## Notre Dame Journal of Formal Logic

### Variations on a Theme of Curry

Lloyd Humberstone

#### Abstract

After an introduction to set the stage, we consider some variations on the reasoning behind Curry's Paradox arising against the background of classical propositional logic and of BCI logic and one of its extensions, in the latter case treating the "paradoxicality" as a matter of nonconservative extension rather than outright inconsistency. A question about the relation of this extension and a differently described (though possibly identical) logic intermediate between BCI and BCK is raised in a final section, which closes with a handful of questions left unanswered by our discussion.

#### Article information

Source
Notre Dame J. Formal Logic Volume 47, Number 1 (2006), 101-131.

Dates
First available in Project Euclid: 27 March 2006

http://projecteuclid.org/euclid.ndjfl/1143468315

Digital Object Identifier
doi:10.1305/ndjfl/1143468315

Mathematical Reviews number (MathSciNet)
MR2211186

Zentralblatt MATH identifier
1107.03018

#### Citation

Humberstone, Lloyd. Variations on a Theme of Curry. Notre Dame J. Formal Logic 47 (2006), no. 1, 101--131. doi:10.1305/ndjfl/1143468315. http://projecteuclid.org/euclid.ndjfl/1143468315.

#### References

• [1] Aitken, W., and J. A. Barrett, "Computer implication and the Curry paradox", Journal of Philosophical Logic, vol. 33 (2004), pp. 631--37.
• [2] Anderson, A. R., R. B. Marcus, and R. M. Martin, editors, The Logical Enterprise, Yale University Press, New Haven, 1975.
• [3] Avron, A., "Some properties of linear logic proved by semantic methods", Journal of Logic and Computation, vol. 4 (1994), pp. 929--38.
• [4] Avron, A., "Formulas for which contraction is admissible", Logic Journal of the IGPL, vol. 6 (1998), pp. 43--48.
• [5] Blamey, S., "Partial logic", pp. 261--354 in Handbook of Philosophical Logic, edited by D. M. Gabbay and F. Guenthner, vol. 5, Kluwer Academic Publishers, Dordrecht, 2d edition, 2002.
• [6] Blok, W. J., and D. Pigozzi, "Algebraizable logics", Memoirs of the American Mathematical Society, vol. 77 (1989), no. 396.
• [7] Boolos, G., The Logic of Provability, Cambridge University Press, Cambridge, 1993.
• [8] Brady, R. T., "The consistency of the axioms of abstraction and extensionality in a three-valued logic", Notre Dame Journal of Formal Logic, vol. 12 (1971), pp. 447--53.
• [9] Brady, R. T., "The simple consistency of a set theory based on the logic $\rm CSQ$", Notre Dame Journal of Formal Logic, vol. 24 (1983), pp. 431--49.
• [10] Brady, R. T., "The non-triviality of dialectical set theory", pp. 437--71 in Paraconsistent Logic: Essays on the Inconsistent, edited by G. Priest, R. Routley, and J. Norman, Analytica, Philosophia Verlag GmbH, Munich, 1989.
• [11] Bunder, M. W., "The answer to a problem of Iséki on BCI-algebras", Mathematics Seminar Notes, Kobe University, vol. 11 (1983), pp. 167--69.
• [12] Bunder, M. W., "BCK and related algebras and their corresponding logics", The Journal of Non-Classical Logic, vol. 2 (1983), pp. 15--24.
• [13] Bunder, M. W., "Tautologies that, with an unrestricted comprehension axiom, lead to inconsistency or triviality", The Journal of Non-Classical Logic, vol. 3 (1986), pp. 5--12.
• [14] Buszkowski, W., "The finite model property for BCI and related systems", Studia Logica, vol. 57 (1996), pp. 303--23.
• [15] Buszkowski, W., "Finite models of some substructural logics", Mathematical Logic Quarterly, vol. 48 (2002), pp. 63--72.
• [16] Geach, P. T., "On insolubilia", Analysis, vol. 15 (1954), pp. 71--72. reprinted in [gea72?].
• [17] Geach, P. T., Logic Matters, University of California Press, Berkeley, 1972.
• [18] Hazen, A., "A variation on a paradox", Analysis, vol. 50 (1990), pp. 7--8.
• [19] Humberstone, L., "Contra-classical logics", Australasian Journal of Philosophy, vol. 78 (2000), pp. 437--74.
• [20] Humberstone, L., "The pleasures of anticipation: Enriching intuitionistic logic", Journal of Philosophical Logic, vol. 30 (2001), pp. 395--438.
• [21] Humberstone, L., "Implicational converses", Logique et Analyse. Nouvelle Série, vol. 45 (2002), pp. 61--79.
• [22] Kabziński, J. K., "Basic properties of the equivalence", Studia Logica, vol. 41 (1982), pp. 17--40.
• [23] Kabziński, J. K., "BCI-algebras from the point of view of logic", Bulletin of the Section of Logic, vol. 12 (1983), pp. 126--29.
• [24] Kabziński, J. K., "Abelian groups and identity connective", Bulletin of the Section of Logic, vol. 22 (1993), pp. 66--71.
• [25] Lemmon, E. J., Beginning Logic, 1st edition, edited by G. W. D. Berry, Nelson, London, 1965.
• [26] Łukasiewicz, J., Selected Works, edited by L. Borkowski, Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam, 1970.
