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Some consistency proofs and a characterization of inconsistency proofs in illative combinatory logic

Published online by Cambridge University Press:  12 March 2014

M. W. Bunder
M. W. Bunder*
Affiliation:
Department of Mathematics, University of Wollongong, Wollongong, New South Wales 2500, Australia
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Extract

It is well known that combinatory logic with unrestricted introduction and elimination rules for implication is inconsistent in the strong sense that an arbitrary term Y is provable. The simplest proof of this, now usually called Curry's paradox, involves for an arbitrary term Y, a term X defined by X = Y(CPy).

The fact that X = P XY = X ⊃ Y is an essential part of the proof.

The paradox can be avoided by placing restrictions on the implication introduction rule or on the axioms from which it can be proved.

In this paper we determine the forms that must be taken by inconsistency proofs of systems of propositional calculus based on combinatory logic, with arbitrary restrictions on both the introduction and elimination rules for the connectives. Generally such a proof will involve terms without normal form and cut formulas, i.e. formulas formed by an introduction rule that are immediately removed by an elimination with at most some equality steps intervening. (Such a sequence of steps we call a “cut”.)

The above applies not only to the strong form of inconsistency defined above, but also to the weak form of inconsistency defined by: all propositions are provable, if this can be represented in the system.

Any inconsistency proof of this kind of system can be reduced to one where the major premise of the elimination rule involved in the cut and its deduction must also appear in the deduction of the minor premise involved in the cut.

We can, using this characterization of inconsistency proofs, put appropriate restrictions on certain introduction rules so that the systems, including a full classical propositional one, become provably consistent.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 52 , Issue 1 , March 1987 , pp. 89 - 110
Copyright
Copyright © Association for Symbolic Logic 1987

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References

REFERENCES

[1] Aczel, P., Frege structures and the notions of proposition, truth and set, The Kleene symposium. A collection of papers on recursion theory and intuitionism (Barwise, J. et al., editors), North-Holland, Amsterdam, 1980, pp. 31–59.Google Scholar
[2] Bunder, M. W., Propositional and predicate calculus based on combinatory logic, Notre Dame Journal of Formal Logic, vol. 15 (1974), pp. 25–32.CrossRefGoogle Scholar
[3] Bunder, M. W., Consistency notions in illative combinatory logic, this Journal, vol. 42 (1977), pp. 527–529.Google Scholar
[4] Bunder, M. W. and Seldin, J. P., Some anomalies in Fitch's system QD, this Journal, vol. 43 (1978), pp. 247–249.Google Scholar
[4a] Bunder, M. W., A generalized Kleene-Rosser paradox, Notre Dame Journal of Formal Logic, vol. 14(1973), pp. 53–54.CrossRefGoogle Scholar
[5] Bunder, M. W., The consistency of a higher order predicate calculus and set theory based on combinatory logic, Mathematical logic in Latin America ( proceedings of the fourth symposium ; Arruda, A. I. et al., editors), North-Holland, Amsterdam, 1980, pp. 73–82.Google Scholar
[6] Bunder, M. W., A one axiom set theory based on higher order predicate logic, Archiv für Mathematische Logik und Grundlagenforschung, vol. 23 (1983), pp. 99–107.CrossRefGoogle Scholar
[7] Bunder, M. W., Conjunction without conditions in illative combinatory logic, Polish Academy of Sciences, Institute of Philosophy and Sociology, Bulletin of the Section of Logic, vol. 13 (1984), pp. 207–212.Google Scholar
[8] Bunder, M. W., A weak consistency proof for some systems of illative combinatory logic, this Journal, vol. 48 (1983), pp. 771–776.Google Scholar
[9] Church, A., A set of postulates for the foundation of logic. I, II, Annals of Mathematics, ser. 2, vol. 33 (1932), pp. 346–366, and vol. 34 (1933), pp. 839–864.CrossRefGoogle Scholar
[10] Curry, H. B. and Feys, R., Combinatory logic. Vol. I, North-Holland, Amsterdam, 1959.Google Scholar
[11] Fitch, F. B., Elements of combinatory logic, Yale University Press, New Haven, Connecticut, 1974.Google Scholar
[12] Kleene, S. C. and Rosser, J. B., The inconsistency of certain formal logics, Annals of Mathematics, ser. 2, vol. 36 (1935), pp. 630–636.CrossRefGoogle Scholar
[13] Kuzichev, A. S., Deductive operators of combinatory logic, Moscow University Mathematics Bulletin, vol. 29 (1974), no. 3/4, pp. 8–14.Google Scholar
[14] Kuzichev, A. S., The expressive potentialities of deductive systems of λ-conversion and combinatory logic, Moscow University Mathematics Bulletin, vol. 29 (1974), no. 5/6, pp. 58–64.Google Scholar
[15] Sanchis, L., Types in combinatory logic, Notre Dame Journal of Formal Logic, vol. 5 (1964), pp. 161–180.CrossRefGoogle Scholar
[16] Seldin, J. P., On Martin Bunder's consistency proof, unpublished notes.Google Scholar