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Extending the first-order theory of combinators with self-referential truth

Published online by Cambridge University Press:  12 March 2014

Andrea Cantini
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Abstract

The aim of this paper is to introduce a formal system STW of self-referential truth, which extends the classical first-order theory of pure combinators with a truth predicate and certain approximation axioms. STW naturally embodies the mechanisms of general predicate application/abstractionon a par with function application/abstraction; in addition, it allows non-trivial constructions, inspired by generalized recursion theory. As a consequence, STW provides a smooth inner model for Myhill's systems with levels of implication.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 2 , June 1993 , pp. 477 - 513
Copyright
Copyright © Association for Symbolic Logic 1993

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References

REFERENCES

[1]Aczel, P., The strength of Martin-Löf's intuitionistic type theory with one universes, Proceedings of the symposium in mathematical logic in Oulu 1974 and Helsinki 1975 (Miettinen, S. and Väänänen, J., editors), Report no. 2, 1977, Department of Philosophy, University of Helsinki, pp. 1–32.Google Scholar
[2]Aczel, P., Frege structures and the notions of proposition, truth and set (Barwise, J., Keisler, H. J., Kunen, K., editors), The Kleene symposium, North Holland, Amsterdam, 1980, pp. 31–59.CrossRefGoogle Scholar
[3]Aczel, P. and Feferman, S., Consistency of the unrestricted abstraction principle using an intensional equivalence operator (Seldin, J. P. and Hindley, J. R., editors), To H. B. Curry: Essays on combinatory logic, Lambda calculus and formalism, Academic Press, New York, 1980, pp. 67–98.Google Scholar
[4]Barendregt, H., The lambda calculus: its syntax and semantics, North Holland, Amsterdam, 1984.Google Scholar
[5]Beeson, M., Foundations of constructive mathematics, Springer, Berlin, 1985.CrossRefGoogle Scholar
[6]Behmann, H., Zu den Widersprüchen der Logik und der Mengenlehre, Jahresberichte der Deutschen Mathematischen Vereinigung, vol. 40 (1931), pp. 37–48.Google Scholar
[7]Behmann, H., Der Prädikatenkalkül mit limitierten Variablen: Grundlegung einer natürlichen exakten Logik, this Journal, vol. 24 (1959), pp. 112–140.Google Scholar
[8]Birkhoff, G., Lattice theory, 3rd edition, American Mathematical Society Colloquium Publications, vol. XXV, American Mathematical Society, Providence, RI, 1967.Google Scholar
[9]Cantini, A., A note on three-valued logic and Tarski theorem on truth definitions, Studia Logica, vol. 39 (1980), pp. 405–415.CrossRefGoogle Scholar
[10]Cantini, A., Nonextensional theories of predicative classes over PA, Rendiconti del Seminario Matematico dell' Università e del Politecnico di Torino, vol. 40 (1982), pp. 47–79.Google Scholar
[11]Cantini, A., Notes on formal theories of truth, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 35 (1989), pp. 97–130.CrossRefGoogle Scholar
[12]Cantini, A., Levels of implication and type free theories of partial classifications with approximation operator, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 38 (1992), pp. 107–141.CrossRefGoogle Scholar
[13]Feferman, S., Nonextensional type free theories of partial operations and classifications. I (Diller, J. and Müller, G. H., editors), Proof theory symposium, Lecture Notes in Mathematics, vol. 500, Springer, Berlin, 1974, pp. 73–118.CrossRefGoogle Scholar
[14]Feferman, S., Towards useful type free theories. I, this Journal, vol. 49 (1984), pp. 75–111.Google Scholar
[15]Feferman, S., Reflecting on incompleteness, the Journal, vol. 56 (1991), pp. 1–49.Google Scholar
[16]Fitch, F., An extension of basic logic, this Journal, vol. 13 (1948), pp. 95–106.Google Scholar
[17]Fitch, F., A consistent combinatory logic with an inverse to equality, this Journal, vol. 45 (1980), pp. 529–543.Google Scholar
[18]Fitting, M., Fundamentals of generalized recursion theory, North Holland, Amsterdam, 1981.Google Scholar
[19]Flagg, R. and Myhill, J., Implication and analysis in classical Frege structures, Annals of Pure and Applied Logic, vol. 34 (1987), pp. 33–85.CrossRefGoogle Scholar
[20]Gilmore, P., The consistency of partial set theory without extensionality, Axiomatic set theory (Jech, T., editor), vol. II, Proceedings of Symposia in Pure Mathematics, vol. 13, American Mathematical Society, Providence, RI, 1974, pp. 147–153.CrossRefGoogle Scholar
[21]Hindley, R. and Seldin, J., Introduction to combinators and λ-calculus, London Mathematical Society Student Texts 1, Cambridge University Press, Cambridge, 1986.Google Scholar
[22]Kripke, S., Outline of a theory of truth, Journal of Philosophy, vol. 72 (1975), pp. 690–716.CrossRefGoogle Scholar
[23]Minari, P. L., Unpublished notes, Florence, 1987.Google Scholar
[24]Moschovakis, Y. N., Elementary induction on abstract structures, North Holland, Amsterdam, 1974.Google Scholar
[25]Myhill, J., Levels of implications, The logical enterprise (Ross, A., Marcus, R. Barcan, Martin, R. M., editors), Yale University Press, New Haven, 1975.Google Scholar
[26]Myhill, J., Paradoxes, Synthèse, vol. 60 (1984), pp. 129–142.CrossRefGoogle Scholar
[27]Scott, D., Combinators and classes, λ-Calculus and computer science (Böhm, C., editor), Lecture Notes in Computer Science, vol. 37, Springer, Berlin, 1975, pp. 1–26.CrossRefGoogle Scholar
[28]Seldin, R., Recent advances in Curry's program, Rendiconti del Seminario Matematico dell' Università e del Politecnico di Torino, vol. 35 (1976), pp. 77–88.Google Scholar
[29]Seldin, R., Curry's program, To H. B. Curry: Essays on combinatory logic, lambda calculus and formalism (Seldin, J. P. and Hindley, J. R., editors), Academic Press, New York, 1980, pp. 3–33.Google Scholar
[30]Smullyan, R., Theory of formal systems, Annals of Mathematical Studies, Princeton University Press, Princeton, NJ, 1961.CrossRefGoogle Scholar
[31]Turner, R., Truth and modality for knowledge representation, Pitman, London, 1990.Google Scholar
[32]Visser, A., Semantics and the liar paradox, Handbook of Philosophical Logic (Gabbay, D. and Guenthner, F., editors), vol. IV, Reidel, Dordrecht, 1989, pp. 617–706.CrossRefGoogle Scholar
[33]Whitman, P. M., Splitting of lattices, American Journal of Mathematics, vol. 65 (1943), pp. 179–196.CrossRefGoogle Scholar