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In Praise of Impredicativity: A Contribution to the Formalization of Meta-Programming

Published online by Cambridge University Press:  25 February 2019

FRANÇOIS BRY
Affiliation:
Institute for Informatics, Ludwig-Maximilian University of Munich, Germany (e-mail: bry@lmu.de)
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Abstract

Processing programs as data is one of the successes of functional and logic programming. Higher-order functions, as program-processing programs are called in functional programming, and meta-programs, as they are called in logic programming, are widespread declarative programming techniques. In logic programming, there is a gap between the meta-programming practice and its theory: The formalizations of meta-programming do not explicitly address its impredicativity and are not fully adequate. This article aims at overcoming this unsatisfactory situation by discussing the relevance of impredicativity to meta-programming, by revisiting former formalizations of meta-programming, and by defining Reflective Predicate Logic, a conservative extension of first-order logic, which provides a simple formalization of meta-programming.

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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Footnotes

*

The author is thankful to Norbert Eisinger, Bob Kowalski, and Antonius Weinzierl for fruitful discussions on the subject of this article. The author acknowledges useful hints from the journal area editor Michael J. Maher, from the anonymous reviewers, and from the audience of the 21st International Conference on Applications of Declarative Programming and Knowledge Management (INAP 2017) during which part of the work reported about in this article has been presented in a talk without associated publication. The author is thankful to Norbert Eisinger and Elke Kroiß for their help in correcting typos and stylistic lapses in drafts of this article.

References

Aczel, P. 1977. An introduction to inductive definitions. In Handbook of Mathematical Logic, Barwise, Jon, Ed. North-Holland, 739782.Google Scholar
Aiello, L. C., Cecchi, C. and Sartini, D. 1986. Representation and use of meta-knowledge. Proceedings of the IEEE 74, 10, 13041321.10.1109/PROC.1986.13633CrossRefGoogle Scholar
Apt, K. R. and Ben-Eliyahu, R. 1996. Meta-variables in logic programming, or in praise of ambivalent syntax. Fundamenta Informaticae 28, 1, 2336.CrossRefGoogle Scholar
Apt, K. R. and van Emden, M. 1982. Contributions to the theory of logic programming. Journal of the Association for Computating Machinery 29, 841862.10.1145/322326.322339CrossRefGoogle Scholar
Backus, J. W. 1959. The syntax and semantics of the proposed international algebraic language of Zürich ACM-GAMM conference. In Proc. of the International Conference on Information Processing, Jun. 1959, Veronese, V., Ed. Paris, France, UNESCO (Paris, France), R. Oldenbourg (München, Germany) and Butterworths (London, Great Britain), 125132.Google Scholar
Barendregt, H. 1993. Lambda Calculi with types. In Handbook of Logic in Computer Science. Vol. 2, Background: Computational Structures. Oxford University Press, 117309.Google Scholar
Barklund, J. 1989. What is a meta-variable in Prolog? In Meta-Programming in Logic Programming, Abramson, Harvey and Rogers, M. H., Eds. MIT Press, 383398.Google Scholar
Barklund, J. 1995. Metaprogramming in logic. Encyclopedia of Computer Science and Technology 33, 205227. Supplement 18: Case-Based Reasoning to User Interface Software Tools.Google Scholar
Barklund, J., Costantini, S., Dell’Acqua, P. and Lanzarone, G. A. 1994. SLD-resolution with reflection. In Logic Programming, Proceedings of the 1994 International Symposium, Bruynooghe, M., Ed. Melbourne, Australia, 554568.Google Scholar
Barklund, J., Costantini, S., Dell’Acqua, P. and Lanzarone, G. A. 1995a. Semantical properties of encodings in logic programming. In Logic Programming – The 1995 International Symposium, Lloyd, J. W., Ed. MIT Press, Portland, Oregon, USA, 288302.Google Scholar
