Higher-Order Logic

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 Lowenheim-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.

[1]  Alonzo Church,et al.  A formulation of the simple theory of types , 1940, Journal of Symbolic Logic.

[2]  Leon Henkin,et al.  Completeness in the theory of types , 1950, Journal of Symbolic Logic.

[3]  Review: Hao Wang, On Zermelo's and von Neumann's Axioms for Set Theory , 1950 .

[4]  Wilhelm Ackermann,et al.  Solvable Cases Of The Decision Problem , 1954 .

[5]  S. Orey.  Model theory for the higher order predicate calculus , 1959 .

[6]  L. Henkin.  A theory of prepositional types , 1963 .

[7]  Perlindström.  First Order Predicate Logic with Generalized Quantifiers , 1966 .

[8]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[9]  M. Rabin.  Decidability of second-order theories and automata on infinite trees. , 1969 .

[10]  H. Enderton.  Finite Partially-Ordered Quantifiers , 1970 .

[11]  J. Bell,et al.  Models and ultraproducts , 1971 .

[12]  Menachem Magidor.  On the role of supercompact and extendible cardinals in logic , 1971 .

[13]  H. Keisler.  Model theory for infinitary logic , 1971 .

[14]  Herbert B. Enderton,et al.  A mathematical introduction to logic , 1972 .

[15]  K. Jon Barwise,et al.  The Hanf number of second order logic , 1972, Journal of Symbolic Logic.

[16]  Jaakko Hintikka,et al.  Quantifiers vs. Quantification Theory , 1973 .

[17]  F. R. Drake,et al.  Set theory : an introduction to large cardinals , 1974 .

[18]  R. Montague.  Formal philosophy; selected papers of Richard Montague , 1974 .

[19]  G. Boolos.  On second-order logic , 1975 .

[20]  Daniel Gallin,et al.  Intensional and Higher-Order Modal Logic , 1975 .

[21]  Jon Barwise,et al.  Admissible sets and structures , 1975 .

[22]  M. H. Lob,et al.  Embedding First Order Predicate Logic in Fragments of Intuitionistic Logic , 1976, J. Symb. Log..

[23]  K. Jon Barwise,et al.  An introduction to recursively saturated and resplendent models , 1976, Journal of Symbolic Logic.

[24]  J. Ressayre.  Models with compactness properties relative to an admissible language , 1977 .

[25]  J. F. A. K. van Benthem.  Modal logic as second-order logic , 1977 .

[26]  J. Malitz,et al.  Compact extensions of L(Q) (part 1a) , 1977 .

[27]  K. Jon Barwise,et al.  On branching quantifiers in English , 1979, J. Philos. Log..

[28]  M. Ajtai.  Isomorphism and higher order equivalence , 1979 .

[29]  Y. Moschovakis.  Descriptive Set Theory , 1980 .

[30]  J. Barwise,et al.  Generalized quantifiers and natural language , 1981 .

[31]  D. Gabbay.  Semantical investigations in Heyting's intuitionistic logic , 1981 .

[32]  Jouko A. Väänänen.  Abstract Logic and Set Theory. II. Large Cardinals , 1982, J. Symb. Log..

[33]  Hartry Field,et al.  Science without Numbers , 1983 .

[34]  W. Hodges.  Elementary Predicate Logic , 1983 .

[35]  Johan van Benthem,et al.  The Logic of Time , 1983 .

[36]  George Boolos,et al.  To Be Is to Be a Value of a Variable (or to Be Some Values of Some Variables) , 1984 .

[37]  Johan van Benthem,et al.  Questions About Quantifiers , 1984, J. Symb. Log..

[38]  Ian A. Mason.  The Metatheory of the Classical Propositional Calculus is not Axiomatizable , 1985, J. Symb. Log..

[39]  Johan van Benthem,et al.  The Ubiquity of Logic in Natural Language , 1986 .

