment of the more ambitious language employed in the logicist program of Frege and Russell. ‘Elementary’ or ‘first-order’, or ‘predicate logic’ only became a recognized stable base for logical theory by 1930, when its interesting and fruitful meta-properties had become clear, such as completeness, compactness and Löwenheim-Skolem. Richer higher-order and type theories receded into the background, to such an extent that the (re-)discovery of useful and interesting extensions and variations upon first-order logic came as a surprise to many logicians in the sixties.
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References
Ackermann, W.: 1968, Solvable Cases of the Decision Problem, North-Holland, Amsterdam.
Ajtai, M.: 1979, ‘Isomorphism and higher-order equivalence’, Annals of Math. Logic 16, 181–203.
Barendregt, H.: 1980, The Lambda Calculus, North-Holland, Amsterdam.
Barwise, K. J.: 1972, ‘The Hanf-number of second-order logic’. J. Symbolic Logic 37, 588–594.
Barwise, K. J.: 1975, Admissible Sets and Structures, Springer, Berlin.
Barwise, K. J. (ed.): 1977, Handbook of Mathematical Logic, North-Holland, Amsterdam.
Barwise, K. J.: 1979, ‘On branching quantifiers in English’, J. Philos. Logic 8, 47–80.
Barwise, K. J. and Cooper, R.: 1981, ‘Generalized quantifiers and natural language’, Linguistics and Philosophy 4, 159–219.
Barwise, K.J. and Schlipf, J.: 1976, ‘An introduction to recursively saturated and resplendent models’, J. Symbolic Logic 41, 531–536.
Barwise, K. J., Kaufman, M. and Makkai, M.: 1978, ‘Stationary logic’, Annals of Math. Logic 13, 171–224. A correction appeared in Annals of Math. Logic 16, 231–232.
Bell, J. L. and Slomson, A. B.: 1969, Models and Ultraproducts, North-Holland, Amsterdam.
Chang, C. C. and Keisler, H. J.: 1973, Model Theory, North-Holland, Amsterdam.
Church, A.: 1940, ‘A formulation of the simple theory of types’, J. Symbolic Logic 5, 56–68.
Copi, I. M.: 1971, The Logical Theory of Types, Routledge & Kegan Paul, London.
Drake, F. R.: 1974, Set Theory. An Introduction to Large Cardinals, North-Holland, Amsterdam.
Enderton, H. B.: 1970, ‘Finite partially-ordered quantifiers’, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 16, 393–397.
Enderton, H. B.: 1972, A Mathematical Introduction to Logic, Academic Press, New York.
Gallin, D.: 1975, Intensional and Higher-Order Modal Logic, North-Holland, Amsterdam.
Garland, S. J.: 1974, ‘Second-order cardinal characterizability’, in Proceedings of Symposia in Pure Mathematics, A.M.S., vol. 13, part II, 127–146.
Henkin, L. A.: 1950, ‘Completeness in the theory of types’, J. Symbolic Logic 15, 81–91.
Henkin, L. A.: 1961, ‘Some remarks on infinitely long formulas’, in Infinitistic Methods. Proceedings of a Symposium on the Foundations of Mathematics, Pergamon Press, London, pp. 167–183.
Henkin, L. A.: 1963, ‘A theory of propositional types’, Fundamenta Math. 52, 323–344.
Hintikka, K. J. J.: 1955, ‘Reductions in the theory of types’, Acta Philosophica Fennica 8, 61–115.
Hintikka, K. J. J.: 1973, ‘Quantifiers versus quantification theory’, Dialectica 27, 329–358.
Hodges, W.: 1983, Elementary predicate logic, Chapter I.1, this Handbook.
Keisler, H. J.: 1971, Model Theory for Infinitary Logic, North-Holland, Amsterdam.
Kemeny, J.: 1950, ‘Type theory vs. set theory’, J. Symbolic Logic 15, 78.
Kleene, S. C.: 1952, ‘Finite axiomatizability of theories in the predicate calculus using additional predicate symbols’, in Two Papers on the Predicate Calculus, Memoirs of the Amer. Math. Soc. vol. 10, pp. 27–68.
Kunen, K.: 1971, ‘Indescribability and the continuum’, in Proceedings of Symposia in Pure Mathematics, A.M.S., vol. 13, part I, 199–204.
Lindström, P.: 1969, ‘On extensions of elementary logic’, Theoria 35, 1–11.
Magidor, M.: 1971, ‘On the role of supercompact and extendible cardinals in logic’, Israel J. Math. 10, 147–157.
Magidor, M. and Malitz, J.: 1977, ‘Compact extensions of LQ’, Annals of Math. Logic 11, 217–261.
Monk, J. D.: 1976, Mathematical Logic, Springer, Berlin.
Montague, R.: 1974, in R. H. Thomason (ed.), Formal Philosophy: Selected Papers of Richard Montague, Yale University Press, New Haven.
Moschovakis, Y. N.: 1980, Descriptive Set Theory, North-Holland, Amsterdam.
Myhill, J. and Scott, D. S.: 1971, ‘Ordinal definability’, in Proceedings in Symposia in Pure Mathematics, A.M.S., vol. 13, part I, 271–278.
Orey, S.: 1959, ‘Model theory for the higher-order predicate calculus’, Transactions of the American Mathematical Society 92, 72–84.
Rabin, M. O.: 1969, ‘Decidability of second-order theories and automata on infinite trees’, Transactions of the American Mathematical Society 141, 1–35.
Ressayre, J. P.: 1977, ‘Models with compactness properties relative to an admissible language’, Annals of Math. Logic 11, 31–55.
Svenonius, L.: 1965, ‘On the denumerable models of theories with extra predicates’, in The Theory of Models, North-Holland, Amsterdam, pp. 376–389.
Väänänen, J.: 1982, ‘Abstract logic and set theory: II Large cardinals’, J. Symbolic Logic 47, 335–346.
Van Benthem, J. F. A. K.: 1977, ‘Modal logic as second-order logic’, report 77-04, Mathematisch Instituut, University of Amsterdam.
Van Benthem, J. F. A. K.: 1982, Questions about Quantifiers, to appear in J. Symbolic Logic.
Van Benthem, J. F. A. K.: 1983, Modal Logic and Classical Logic, Bibliopolis, Naples.
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Van Benthem, J., Doets, K. (1983). Higher-Order Logic. In: Gabbay, D., Guenthner, F. (eds) Handbook of Philosophical Logic. Synthese Library, vol 164. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-7066-3_4
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