Barklund, J., Dell’Acqua, P., Costantini, S. and Lanzarone, G. A. 1995b. Semantical properties of SLD-resolution with reflection. In Logic Programming, Proceedings of the Twelfth International Conference on Logic Programming, Sterling, L., Ed. MIT Press, Tokyo, Japan, 830.Google Scholar
Barklund, J., Dell’Acqua, P., Costantini, S. and Lanzarone, G. A. 2000. Reflection principles in computational logic. Journal of Logic and Computation 10, 743786.Google Scholar
Benjamin, P. 1990. A meta level manifesto. In Machine Learning, Meta-reasoning and Logics, Brazdil, P. B. and Konolige, K., Eds. Kluwer Academic Publishers, 317.CrossRefGoogle Scholar
Bezem, M. 1999. Extensionality of simply typed logic programs. In Proc. of the 16th International Conference on Logic Programming (ICLP), de , D.Schreye, Ed. MIT Press, 395410.Google Scholar
Bezem, M. 2001. An improved extensionality criterion for higher-order logic programs. In Computer Science Logic, 15th International Workshop, CSL. Vol. 2142, LNCS. Fribourg, L., Ed. Springer-Verlag, Paris, France, 203216.CrossRefGoogle Scholar
Böhm, C. and Jacopini, G. 1966. Flow diagrams, Turing machines and languages with only two formation rules. Communications of the ACM, 9, 5, 366371.CrossRefGoogle Scholar
Boole, G. 1854. An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities. Macmillan. Reprinted by Dover Publications, New York, 1958, and by Cambridge University Press, 2009.Google Scholar
Bowen, K. and Kowalski, R. 1982. Amalgamating language and metalanguage in logic programming. In Logic Programming, Clark, K. and Tärnlund, S. A., Eds. Academic Press, 153173.Google Scholar
Bowen, K. A. 1985. Meta-level programming and knowledge representation. New Generation Computing 3, 359383.CrossRefGoogle Scholar
Bowen, K. A. and Weinberg, T. 1985. A meta-level extension of Prolog. In Proc. of the IEEE Symposium on Logic Programming, Cohen, J. and Conery, J., Eds. IEEE, Boston, Massachusetts, USA, 669675.Google Scholar
Burgess, J. P. 2005. Fixing Frege. Princeton University Press, Princeton, NJ, USA.Google Scholar
Cardelli, L. 2004. Type systems. In CRC Handbook of Computer Science and Engineering, Tucker, A. B., Ed. CRC Press, 22082236.Google Scholar
Cardelli, L. and Wegner, P. 1985. On understanding types, data abstraction, and polymorphism. ACM Computing Surveys 17, 4 (December), 471522.Google Scholar
Cervesato, I. and Rossi, G. 1992. Logic meta-programming facilities in ‘LOG. Research Showcase @ CMU 6-1992, Carnegie Mellon University, School of Computer Science, Computer Science Department.Google Scholar
Chang, C.-L. and Lee, R. C.-T. 1997. Symbolic Logic and Mechanical Theorem Proving. Academic Press.Google Scholar
Charalambidis, A., Handjopoulos, K., Rondogiannis, P. and Wadge, william L.. 2013. Extensional higher-order logic programming. ACM Transactions on Computational Logic 14, 3, 91103.CrossRefGoogle Scholar
Chen, W. 1987. A theory of modules based on second-order logic. In Proc. IEEE Symposium on Logic Programming, IEEE Computer Society, San Francisco, California, USA, 2433.Google Scholar
Chen, W., Kifer, M. and Warren, D. S. 1993. HiLog: A foundation for higher-order logic programming. Journal of Logic Programming 15, 3, 187230.CrossRefGoogle Scholar
Church, A. 1932. A set of postulates for the foundation of logic. Annals of Mathematics – Series 233, 2, 346366.CrossRefGoogle Scholar
Church, A. 1940. A formulation of the simple theory of types. Journal of Symbolic Logic 5, 5668.CrossRefGoogle Scholar
Chwistek, L. 1921. Antynomje logikiformalnej. Przeglad Filozoficzny 24, 164171. In Polish, English translation: Chwistek (1967).Google Scholar
Chwistek, L. 1967. Antinomies of formal logic. In Polish Logic: 1920–1939, McCall, S., Ed. Clarendon Press, 338345. English translation by Z. Jordan of Chwistek (1921).Google Scholar
Clocksin, W. F. and Mellish, C. S. 1981. Programming in Prolog. Springer-Verlag.Google Scholar