[40]  J. Benthem.  Essays in Logical Semantics , 1986 .

[41]  J. Roger Hindley,et al.  Introduction to Combinators and Lambda-Calculus , 1986 .

[42]  Harold T. Hodes,et al.  The | lambda-Calculus. , 1988 .

[43]  Dag Westerstaåhl,et al.  Quantifiers in Formal and Natural Languages , 1989 .

[44]  Reinhard Muskens,et al.  A relational formulation of the theory of types , 1989 .

[45]  Paris C. Kanellakis,et al.  Elements of Relational Database Theory , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[46]  Hans Jürgen Ohlbach,et al.  Semantics-Based Translation Methods for Modal Logics , 1991, J. Log. Comput..

[47]  Yde Venema,et al.  Many-dimensional Modal Logic , 1991 .

[48]  Victor Manual Sánchez Valencia,et al.  Studies on natural logic and categorial grammar , 1991 .

[49]  Michael Zakharyaschev,et al.  Canonical formulas for K4. Part I: Basic results , 1992, Journal of Symbolic Logic.

[50]  Serge Lapierre.  A Functional Partial Semantics for Intensional Logic , 1992, Notre Dame J. Formal Log..

[51]  Andrew M. Pitts,et al.  On an interpretation of second order quantification in first order intuitionistic propositional logic , 1992, Journal of Symbolic Logic.

[52]  François Lepage.  Partial Functions in Type Theory , 1992, Notre Dame J. Formal Log..

[53]  Wilfrid Hodges,et al.  Model Theory: The existential case , 1993 .

[54]  M. de Rijke.  Extending modal logic , 1993 .

[55]  Philip Kremer,et al.  Quantifying over propositions in relevance logic: nonaxiomatisability of primary interpretations of ∀p and ∃p , 1993, Journal of Symbolic Logic.

[56]  Daniel Leivant,et al.  Higher order logic , 1994, Handbook of Logic in Artificial Intelligence and Logic Programming.

[57]  Marcin Mostowski,et al.  Quantifiers, Some Problems and Ideas , 1995 .

[58]  J.F.A.K. van Benthem,et al.  Language in Action: Categories, Lambdas and Dynamic Logic , 1997 .

[59]  M. Mostowski.  Quantifiers Definable by Second Order Means , 1995 .

[60]  Jörg Flum,et al.  Finite model theory , 1995, Perspectives in Mathematical Logic.

[61]  Johan van Benthem,et al.  Directions in generalized quantifier theory , 1995, Stud Logica.

[62]  Kees Doets,et al.  Basic model theory , 1996, Studies in logic, language and information.

[63]  Helmut Schwichtenberg,et al.  Basic proof theory , 1996, Cambridge tracts in theoretical computer science.

[64]  Leon Henkin,et al.  The Discovery of My Completeness Proofs , 1996, Bulletin of Symbolic Logic.

[65]  Michael Zakharyaschev,et al.  Canonical formulas for K4. Part II: Cofinal subframe logics , 1996, Journal of Symbolic Logic.

[66]  John C. Mitchell,et al.  Foundations for programming languages , 1996, Foundation of computing series.

[67]  Johan van Benthem.  Content versus wrapping: an essay in semantic complexity , 1997 .

[68]  Maarten Marx,et al.  Multi-dimensional modal logic , 1997, Applied logic series.

[69]  Godehard Link.  Algebraic semantics in language and philosophy , 1997 .

[70]  Dag Westerståhl,et al.  Generalized Quantifiers in Linguistics and Logic , 1997, Handbook of Logic and Language.

[71]  Jan Tore Lønning,et al.  Plurals and Collectivity , 1997, Handbook of Logic and Language.

[72]  Johan van Benthem,et al.  Interpolation, preservation, and pebble games , 1999, Journal of Symbolic Logic.

[73]  Jerzy Tiuryn,et al.  Dynamic logic , 2001, SIGA.