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Logics without the contraction rule. (English) Zbl 0583.03018
Syntactical and semantical properties of propositional logics weaker than the intuitionistic, in which the contraction rule does not hold, are studied. Main tools of this paper are Gentzen-type formulation on the one hand, and Kripke-type semantics using partially ordered monoids on the other hand. These tools give us a uniform way of treating a lot of nonclassical logics, e.g. \({\L}ukasiewicz's\) many-valued logics, relevant logics, the intuitionistic logic and logics related to BCK-algebras, as shown in § 6 and § 7. As an application, some algebraic results on BCK and related algebras are obtained in § 8.
On this occasion, we notice that Problem 1 in § 9 was solved affirmatively by Wroński and Idziak, and that Idziak solved also Problem 7, by giving the necessary and sufficient condition for the class of D-BCC-algebras to form a variety, after the publication of this paper.

MSC:
03B60 Other nonclassical logic
03B20 Subsystems of classical logic (including intuitionistic logic)
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[1] Proof theory 81 (1975) · Zbl 0354.02027
[2] Semantics for relevant logics 37 pp 159– (1972)
[3] Truth, syntax and modality pp 199– (1973)
[4] Mathematica Japonica 23 pp 1– (1978)
[5] Mathematica Japonica 28 pp 157– (1983)
[6] Journal of Tsuda College 5 pp 67– (1973)
[7] Journal of the Faculty of Science, University of Tokyo, Section 1 12 pp 223– (1966)
[8] The separation theorem of intuitionist propositional logics 27 pp 391– (1962)
[9] General lattice theory (1978)
[10] Journal of Philosophical Logic 3 pp 347– (1974)
[11] Transactions of the American Mathematical Society 93 pp 74– (1959)
[12] DOI: 10.1090/S0002-9947-1958-0094299-1 · doi:10.1090/S0002-9947-1958-0094299-1
[13] Algebra i Logika 16 pp 643– (1977)
[14] Polish Academy of Sciences, Institute of Philosophy and Sociology, Bulletin of the Section of Logic 12 pp 50– (1983)
[15] Formal systems and recursive functions pp 92– (1965)
[16] Reports of Faculty of Science, Shizuoka University 17 pp 13– (1983)
[17] Mathematica Japonica 29 (1984)
[18] Nagoya Mathematical Journal 84 pp 119– (1981) · Zbl 0482.03007 · doi:10.1017/S0027763000019577
[19] On a decision procedure for free lo-algebraic systems (1974)
[20] Polish Academy of Sciences, Institute of Philosophy and Sociology, Bulletin of the Section of Logic 6 pp 2– (1977)
[21] Mathematica Japonica 29 pp 115– (1984)
[22] Mathematica Japonica 28 pp 215– (1983)
[23] Mathematica Japonica 28 pp 211– (1983)
[24] DOI: 10.1002/malq.19730190703 · Zbl 0301.02019 · doi:10.1002/malq.19730190703
[25] DOI: 10.1090/pspum/025/0363802 · doi:10.1090/pspum/025/0363802
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