Costantini, S. 2002. Meta-Reasoning: A survey. In Computational Logic: Logic Programming and Beyond (Festschrift in honour of Robert Kowalski), Kakas, A. C. and Sadri, F., Eds. Vol. 2408, LNCS. Springer-Verlag, 254288.Google Scholar
Costantini, S. and Lanzarone, G. A. 1989. A metalogic programming language. In Proc. of the International Conference on Logic Programming, Jun. 1989, Levi, G. and Martelli, M., Ed. Lisbon, Portugal, 218233.Google Scholar
Costantini, S. and Lanzarone, G. A. 1994. A metalogic programming approach: Language, semantics and applications. Journal of Experimental & Theoretical Artificial Intelligence 6, 3, 239287.CrossRefGoogle Scholar
Damas, L. 1985. Type Assignment in Programming Languages. Ph.D. thesis, Report No. CST-33-85, University of Edinburgh.Google Scholar
Damas, L. and Milner, R. 1982. Principal type-schemes for functional programs. In Proc. of the 9th Symposium on Principles of Programming Languages (POPL), DeMillo, R., Ed. ACM, Albuquerque, New Mexico, USA, 207212.Google Scholar
De Schreye, D. and Martens, B. 1992. A sensible least Herbrand semantics for untyped vanilla meta-programming and its extension to a limited form of amalgamation. In Meta-Programming in Logic, Proc. of the 3rd International Workshop on Meta-Programming (META), Jun. 1992, Peterossi, A., Ed. LNCS. Springer-Verlag, Uppsala, Sweden, 192204.Google Scholar
Dijkstra, E. W. 1968. Letters to the editor: Go to statement considered harmful. Communications of the ACM 11, 3, 147148.10.1145/362929.362947CrossRefGoogle Scholar
Eshghi, K. 1986. Meta-language in Logic Programming. Ph.D. thesis, Department of Computing, Imperial College of Science and Technnology, University of London, UK.Google Scholar
Feferman, S. 2005. Predicativity. In The Oxford Handbook of Philosophy of Mathematics and Logic, Shapiro, S., Ed. Oxford University Press, 590624.CrossRefGoogle Scholar
Frege, G. 1879. Begriffsschrift – Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Verlag von Louis Nebert, Halle an der Saale. In German, English translation: Frege (1967, 2002).Google Scholar
Frege, G. 1893. Grundgesetze der Arithmetik, Band I. Verlag Herman Pohle, Jena. In German, partial English translation: Frege (1964).Google Scholar
Frege, G. 1903. Grundgesetze der Arithmetik, Band II. Verlag Herman Pohle, Jena. In German, partial English translation: Frege (1964).Google Scholar
Frege, G. 1964. The Basic Laws of Arithmetic. University of California Press, Berkeley, CA, USA. Partial English translation of Frege (1893, 1903) by Montgomery Furth.Google Scholar
Frege, G. 1967, 2002. A concept notation: A formula language of pure thought, modelled upon that of arithmetic. In From Frege to Gödel: A Sourcebook in Mathematical Logic, 1879-1931, van Heijenoort, J., Ed. Harvard University Press, 1–82, English translation of Frege (1879).Google Scholar
Gelfond, M. and Lifschitz, V. 1988. The stable model semantics for logic programming. In Proc. of the Fifth International Conference on Logic Programming (ICLP), Aug. 1988, Bowen, K. A. and Kowalski, R. A., Eds. MIT Press, Seattle, Washington, 10701080.Google Scholar
Gentzen, G. 1934. Untersuchungen über das logische Schließen I. Mathematische Zeitschrift 39, 2, 176210. In German. English translations: Gentzen (1964), Gentzen (1969).CrossRefGoogle Scholar
Gentzen, G. 1964. Investigations into logical deduction. American Philosophical Quarterly 1, 4 (October), 288306. English translation of Gentzen (1934).Google Scholar
Gentzen, G. 1969. Investigations into logical deduction. In The Collected Works of Gerhard Gentzen, Szabo, M. E., Ed. Studies in logic and the foundations of mathematics. North-Holland, 68131. English translation of Gentzen (1934).Google Scholar
Giordano, L., Martelli, A. and Rossi, G. 1994. Structured Prolog: A language for structured logic programming. Software-Concepts and Tools 15, 125145.Google Scholar
