Abstract
Medvedev's intermediate logic (MV) can be defined by means of Kripke semantics as the family of Kripke frames given by finite Boolean algebras without units as partially ordered sets. The aim of this paper is to present a proof of the theorem: For every set of connectivesΦ such that\(\{ \to , \vee , \urcorner \} \not \subseteq \Phi \subseteq \{ \to , \wedge , \urcorner \} \) theΦ-fragment ofMV equals theΦ fragment of intuitionistic logic. The final part of the paper brings the negative solution to the problem set forth by T. Hosoi and H. Ono, namely: is an intermediate logic based on the axiom (⌝a→b∨c) →(⌉a→b)∨(⌝a → c) separable?
Similar content being viewed by others
References
P. Crawley andR. P. Dilworth,Algebraic theory of lattices, Prentice-Hall, INC. Englewood Cliffs, New Jersey 1973.
M. C. Fitting,Intuitionistic Logic, Modal Theory and Forcing, North-Holland, Amsterdam 1969.
D. M. Gabbay,The decidability of the Kreisel-Putnam system,The Journal of Symbolic Logic 35 (1970), pp. 431–437.
A. Grzegorczyk,Some relational systems and the associated topological spaces,Fundamenta Mathematicae 60 (1967), pp. 223–231.
T. Hosoi andH. Ono,Intermediate propositional logics (A survey),Journal of Tsuda College 5 (1973), pp. 67–82.
V. A. Jankov,Some superconstructive propositional calculi,Doklady Akademii Nauk SSSR 151 (1963), pp. 796–797Soviet Mathematics Doklady 4 (1963), pp. 1103–1105.
—,The calculus of the “law of excluded middle”,Izviestija AkademiiNauk SSSR. 32 (1968), pp. 1044–1051Mathematics USSR — Izvestija 2 (1968), pp.997–1004.
S. Jaśkowski,Recherches sur le systéme de la logique intuitioniste,Actualités: Scientifiques et Industrielles 393 (1936), pp. 58–61.
D. H. J. de Jongh andA. S. Troelstra,On the connection of partially ordered sets with some pseudo-Boolean algebras,Indagationes Mathematicae 28 (1966), pp. 317–329.
L. L. Maksimova, Skvorcov D. P. andV. B. Šehtman,Neyozmožnost' konečnoi aksiomatizacii logiki finitnyh zadač Medvedeva,Doklady Akademii Nauk SSSR 245 (1979), pp. 1051–1054.
C. G. McKay,A note on the Jaśkowski sequence,Zeitschrift für mathematische Logik und Grundlagen der Mathematik 13 (1967), pp. 95–96.
T. J. Medvedev,Interpretacija logičeskih formul posredstvom finitnyh zadač i svjaz' ee s teoriei realizuemosti,Doklady Akademii Nauk SSSR 148 (1963), pp. 771–774.
—,Finitnye zadači,Doklady Akademii Nauk SSSR 142 (1962), pp. 1015–1018.
H. Ono,Kripke models and intermediate logics,Publications of the Research Institute for Mathematical Sciences 6 (1970–71), pp. 461–476.
H. Rasiowa andR. Sikorski,The Metamathematics of Mathematics PWN Warszawa 1970.
K. Segerberg,Proof of conjecture of McKay,Fundamenta Mathematicae 81 (1974), pp. 267–270.
D. P. Skvorcov,O realizuemosti i finitnoi obščeznačimosti propozicionalnyh formul s ograničenijami na vhoždenija implikacii,Matematičeskie Zametki 25 (1979), pp. 919–931.
—,O vhoždenii implikacii v finitno obščeznačimye intuicionistski nedokazuemye formuly logiki vyskazyvanii,Matematičeskie Zametki 20 (1976), pp. 383–390.
C. Smoryński,Investigations of the intuitionistic formal systems by means of Kripke models. Doctoral dissertation, Stanford University 1972.
A. S. Troelstra,On intermediate propositional logics,Indagationes Mathematicae 27 (1965), pp. 141–152.
T. Umezawa,Über Zwischensysteme der Aussagenlogik,Nagoya Mathematic Journal 9 (1955), pp. 181–189.
Author information
Authors and Affiliations
Additional information
The author is much obliged to Professor Andrzej Wroński for his precious suggestions which were of great help in writing this paper.
Rights and permissions
About this article
Cite this article
Szatkowski, M. On fragments of Medvedev's logic. Stud Logica 40, 39–54 (1981). https://doi.org/10.1007/BF01837554
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01837554