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\bibitem{Bunder1969}
Martin~W. Bunder.
\newblock {\em Set theory based on combinatory logic}.
\newblock PhD thesis, University of Amsterdam, 1969.

\bibitem{Bunder1986}
Martin~W. Bunder.
\newblock Some consistency proofs and a characterization of inconsistency
  proofs in liiative combinatory logic.
\newblock {\em Journal of Symbolic Logic}, 52:89--110, 1986.

\bibitem{Burali-Forti1987}
Cesare Burali-Forti.
\newblock Una questione sui numeri transfiniti.
\newblock {\em Rendiconti del Circolo matematico di Palermo}, 11:154--164,
  1897.

\bibitem{Church1932}
Alonzo Church.
\newblock A set of postulates for the foundation of logic {I}.
\newblock {\em Annals of Mathematics, ser. 2}, 33:346--366, 1932.

\bibitem{Church1933}
Alonzo Church.
\newblock A set of postulates for the foundation of logic {II}.
\newblock {\em Annals of Mathematics, ser. 2}, 34:839--864, 1933.

\bibitem{Bunder1998}
Wil Dekkers, Martin Bunder, and Henk Barendregt.
\newblock Completeness of two systems of illative combinatory logic for
  first-order propositional and predicate calculus.
\newblock {\em Archive for Mathematical Logic}, 37:327--341, 1998.

\bibitem{Fraenkel1922}
Abraham~A. Fraenkel.
\newblock Der {B}egriff ``definit'' und die {U}nabh\"{a}ngigkeit des
  {A}uswahlsaxioms.
\newblock {\em Sitzungsberichte der Preussisehen Akademie der Wissenschaften,
  Physikalisch-mathematische Klasse}, pages 253--257, 1922.

\bibitem{Fraenkel1973}
Abraham~A. Fraenkel, Yehoshua Bar-Hillel, and Azriel Levy.
\newblock {\em Foundation Set Theory}, volume~67 of {\em Studies in Logic and
  the Foundations of Mathematics}.
\newblock North-Holland, second edition, 1973.

\bibitem{Frege02}
Gottlob Frege.
\newblock Letter to {R}ussell.
\newblock Translated in \cite{Heijenoort}, 1902.

\bibitem{Gentzen}
Gerhard Gentzen.
\newblock Untersuchungen \"{u}ber das logische {S}chliessen.
\newblock {\em Mathematische Zeitschrift}, 39:176--210, 405--431, 1935.
\newblock Translated in \cite{Szabo}.

\bibitem{Grisin1974}
V.~N. Gri\v{s}in.
\newblock A nonstandard logic, and its application to set theory.
\newblock In {\em Studies in Formalized Languages and Nonclassical Logics},
  pages 135--171. ``Nauka'', Moscow, 1974.
\newblock (Russian).

\bibitem{Grisin1982}
V.~N. Gri\v{s}in.
\newblock Predicate and set-theoretic calculi based on logic without
  contractions.
\newblock {\em Math. USSR Izvestija}, 18:41--59, 1982.

\bibitem{Hindley2}
J.~Roger Hindley.
\newblock {\em Basic Simple Type Theory}, volume~42 of {\em Cambridge Tracts in
  Theoretical Computer Science}.
\newblock Cambridge University Press, 1997.

\bibitem{Hindley}
J.~Roger Hindley and Jonathan~P. Seldin.
\newblock {\em Introduction to combinators and $\lambda$--calculus}, volume~1
  of {\em London Mathematical Society Student Texts}.
\newblock Cambridge University Press, 1986.

\bibitem{Kleene1952}
Stephen~C. Kleene.
\newblock {\em Two papers on the predicate calculus}.
\newblock Number~10 in Memoirs of the American Mathematical Society. American
  Mathematical Society, 1952.

\bibitem{Kleene1935}
Stephen~C. Kleene and J.~Barkley Rosser.
\newblock The inconsistency of certain formal logics.
\newblock {\em Annals of Mathematics}, 36:630--636, 1935.

\bibitem{Komori89}
Yuichi Komori.
\newblock Illative combinatory logic baed on {BCK}--logic.
\newblock {\em Mathematica Japonica}, 34:585--596, 1989.

\bibitem{OnoKomo}
Hiroakira Ono and Yuichi Komori.
\newblock Logics without the contraction rule.
\newblock {\em Journal of Symbolic Logic}, 50:169--201, 1985.

\bibitem{Petersen2000}
Uwe Petersen.
\newblock Logic without contraction as based on inclusion and unrestricted
  abstraction.
\newblock {\em Studia Logica}, 64:365--403, 2000.

\bibitem{Rose1958}
Alan Rose and J.~Barkley Rosser.
\newblock Fragments of many-valued statement calculi.
\newblock {\em Trans. Amer. Math. Soc.}, 87:1--53, 1958.

\bibitem{Russell02}
Bertrand Russell.
\newblock Letter to {F}rege.
\newblock Translated in \cite{Heijenoort}, 1902.

\bibitem{Substructural93}
Peter Schroeder-Heister and Kosta Do\v{s}en, editors.
\newblock {\em Substructural Logics}, volume~2 of {\em Studies in Logic and
  Computation}.
\newblock Oxford Univ. Press, 1993.

\bibitem{Shaw-Kwei1954}
Moh Shaw-Kwei.
\newblock Logical paradoxes for many valued systems.
\newblock {\em Journal of Symbolic Logic}, 19:37--40, 1954.

\bibitem{Szabo}
Manfred~E. Szabo.
\newblock {\em The collected papers of Gerhard Gentzen}.
\newblock North-Holland, Amsterdam, 1969.

\bibitem{Lukasiewicz1920}
Jan~\L ukasiewicz.
\newblock O logice tr\'{o}jwarto\'{s}ciowej.
\newblock {\em Ruch filozoficzny}, 5:169--171, 1920.

\bibitem{Lukasiewicz1930}
Jan~\L ukasiewicz and Alfred Tarski.
\newblock Untersuchungen \"{u}ber den {A}ussagenkalk\"{u}l.
\newblock {\em Comptes rendus des s\'{e}ances de la Soci\'{e}t\'{e} des
  Sciences et des Lettres de Varsovie, Classe III}, 23:30--50, 1930.

\bibitem{Heijenoort}
Jean van Heijenoort.
\newblock {\em From Frege to {G\"{o}del}}.
\newblock Harvard University Press, third edition, 1976.

\bibitem{White1979}
Richard~B. White.
\newblock The consistency of the axiom of comprehension in the infinite-valued
  predicate logic of {{\L ukasiewicz}}.
\newblock {\em Journal of Philosophical Logic}, 8:509--534, 1979.

\bibitem{White1987}
Richard~B. White.
\newblock A demonstrably consistent type-free extension of the logic {BCK}.
\newblock {\em Mathematica Japonica}, 32:149--169, 1987.

\bibitem{Zermelo1908}
Ernst Zermelo.
\newblock Untersuchungen \"{u}ber die {G}rundlagen der {M}engenlehre {I}.
\newblock {\em Mathematische Annalen}, 65:261--281, 1908.

\end{thebibliography}