Giordano, L. and Olivetti, N. 1994. Combining negation as failure and embedded implications in logic programs. Journal of Logic Programming 19, 20, 1679.Google Scholar
Gödel, K. 1930. Über die Vollständigkeit des Logikkalküls. Doctoral thesis, University of Vienna, Austria. In German, English translation: Gödel (2002).Google Scholar
Gödel, K. 1931. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatsheft für Mathematik und Physik 38, 173198. In German, English translation: Gödel (1986).CrossRefGoogle Scholar
Gödel, K. 1944. Russell’s mathematical logic. In The Philosophy of Bertrand Russell, Schilpp, P. A., Ed. Northwestern University, Chicago, IL, USA, 123153.Google Scholar
Gödel, K. 1967, 2002. The completeness of the Axioms of the functional calculus of logic. In From Frege to Gödel: A Sourcebook in Mathematical Logic, 1879-1931. van Heijenoort, J., Ed. Harvard University Press, 582591.Google Scholar
Gödel, K. 1986. On Formally Undecidable Propositions of Principia Mathematica and Related Systems I. Vol. I Publications 19291936. Oxford University Press. English translation of Gödel (1930).Google Scholar
Graf, P. 1996. Term Indexing. Vol. 1053, LNCS. Springer-Verlag. Doctoral thesis, Saarland University, Germany, 1995.Google Scholar
Haemmerlé, R. and Fages, F. 2006. Modules for Prolog revisited. In Logic Programming – International Conference on Logic Programming (ICLP), Etalle, S. and Truszczynski, M., Ed. Springer-Verlag, Seattle, Washington, USA. 4155.Google Scholar
Halpern, J. Y. and Moses, Y. 1985. A guide to the modal logics of knowledge and belief. In Proc. 9th International Joint Conference on Artificial Intelligence (IJCAI), Joshi, A., Ed. Vol. 1, Morgan Kaufmann, Los Angeles, California, USA, 480490.Google Scholar
Harper, R., Honsell, F. and Plotkin, G. 1993. A framework for defining logics. Journal of the ACM (JACM) 40, 1, 143184.CrossRefGoogle Scholar
Higgins, C. P. 1996. On the declarative and procedural semantics of definite metalogic programs. Journal of Logic and Computation 6, 3, 363407.CrossRefGoogle Scholar
Hill, P. and Lloyd, J. W. 1994. The Gödel Programming Language. MIT Press.Google Scholar
Hill, P. M. and Lloyd, J. W. 1988. Analysis of meta-programs. In Meta-Programming in Logic Programming, Workshop on Meta-Programming in Logic (META-88), Jun. 1988, Abramson, H. and Rogers, M. H., Eds. MIT Press, Bristol, Great Britain, 2351.Google Scholar
Hindley, R. J. 1969. The principal type-scheme of an object in combinatory logic. Transactions of the American Mathematical Society 146, Bristol, Great Britain, 2960.Google Scholar
Hintikka, J. 1964. Knowledge and Belief – An Introduction to the Logic of the Two Notions, 2nd ed. Cornell University Press.Google Scholar
Huet, G. P. 1976. Résolution d’équations dans des langages d’ordre 1, 2, …, ω. Doctoral thesis, Mathématiques, Université Paris VII.Google Scholar
Jiang, Y. 1994. Ambivalent Logic as the semantic basis of logic programming. In Proc. International Conference on Logic Programming, Hentenryck, P. V., Ed. MIT Press, Santa Marherita Ligure, Italy, 387401.Google Scholar
Kalsbeek, M. B. 1993. The Vanilla Metainterpreter for Definite Logic Programs and Ambivalent Syntax. Tech. Rep. CT 93-01, Institute for Logic Language and Computation, University of Amsterdam.Google Scholar
Kalsbeek, M. B. and Jiang, Y. 1995. A vademecum of ambivalent logic. In Meta-Logics and Logic Programming, Apt, K. R. and Turini, F., Ed. MIT Press, 2756.Google Scholar
Kowalski, R. 1973. Predicate logic as a programming language. Memo 70, Department of Artificial Intelligence, Edinburgh University. Also in: Proc. IFIP Congress, Stockholm, North Holland, 1974, 569–574.Google Scholar
Kowalski, R. 1979. Logic For Problem Solving. North Holland.Google Scholar
Kowalski, R. and Kim, J.-S. 1991. A metalogic approach to multi-agent knowledge and belief. In Artificial and Mathematical Theory of Computation: Papers in Honor of John McCarthy, Lipschitz, V., Ed. Academic Press, 231246.10.1016/B978-0-12-450010-5.50019-0CrossRefGoogle Scholar