• [27] Makinson, D., "A warning about the choice of primitive operators in modal logic", Journal of Philosophical Logic, vol. 2 (1973), pp. 193--96.
• [28] Meredith, C. A., and A. N. Prior, "Equational logic", Notre Dame Journal of Formal Logic, vol. 9 (1968), pp. 212--26.
• [29] Meyer, R. K., R. Routley, and J. M. Dunn, "Curry's Paradox", Analysis, vol. 39 (1979), pp. 124--28.
• [30] Meyer, R. K., and J. K. Slaney, "Abelian logic (from A to Z)", pp. 245--88 in Paraconsistent Logic, Essays on the Inconsistent, edited by G. Priest, R. Routley, and J. Norman, Analytica, Philosophia Verlag GmbH, Munich, 1989.
• [31] Meyer, R. K., "Peirced clean through", Bulletin of the Section of Logic, vol. 19 (1990), pp. 100--101.
• [32] Meyer, R. K., and H. Ono, "The finite model property for $\it BCK$" and $\it BCIW$, Studia Logica, vol. 53 (1994), pp. 107--18.
• [33] Paoli, F., Substructural Logics: A Primer, vol. 13 of Trends in Logic---Studia Logica Library, Kluwer Academic Publishers, Dordrecht, 2002.
• [34] Petersen, U., "Logic without contraction as based on inclusion and unrestricted abstraction", Studia Logica, vol. 64 (2000), pp. 365--403.
• [35] Poernomo, I., Variations on a Theme of Curry and Howard: Adapting the Curry-Howard Isomorphism for the Synthesis of Imperative and Structural Programs, Ph.D. thesis, Monash University, 2003.
• [36] Prawitz, D., Natural Deduction. A Proof-Theoretical Study, vol. 3 of Stockholm Studies in Philosophy, Almqvist & Wiksell, Stockholm, 1965.
• [37] Prior, A. N., "Curry's paradox and three-valued logic", Australasian Journal of Philosophy, vol. 33 (1955), pp. 177--82.
• [38] Raftery, J. G., and C. J. van Alten, "Residuation in commutative ordered monoids with minimal zero", Reports on Mathematical Logic, vol. 34 (2000), pp. 23--57. Corrigendum: Reports on Mathematical Logic, vol. 39 (2005), pp. 133--35.
• [39] Restall, G., "How to be really contraction free", Studia Logica, vol. 52 (1993), pp. 381--91.
• [40] Restall, G., An Introduction to Substructural Logics, Routledge, London, 2000.
• [41] Rogerson, S., and S. Butchart, "Naï ve comprehension and contracting implications", Studia Logica, vol. 71 (2002), pp. 119--32.
• [42] Rogerson, S., and G. Restall, "Routes to triviality", Journal of Philosophical Logic, vol. 33 (2004), pp. 421--36.
• [43] Shaw-Kwei, M., "Logical paradoxes for many-valued systems", The Journal of Symbolic Logic, vol. 19 (1954), pp. 37--40.
• [44] Skolem, T., "A set theory based on a certain $3$"-valued logic, Mathematica Scandinavica, vol. 8 (1960), pp. 127--36.
• [45] Skolem, T., "Studies on the axiom of comprehension", Notre Dame Journal of Formal Logic, vol. 4 (1963), pp. 162--70.
• [46] Skolem, T., "Investigations on a comprehension axiom without negation in the defining propositional functions", Notre Dame Journal of Formal Logic, vol. 1 (1960), pp. 13--22.
• [47] Slaney, J., "A simple proof of consistency for a naive axiom of propositional comprehension", The Journal of Non-Classical Logic, vol. 4 (1987), pp. 23--31.
• [48] Slaney, J. K., "RWX is not Curry paraconsistent", pp. 472--80 in Paraconsistent Logic: Essays on the Inconsistent, edited by G. Priest, R. Routley, and J. Norman, Analytica, Philosophia Verlag GmbH, Munich, 1989.
• [49] Slaney, J. K., "A general logic", Australasian Journal of Philosophy, vol. 68 (1990), p. 74--88.
• [50] Smiley, T., "Relative necessity", The Journal of Symbolic Logic, vol. 28 (1963), pp. 113--34.
• [51] Sobociński, B., "Note about Łukasiewicz's theorem concerning the system of axioms of the implicational propositional calculus", Notre Dame Journal of Formal Logic, vol. 19 (1978), pp. 457--60.
• [52] Terui, K., "Light affine set theory: A naï ve set theory of polynomial time", Studia Logica, vol. 77 (2004), pp. 9--40.
• [53] Thistlewaite, P. B., M. A. McRobbie, and R. K. Meyer, Automated Theorem-Proving in Nonclassical Logics, Research Notes in Theoretical Computer Science, Pitman Publishing Ltd., London, 1988.
• [54] Thomas, I., "A proof of a theorem of Łukasiewicz", Notre Dame Journal of Formal Logic, vol. 12 (1971), pp. 507--508.
• [55] van Benthem, J. F. A. K., "Four paradoxes", Journal of Philosophical Logic, vol. 7 (1978), pp. 49--72.
• [56] White, R. B., "The consistency of the axiom of comprehension in the infinite-valued predicate logic of Łukasiewicz", Journal of Philosophical Logic, vol. 8 (1979), pp. 509--34.
• [57] White, R. B., "A demonstrably consistent type-free extension of the logic $\it BCK$", Mathematica Japonica, vol. 32 (1987), pp. 149--69.
• [58] White, R. B., "A consistent theory of attributes in a logic without contraction", Studia Logica, vol. 52 (1993), pp. 113--42.
• [59] Zucker, J., "The correspondence between cut-elimination and normalization. Part I. Intuitionistic predicate logic", Annals of Mathematical Logic, vol. 7 (1974), pp. 1--112; errata, ibid., p. 156.