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A note on finite axiomatization of partial propositional calculi

Published online by Cambridge University Press:  12 March 2014

W. E. Singletary*
Affiliation:
The Pennsylvania State University

Extract

At the Princeton Bicentennial in 1946 Tarski posed the question as to whether or not certain problems connected with partial propositional calculi were recursively solvable. Since that time the specific problems mentioned by Tarski as well as a number of related problems have been shown to be recursively unsolvable. Such results are due to Post and Linial [5], Yntema [9], Gladstone [2], Ihrig [3] and Singletary [7], [8].

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1967

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References

[1]Church, A., Introduction to mathematical logic, vol. 1, Princeton University Press, Princeton, N. J., 1956.Google Scholar
[2]Gladstone, M. D., Some ways of constructing a propositional calculus of any required degree of unsolvability, Transactions of the American Mathematical Society, vol. 118 (1965), pp. 192210.CrossRefGoogle Scholar
[3]Harrop, R., A relativization procedure for propositional calculi with an application to a generalized form of Post's theorem. Proceedings of the London Mathematical Society, vol. 14 (1964), pp. 595617.CrossRefGoogle Scholar
[4]Ihrig, A. H., The Post Lineal theorems for arbitrary recursively enumerable degrees of unsolvability, Notre Dame journal of formal logic, vol. 4 (1965), pp. 5471.Google Scholar
[5]Kleene, S. C., Introduction to metamathematics, D. Van Nostrana Co., Inc., New York, 1952.Google Scholar
[6]Linial, S. and Post, E. L., Recursive unsolvability of the deducibility, Tarsk's completeness and independence of axioms problems of the propositional calculus, Bulletin of the American Mathematical Society, vol. 55 (1949), p. 50 (Abstract).Google Scholar
[7]Lukasiewicz, J. and Tarski, A., Untersuchungen über den Assagenkalkül, Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III, vol. 23 (1930), pp. 3050.Google Scholar
[8]Singletary, W. E., A Complex of problems proposed by Post, Bulletin of the American Mathematical Society, vol. 70 (1964), pp. 105109.CrossRefGoogle Scholar
[9]Singletary, W. E., Doctoral Dissertation, University of Illinois, Urbana, Ill., 1964.Google Scholar
[10]Yntema, M. K., A detailed argument for the Post-Lineal theorems, Notre Dame journal of formal logic, vol. 5 (1964), pp. 3750.CrossRefGoogle Scholar