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Toward useful type-free theories. I

Published online by Cambridge University Press:  12 March 2014

Solomon Feferman
Solomon Feferman*
Affiliation:
Stanford University, Stanford, California 94305
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Extract

There is a distinction between semantical paradoxes on the one hand and logical or mathematical paradoxes on the other, going back to Ramsey [1925]. Those falling under the first heading have to do with such notions as truth, assertion (or proposition), definition, etc., while those falling under the second have to do with membership, class, relation, function (and derivative notions such as cardinal and ordinal number), etc. There are a number of compelling reasons for maintaining this separation but, as we shall see, there are also many close parallels from the logical point of view.

The initial solutions to the paradoxes on each side—namely Russell's theory of types for mathematics and Tarski's hierarchy of language levels for semantics— were early recognized to be excessively restrictive. The first really workable solution to the mathematical paradoxes was provided by Zermelo's theory of sets, subsequently improved by Fraenkel. The informal argument that the paradoxes are blocked in ZF is that its axioms are true in the cumulative hierarchy of sets where (i) unlike the theory of types, a set may have members of various (ordinal) levels, but (ii) as in the theory of types, the level of a set is greater than that of each of its members. Thus in ZF there is no set of all sets, nor any Russell set {x∣x∉x} (which would be universal since ∀x(x∉x) holds in ZF). Nor is there a set of all ordinal numbers (and so the Burali-Forti paradox is blocked).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 1 , March 1984 , pp. 75 - 111
Copyright
Copyright © Association for Symbolic Logic 1984