Levi, G. and Ramundo, D. 1993. A formalization of meta-programming for real. In Proc. of the 10th International Conference on Logic Programming, Jun. 1993, Warren, D. S., Ed. MIT Press, Budapest, Hungary, 354373.Google Scholar
Levy, J. and Veanes, M. 2000. On the undecidability of second-order unification. Information and Computation 159, 1–2, 125150.CrossRefGoogle Scholar
Link, G., Ed. 2004. One Hundred Years of Russell’s Paradox: Mathematics, Logic, Philosophy. de Gruyter.Google Scholar
Liskov, B. and Zilles, S. 1974. Programming with abstract data types. ACM SIGPLAN Notices 9, 4, 5059.Google Scholar
Lismont, L. and Mongin, P. 1994. On the logic of common belief and common knowledge. Theory and Decision 37, 75106.CrossRefGoogle Scholar
Lloyd, J. W. 1987. Foundation of Logic Programming, 2nd ed. Springer-Verlag.CrossRefGoogle Scholar
Martelli, A. and Rossi, G. 1988. Enhancing Prolog to support Prolog programming environments. In ESOP’88: 2nd European Symposium on Programming, Mar. 1988, Vol. 300, LNCS. Ganzinger, H., Ed. Springer-Verlag, Nancy, France, 317327.Google Scholar
McCarthy, J. and Levin, M. I. 1965. Lisp 1.5 Programmer’s Manual. Tech. Rep., Computation Center, Massachusetts Institute of Technology, USA.Google Scholar
McCarthy, J., Sato, M., Hayashi, T. and Igrashi, S. 1978. On the Model Theory of Knowledge. Tech. Rep., AIM-312, Stanford University.CrossRefGoogle Scholar
Miller, D. 1989. A logical analysis of modules in logic programming. Journal of Logic Programming 6, 1–2, 79108.CrossRefGoogle Scholar
Miller, D. and Nadathur, G. 1988. An overview of Lambda-Prolog. In Proc. Fifth International Conference and Symposium on Logic Programming, Kowalski, R. A. and Bowen, K. A., Eds. MIT Press, Seattle, Washington, USA, 810827.Google Scholar
Miller, D. and Nadathur, G. 2012. Programming with Higher-Order Logic. Cambridge University Press.CrossRefGoogle Scholar
Milner, R. 1978. A theory of type polymorphism in programming. Journal of Computer and System Science 17, 348374.CrossRefGoogle Scholar
Moschovakis, J. 1999, 2015. Intuitionistic Logic. The Metaphysics Research Lab, Center for the Study of Language and Information, Stanford University, Stanford, CA 94305-4115. https://plato.stanford.edu/entries/logic-intuitionistic/.Google Scholar
O’Keefe, R. 1990. The Craft of Prolog. MIT Press.Google Scholar
Perlis, A. J. and Samelson, K. 1958. Preliminary report: International algebraic language. Communications of the ACM 1, 12, 822.Google Scholar
Perlis, D. 1985. Languages with self-reference i: Foundations. Artificial Intelligence 25, 3, 307322.Google Scholar
Perlis, D. 1988a. Languages with self-reference ii: Knowledge, belief, and modality. Artificial Intelligence 34, 2, 179212.10.1016/0004-3702(88)90038-0CrossRefGoogle Scholar
Perlis, D. 1988b. Meta in logic. In Meta-level Architectures and Reflection, Maes, P. and Nardi, D., Ed. North-Holland, 3749.Google Scholar
Pfenning, F. 1991. Logic Programming in the LF Logical Framework. Research Rep., School of Computer Science, Carnegie Mellon University.CrossRefGoogle Scholar
Pfenning, F. and Schürmann, C. 1999. System description: Twelf – A meta-logical framework for deductive systems. In Proc. of the 16th International Conference on Automated Deduction (CADE). Vol. 1632, LNAI. Ganzinger, H., Ed. Springer-Verlag, Trento, Italy, 202206.CrossRefGoogle Scholar
Pierce, B. C. 2002. Types and Programming Languages. MIT Press.Google Scholar
Ramsey, F. P. 1926. The foundations of mathematics. Proceedings of the London Mathematical Society, Series 225, 338384.Google Scholar
Robinson, A. and Voronkov, A., Eds. 2001. Handbook of Automated Reasoning. Vol. 1. ScienceDirect.Google Scholar
Robinson, G. and Wos, L. 1968. Completeness of paramodulation. In Proc. of the Spring 1968 Meeting of the Association for Symbolic Logic, Journal of Symbolic Logic, Vol. 34, (1969), 102–103.Google Scholar
