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Deducibility and many-valuedness

  • D. J. Shoesmith (a1) and T. J. Smiley (a1)


Lindenbaum's construction of a matrix for a propositional calculus, in which the wffs themselves are taken as elements and the theorems as the designated elements, immediately establishes two general results: that every prepositional calculus is many-valued, and that every many-valued propositional calculus is also ℵ0-valued. These results are however concerned exclusively with theoremhood, the inferential structure of the calculus being relevant only incidentally, in that it may serve to determine the set of theorems. We therefore ask what happens when deducibility is taken into consideration on a par with theoremhood. The answer is that in general the Lindenbaum construction is no longer adequate and both results fail.



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[1]Anderson, Alan Ross, On the interpretation of a modal system of Łukasiewicz, The journal of computing systems, vol. 1, no. 4 (1954), pp. 209210.
[2]Åqvist, Lennart, Results concerning some modal systems that contain S2, this Journal, vol. 29 (1964), pp. 7987.
[3]Glivenko, V., Sur quelques points de la logique de M. Brouwer, Académie Royale de Belgique, Bulletins de la classe des sciences, ser. 5, vol. 15 (1929), pp. 183188.
[4]Gödel, Kurt, Zum intuitionistischen Aussagenkalkül, Ergebnisse eines mathematischen Kolloquiums, Heft 4 (1931/1932), p. 40.
[5]Halldén, Sören, On the semantic non-completeness of certain Lewis calculi, this Journal, vol. 16 (1951), pp. 127129.
[6]Harrop, Ronald, Some structure results for propositional calculi, this Journal, vol. 30 (1965), pp. 271292.
[7]Hay, Louise Schmir, Axiomatization of the infinite-valued predicate calculus, this Journal, vol. 28 (1963), pp. 7786.
[8]Heyting, Arend, Die formalen Regeln der intuitionistischen Logik, Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse, 1930, pp. 4256.
[9]Hilbert, D. and Bernays, P., Grundlagen der Mathematik, vol. 2, Springer, Berlin, 1939.
[10]Johansson, Ingebrigt, Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus, Compositio mathematica, vol. 4 (1936), pp. 119136.
[11]Kleene, Stephen Cole, Introduction to metamathematics, North-Holland, Amsterdam, 1952.
[12]Kreisel, G. and Krivine, J. L., Elements of mathematical logic, North-Holland, Amsterdam, 1967.
[13]Lemmon, E. J., A note on Halldén-incompleteness, Notre Dame journal of formal logic, vol. 7 (1966), pp. 296300.
[14]Lewis, Clarence Irving and Lanoford, Cooper Harold, Symbolic logic, 2nd edition, Dover, New York, 1959.
[15]Łukasiewicz, Jan, A system of modal logic, The journal of computing systems, vol. 1, no. 3 (1953), pp. 111149.
[16]Łukasiewicz, J. and Tarski, A., Untersuchungen über den Aussagenkalkül, Comptes rendus des séances de la Société des sciences et des lettres de Varsovie, Classe III, vol. 23 (1930), pp. 3050.
[17]McKinsey, J. C. C., Systems of modal logic which are not unreasonable in the sense of Halldén, this Journal, vol. 18 (1953), pp. 109113.
[18]McNaughton, Robert, A theorem about infinite-valued sentential logic, this Journal, vol. 16 (1951), pp. 113.
[19]Rogers, Hartley Jr., Theory of recursive functions and effective Calculability, McGraw-Hill, New York, 1967.
[20]Rose, Alan and Rosser, J. Barkley, Fragments of many-valued statement calculi, Transactions of the American Mathematical Society, vol. 87 (1958), pp. 153.
[21]Rosser, J. B. and Turquette, A. R., Many-valued logics, North-Holland, Amsterdam, 1952.
[22]Smiley, Timothy, On Łukasiewicz's Ł-modal system, Notre Dame journal of formal logic, vol. 2 (1961), pp. 149153.

Deducibility and many-valuedness

  • D. J. Shoesmith (a1) and T. J. Smiley (a1)


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