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References

REFERENCES

Ackermann, W. [1950] Widerspruchsfreier Aufbau der Logik. Typenfreies System ohne Tertium non datur, this Journal, vol. 15, pp. 33–37.Google Scholar
Ackermann, W. [1958] Ein typenfreies System der Logik mit ausreichender mathematischer Anwendungsfähigkeit. I, Archiv für Mathematische Logik und Grundlagenforschung vol. 4, pp. 1–26.CrossRefGoogle Scholar
Aczel, P. [1980] Frege structures and the notions of proposition, truth and set, The Kleene symposium (Barwise, J., Keisler, H. J. and Kunen, K., editors), North-Holland, Amsterdam, pp. 31–59.CrossRefGoogle Scholar
Aczel, P. and Feferman, S. [1980] Consistency of the unrestricted abstraction principle using an intensional equivalence operator, in Seldin/Hindley [1980], pp. 67–98.Google Scholar
Behmann, H. [1931] Zu den Widersprüchen der Logik und aer Mengenlehre, Jahresbericht der Deutschen Mathematiker-Vereinigung vol. 40, pp. 37–48.Google Scholar
Behmann, H. [1959] Der Prädikatenkalkül mit limitierten Variablen: Grundlegung einer natürlichen exakten Logik, this Journal, vol. 24, pp. 112–140.Google Scholar
Bochvar, D. A. [1981] On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus, Matematičeskiĭ Sbornik, vol. 4 (46) (1939), pp. 287–308; English translation in History and Philosophy of Logic, vol. 2 (1981), pp. 87–112.Google Scholar
Brady, R. T. [1971] The consistency of the axioms of abstraction and extensionality in three-valued logic, Notre Dame Journal of Formal Logic, vol. 12, pp. 447–453.CrossRefGoogle Scholar
Bunder, M. W. [1980] The naturalness of illative combinatory logic as a basis for mathematics, in Seldin/Hindley [1980], pp. 55–64.CrossRefGoogle Scholar
Bunder, M. W. [1982] Some results in Aczel-F eferman logic and set theory, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 28, pp. 269–276.CrossRefGoogle Scholar
Burce, T. [1979] Semantical paradox, Journal of Philosophy, vol. 76, pp. 169–198.Google Scholar
Cantini, A. [1979a] A note on three-valued logic and Tarski theorem on truth definitions, preprint, Mathematisches Institut, München, 16 pp. (published in Studia Logica, vol. 39 (1980), pp. 405–414).CrossRefGoogle Scholar
Cantini, A. [1979b] “Tarsfci extensions” of theories, preprint, Mathematisches Institut, München, 18 pp.Google Scholar
Curry, H. B. [1942] The inconsistency of certain formal logics, this Journal, vol. 7, pp. 115–117.Google Scholar
Curry, H. B. [1980] Some philosophical aspects of combinatory logic, The Kleene symposium (Barwise, J., Keisler, H. J. and Kunen, K., editors), North-Holland, Amsterdam, pp. 85–101.CrossRefGoogle Scholar
Feferman, S. [1967] Set-theoretical foundations of category theory (with an appendix by Kreisel, G.), Reports of the Midwest Category Seminar. III, Lecture Notes in Mathematics, vol. 106, Springer-Verlag, Berlin, 1969, pp. 207–246.CrossRefGoogle Scholar
Feferman, S. [1975a] A language and axioms for explicit mathematics, Algebra and logic, Lecture Notes in Mathematics, vol. 450, Springer-Verlag, Berlin, pp. 87–139.CrossRefGoogle Scholar
Feferman, S. [1975b] Investigative logic for theories of partial functions and relations. I and II, unpublished notes, Stanford University, Stanford, California, 21 pp. and 13 pp.Google Scholar
Feferman, S. [1975c] Non-extensional type-free theories of partial operations and classifications. I, ⊧ ISILC Proof Theory Symposion, Kiel, 1974, Lecture Notes in Mathematics, vol. 500, Springer-Verlag, Berlin, pp. 73–118.Google Scholar
Feferman, S. [1976] Comparison of some type-free semantic and mathematical theories, unpublished notes, Stanford University, Stanford, California, 18 pp.Google Scholar
Feferman, S. [1977] Categorical foundations and foundations of category theory, Logic, foundations of mathematics and computability theory (Butts, R. and Hintikka, J., editors), Reidel, Dordrecht, pp. 149–169.CrossRefGoogle Scholar
Feferman, S. [1979] Constructive theories of functions and classes, Logic Colloquium '78 (Boffa, M., van Dalen, D. and McAloon, K., editors), North-Holland, Amsterdam, pp. 159–224.Google Scholar
Fitch, F. B. [1948] An extension of basic logic, this Journal, vol. 13, pp. 95–106.Google Scholar
Fitch, F. B. [1963] The system CΔ of combinatory logic, this Journal, vol. 28, pp. 87–97.Google Scholar
Fitch, F. B. [1966] A consistent modal set theory (abstract), this Journal vol. 31, p. 701.Google Scholar
Fitch, F. B. [1980] A consistent combinatory logic with an inverse to equality, this Journal, vol. 45, pp. 529–543.Google Scholar
Gilmore, P. C. [1974] The consistency of partial set theory without extensionality, Axiomatic set theory, Proceedings of Symposia in Pure Mathematics, vol. 13, Part II, American Mathematical Society, Providence, R.I., pp. 147–153.CrossRefGoogle Scholar
Gilmore, P. C. [1980] Combining unrestricted abstraction with universal quantification, in Seldin/Hindley [1980], pp. 99–123.Google Scholar
Gödel, K. [1931] Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. I, Monatshefte für Mathematik und Physik, vol. 38, pp. 173–198.CrossRefGoogle Scholar
Halldén, S. [1949] The logic of nonsense, Uppsala Universitets Årsskrift, vol. 1949, no. 9.Google Scholar