Robinson, G. and Wos, L. 1969. Paramodulation and theorem proving in first order theories with equality. In Machine Intelligence 4. Edinburgh University Press, 135150. Proc. of the Fourth Annual Machine Intelligence Workshop. Edinburgh, 1968. Reprinted in Siekmann and Wrightson (1983, pp. 98–3134).Google Scholar
Rossi, G. 1989. Meta-programming facilities in an extended Prolog. In Proc. of the Fifth International Conference on Artificial Intelligence and Information-Control Systems of Robots – 89, Plander, I., Ed. World Scientific, distributed by North Holland, Štrbské Pleso, Czechoslovakia Google Scholar
Rossi, G. 1992. Programs as data in an extended Prolog. The Computer Journal 36, 3, 217226.CrossRefGoogle Scholar
Russell, B. 1907. On some difficulties in the theory of transfinite numbers and order types. Proceedings of the London Mathematical Society s2–4, 1, 2953.Google Scholar
Russell, B. 1908. Mathematical logic as based on the theory of types. American Journal of Mathematics 30, 222262.CrossRefGoogle Scholar
Russell, B. 1986. The Collected Papers of Bertrand Russell. Vol. 6, Logical and Philosophical Papers 1909–13. George Allen & Unwin.Google Scholar
Siekmann, J. H. and Wrightson, G., Eds. 1983. Automation of Reasoning. Symbolic Computation (Artificial Intelligence). Vol. 1, Classical Papers on Computational Logic 19571966. Springer-Verlag.Google Scholar
Siklóssy, L. 1976. Let’s Talk Lisp. Prentice-Hall.Google Scholar
Sterling, L. S. and Shapiro, E. Y. 1994. The Art of Prolog, 2nd ed. MIT Press.Google Scholar
Sugano, H. 1989. Reflective Computation in Logic Language and Its Semantics. Tech. Rep., International Institute for Advanced Study of Social and Information Science, Fujitsu Limited.Google Scholar
Sugano, H. 1990. Meta and reflective computation in logic programming and its semantics. In Proc. of the 2nd Workshop on Meta-Programming in Logic Programming (META), Bruynooghe, M., Ed. Springer-Verlag, Leuven, Belgium, 1934.Google Scholar
Tarski, A. 1935. Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica: commentarii Societatis Philosophicae Polonorum 146, 261405. In German, English translation: Tarski (1935).Google Scholar
Tarski, A. 1935. The concept of truth in formalized languages. In Logic, Semantics, Metamathematics, Papers from 1923 to 1938, 2nd ed. Corcoran, J., Eds. Hackett Publishing Company, Inc., 152278.Google Scholar
Trump, D. 2017. “Why would Kim Jong-un insult me by calling me ‘old’, when I would NEVER call him ‘short and fat’? Oh well, I try so hard to be his friend – And maybe someday that will happen!”. Twitter Web Client https://twitter.com/realdonaldtrump/status/929511061954297857.Google Scholar
van Ditmarsch, H., Halpern, J. Y., van der Hoek, W. and Kooi, B., Eds. 2015. Handbook of Epistemic Logic. College Publications.Google Scholar
van Emden, M. and Kowalski, R. 1976. The semantics of predicate logic as a programming language. Journal of the Association for Computating Machinery 23, 733742.CrossRefGoogle Scholar
van Harmelen, F. 1989. A classification of metalevel architectures. In Logic-based Knowledge Representation, Jackson, P. M., Reichgelt, J., and van Harmelen, F., Eds. MIT Press, 135.Google Scholar
van Harmelen, F. 1992. Definable naming relations in meta-level systems. In Proc. of the Third Workshop on Meta-Programming in Logic (META), Uppsala, Sweden, Vol. 649, LNCS, Pettorossi, A., Ed. Springer-Verlag, 89104.CrossRefGoogle Scholar
von Kutschera, F. 1989. Gottlob Frege: Eine Einführung in sein Werk, de Gruyter, Berlin. In German.10.1515/9783110853933CrossRefGoogle Scholar
Wadge, W. W. 1991. Higher-order horn-logic programming. In Logic Programming: Proceedings of the 1991 International Symposium, Saraswat, V. A. and Ueda, K., Eds. MIT Press, 289303.Google Scholar
Weyhrauch, R. W. 1980. Prolegomena to a theory of mechanized formal reasoning. Artificial Intelligence 13, 1–2, 133170.CrossRefGoogle Scholar
Whitehead, A. N. and Russell, B. 1910, 1912, 1913. Principia Mathematica. Vols. 1–3. Cambridge University Press.Google Scholar
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