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A note on some intermediate propositional calculi

Published online by Cambridge University Press:  12 March 2014

Branislav R. Boričić
Branislav R. Boričić*
Affiliation:
University of Beograd, Kamenička 6, 11000 Beograd, Yugoslavia
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Extract

This note is written in reply to López-Escobar's paper [L-E] where a “sequence” of intermediate propositional systems NLCn (n ≥ 1) and corresponding implicative propositional systems NLICn (n ≥ 1) is given. We will show that the “sequence” NLCn contains three different systems only. These are the classical propositional calculus NLC1, Dummett's system NLC2 and the system NLC3. Accordingly (see [C], [Hs2], [Hs3], [B 1], [B2], [Hs4], [L-E]), the problem posed in the paper [L-E] can be formulated as follows: is NLC3a conservative extension of NLIC3? Having in mind investigations of intermediate propositional calculi that give more general results of this type (see V. I. Homič [H1], [H2], C. G. McKay [Mc], T. Hosoi [Hs 1]), in this note, using a result of Homič (Theorem 2, [H1]), we will give a positive solution to this problem.

NLICnand NLCn. If X and Y are propositional logical systems, by X ⊆ Y we mean that the set of all provable formulas of X is included in that of Y. And X = Y means that X ⊆ Y and Y ⊆ X. A(P1/B1, …, Pn/Bn) is the formula (or the sequent) obtained from the formula (or the sequent) A by substituting simultaneously B1, …, Bn for the distinct propositional variables P1, …, Pn in A.

Let Cn(n ≥ 1) be the string of the following sequents:

Having in mind that the calculi of sequents can be understood as meta-calculi for the deducibility relation in the corresponding systems of natural deduction (see [P]), the systems of natural deductions NLCn and NLICn (n ≥ 1), introduced in [L-E], can be identified with the calculi of sequents obtained by adding the sequents Cn as axioms to a sequential formulation of the Heyting propositional calculus and to a system of positive implication, respectively (see [C], [Ch], [K], [P]).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 2 , June 1984 , pp. 329 - 333
Copyright
Copyright © Association for Symbolic Logic 1984

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References

REFERENCES

[B1]Bull, R. A., The implicational fragment of Dummett's LC, this Journal, vol. 27 (1962), pp. 189–192.Google Scholar
[B2]Bull, R. A., Some results for implicational calculi, this Journal, vol. 29 (1964), pp. 33–39.Google Scholar
[Ch]Church, A., Introduction to mathematical logic. Vol. I, Princeton University Press, Princeton, New Jersey, 1956.Google Scholar
[C]Curry, H. B., Foundations of mathematical logic, McGraw-Hill, New York, 1963.Google Scholar
[D]Dummett, M., A propositional calculus with denumerable matrix, this Journal, vol. 24 (1959), pp. 96–107.Google Scholar
[H1]Homič, V. I., A separability theorem for superinluilionistic propositional calculi, Soviet Mathematics—Doklady, vol. 17 (1976), pp. 1214–1216.Google Scholar
[H2]Homič, V. I., On the problem of separability for superintuitionistic propositional logics, Soviet Mathematics—Doklady, vol. 22 (1980), pp. 476–479.Google Scholar
[Hr]Horn, A., The separation theorem of intuitionist propositional logics, this Journal, vol. 27 (1962), pp. 391–399.Google Scholar
[Hs 1]Hosoi, T., On the separation theorem of intermediate propositional calculi, Proceedings of the Japan Academy, vol. 42 (1966), pp. 535–538.Google Scholar
[Hs2]Hosoi, T., The separation theorem on the classical system, Journal of the Faculty of Science, University of Tokyo, vol. 12 (1966), pp. 223–230.Google Scholar
[Hs 3]Hosoi, T., Algebraic proof of the separation theorem on classical propositional calculus, Proceedings of the Japan Academy, vol. 42 (1966), pp. 67–69.Google Scholar
[Hs4]Hosoi, T., Algebraic proof of the separation theorem on Dummett's LC, Proceedings of the Japan Academy, vol. 42 (1966), pp. 693–695.Google Scholar
[K]Kleene, S. C., Introduction to metamathematics, North-Holland, Amsterdam, 1952.Google Scholar
[L-E]López-Escobar, E. G. K., Implicational logics in natural deduction systems, this Journal, vol. 47 (1982), pp. 184–186.Google Scholar
[Mc]McKay, C. G., The non-separability of a certain finite extension of Heyting's propositional logic, Koninklijke Nederlandse Akademie van Wetenschappen. Proceedings, Series A, vol. 71 (1968), pp. 312–315.Google Scholar
[Pr]Prawitz, D., Natural deduction, Almqvist & Wiksell, Stockholm, 1965.Google Scholar
[W]Wajsberg, M., Untersuchungen über den Aussagenkalkül von A. Heyting, Wiadomości Matematyczne, vol. 46 (1938), pp. 45–101. (See also the Review by B. Rosser in this Journal, vol. 3 (1938), p. 169.)Google Scholar