Herzberger, H. G. [1970] Paradoxes af grounding in semantics, Journal of Philosophy, vol. 67, pp. 145–167.CrossRefGoogle Scholar
Jeroslow, R. G. [1973] Redundancies in the Hilbert-Bernays derivability conditions for Gödel's second incompleteness theorem, this Journal, vol. 38, pp. 359–367.Google Scholar
Kechris, A. and Moschovakis, Y., [1977] Recursion in higher types, Handbook of Mathematical Logic (Barwise, J., editor), North-Holland, Amsterdam, pp. 681–737.CrossRefGoogle Scholar
Kindt, W. [1976] Über Sprachen mit Wahrheitsprädikat, Sprachdynamik und Sprachstruktur (Habel, C. and Kanngiesser, S., editors), Niemeyer, Tübigen, 1978.Google Scholar
Kleene, S. C. [1952] Introduction to metamathematics, Van Nostrand, Princeton, N.J.Google Scholar
Kleene, S. C. and Vesley, R. [1965] The foundations of intuitionistic mathematics, especially in relation to recursive functions, North-Holland, Amsterdam.Google Scholar
Krasner, M. [1962] Le définitionnisme, Actes du Colloques de Mathématiques, Pascal Tricentenaire. I, Annales de la Faculté des Sciences, Université de Clermont, no. 7, pp. 55–81.Google Scholar
Kreisel, G. and Troelstra, A. S. [1970] Formal systems for some branches of intuitionistic analysis, Annals of Mathematical Logic, vol. 1, pp. 229–387.CrossRefGoogle Scholar
Kripke, S. [1975] Outline of a theory of truth, Journal of Philosophy, vol 72, pp. 690–716.CrossRefGoogle Scholar
Martin, R. L. (Editor) [1970] The paradox of the liar, Yale University Press, New Haven, Connecticut; second edition, Ridgeview Publishing Co., Atascadero, California, 1978.Google Scholar
Martin, R. L. and Woodruff, P. W. [1975] On representing “true-in-L” in L, Philosophia, vol. 5, pp. 213–217.CrossRefGoogle Scholar
McCall, S. (Editor) [1967] Polish logic: 1920–1939, Clarendon Press, Oxford.Google Scholar
MacLane, S. [1971] Categories for the working mathematician, Springer-Verlag, Berlin.Google Scholar
Nepeĭvoda, N. N. [1973] A new notion of predicative truth and definability, Matematičeskie Zametki, vol. 13, pp. 735–745; English translation, Mathematical Notes of the Academy of Sciences of the USSR, vol. 13, pp. 439–445.Google Scholar
Parsons, C. [1974] The liar paradox, Journal of Philosophical Logic, vol. 3, pp. 381–412.CrossRefGoogle Scholar
Prawitz, D. [1965] Natural deduction. A proof-theoretical study, Almqvist and Wiksell, Stockholm.Google Scholar
Prior, A. [1967] Many-valued logic, Encyclopedia of Philosophy, vol. 5, MacMillan, New York, pp. 1–5.Google Scholar
Ramsey, F. P. [1925] The foundations of mathematics, Proceedings of the London Mathematical Society, ser.2, vol. 25, pp. 338–384 (also in Foundations, Humanities Press, 152–212).Google Scholar
Rosser, J. B. and Turquette, A. R. [1952] Many-valued logics, North-Holland, Amsterdam.Google Scholar
Schütte, K. [1953] Zur Widerspruchsfreiheit einer typenfreien Logik, Mathematische Annalen, vol. 125, pp. 394–400.CrossRefGoogle Scholar
Schütte, K. [1960] Beweistheorie, Springer-Verlag, Berlin.Google Scholar
Scott, D. [1960] The notion of rank in set-theory, Summaries of talks presented at the Summer Institute for Symbolic Logic, Cornell University, 1957, 2nd ed., Institute for Defense Analyses, Princeton, N.J., (1960) pp. 267–269.Google Scholar
Scott, D. [1975] Combinators and classes, λ-calcalus and computer science theory, Lecture Notes in Computer Science, vol. 37, Springer-Verlag, Berlin, pp. 1–26.CrossRefGoogle Scholar
Segerberg, K. [1965] A contribution to nonsense-fogies, Theoria, vol. 31, pp. 199–217.CrossRefGoogle Scholar
Seldin, J. P and Hindley, J. R. (Editors) [1980] To H. B. Curry: Essays on combinatory logic, lambda calculus and formalism, Academic Press, New York.Google Scholar
Skyrms, B. [1970] Return of the liar: three-valued logic and the concept of truth, American Philosophical Quarterly, vol. 7, pp. 153–161.Google Scholar
Skolem, T. [1960] A set theory based on a certain three-valued logic, Mathematica Scandinavica, vol. 8, pp. 127–136.CrossRefGoogle Scholar
Skolem, T. [1963] Studies on the axiom of comprehension, Notre Dame Journal of Formal Logic, vol. 4, pp. 162–170.CrossRefGoogle Scholar
Tarski, A. [1956] Logic, semantics and me t ämat hematics. Papers from 1923 to 1938 (Woodger, J. H., editor), Clarendon Press, Oxford.Google Scholar
Thomason, R. [1969] A semantical study of constructive falsity, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 15, pp. 247–257.CrossRefGoogle Scholar
van Fraassen, B. [1968] Presupposition, implication and self-reference, Journal of Philosophy, vol. 65, pp. 135–152.CrossRefGoogle Scholar
Wang, H. [1961] The calculus of partial predicates and its extension to set theory. I, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 7, pp. 283–288.CrossRefGoogle Scholar
Wolf, R.G. [1977] A survey of many-valued logic (1966–1974), Modern uses of multiple-valued logic (Dunn, J. M. and Epstein, G., editors), Reidel, Dordrecht, pp. 167–323.CrossRefGoogle Scholar
Woodruff, P. [1969] Foundations of three-valued logic, Dissertation, Department of Philosophy, University of Pittsburgh, Pittsburgh, Pennsylvania.Google Scholar
Woodruff, P. [1973] On constructive nonsense logic, Modality, morality and other problems of sense and nonsense. Essays dedicated to Sören Halldén, CWK Gleerup Bokförlag, Lund, pp. 192–205.Google Scholar