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Toward useful type-free theories. I

  • Solomon Feferman (a1)

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 {xxx} (which would be universal since x(xx) holds in ZF). Nor is there a set of all ordinal numbers (and so the Burali-Forti paradox is blocked).

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W. Ackermann [1958] Ein typenfreies System der Logik mit ausreichender mathematischer Anwendungsfähigkeit. I, Archiv für Mathematische Logik und Grundlagenforschung vol. 4, pp. 126.

P. Aczel [1980] Frege structures and the notions of proposition, truth and set, The Kleene symposium ( J. Barwise , H. J. Keisler and K. Kunen , editors), North-Holland, Amsterdam, pp. 3159.

R. T. Brady [1971] The consistency of the axioms of abstraction and extensionality in three-valued logic, Notre Dame Journal of Formal Logic, vol. 12, pp. 447453.

M. W. Bunder [1982] Some results in Aczel-F eferman logic and set theory, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 28, pp. 269276.

A. Cantini [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. 405414).

H. B. Curry [1980] Some philosophical aspects of combinatory logic, The Kleene symposium ( J. Barwise , H. J. Keisler and K. Kunen , editors), North-Holland, Amsterdam, pp. 85101.

S. Feferman [1977] Categorical foundations and foundations of category theory, Logic, foundations of mathematics and computability theory ( R. Butts and J. Hintikka , editors), Reidel, Dordrecht, pp. 149169.

H. G. Herzberger [1970] Paradoxes af grounding in semantics, Journal of Philosophy, vol. 67, pp. 145167.

A. Kechris and Y. Moschovakis , [1977] Recursion in higher types, Handbook of Mathematical Logic ( J. Barwise , editor), North-Holland, Amsterdam, pp. 681737.

W. Kindt [1976] Über Sprachen mit Wahrheitsprädikat, Sprachdynamik und Sprachstruktur ( C. Habel and S. Kanngiesser , editors), Niemeyer, Tübigen, 1978.

S. C. Kleene and R. Vesley [1965] The foundations of intuitionistic mathematics, especially in relation to recursive functions, North-Holland, Amsterdam.

G. Kreisel and A. S. Troelstra [1970] Formal systems for some branches of intuitionistic analysis, Annals of Mathematical Logic, vol. 1, pp. 229387.

S. Kripke [1975] Outline of a theory of truth, Journal of Philosophy, vol 72, pp. 690716.

R. L. Martin and P. W. Woodruff [1975] On representing “true-in-L” in L, Philosophia, vol. 5, pp. 213217.

C. Parsons [1974] The liar paradox, Journal of Philosophical Logic, vol. 3, pp. 381412.

K. Schütte [1953] Zur Widerspruchsfreiheit einer typenfreien Logik, Mathematische Annalen, vol. 125, pp. 394400.

D. Scott [1975] Combinators and classes, λ-calcalus and computer science theory, Lecture Notes in Computer Science, vol. 37, Springer-Verlag, Berlin, pp. 126.

K. Segerberg [1965] A contribution to nonsense-fogies, Theoria, vol. 31, pp. 199217.

T. Skolem [1963] Studies on the axiom of comprehension, Notre Dame Journal of Formal Logic, vol. 4, pp. 162170.

R. Thomason [1969] A semantical study of constructive falsity, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 15, pp. 247257.

B. van Fraassen [1968] Presupposition, implication and self-reference, Journal of Philosophy, vol. 65, pp. 135152.

H. Wang [1961] The calculus of partial predicates and its extension to set theory. I, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 7, pp. 283288.

R.G. Wolf [1977] A survey of many-valued logic (1966–1974), Modern uses of multiple-valued logic ( J. M. Dunn and G. Epstein , editors), Reidel, Dordrecht, pp. 167323.

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The Journal of Symbolic Logic
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