There exists an uncountable set of pretabular extensions of the relevant logic R and each logic of this set is generated by a variety of finite height. (English) Zbl 1161.03011

A propositional logic is called pretabular if it is not generated by a finite algebra but every proper extension is. As for such logics, the results by Maksimova are well-known; there exist exactly three pretabular extensions of the intuitionistic logic, and exactly five pretabular extensions of the modal logic S4. On the other hand, the only known pretabular extension of the relevant logic R was the logic RM shown by Dunn, who then proposed a problem to find other pretabular extensions of R. The main results of this paper described by the title, give an answer to the problem showing a sharp contrast to the Maksimova’s results.


03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
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[1] Studia Logica XLII pp 49– (1983)
[2] Theoria 40 pp 52– (1978)
[3] Handbook of philosophical logic 3 pp 117–
[4] Algebraic completeness results for R-mingle and its extensions 35 pp 1– (1970) · Zbl 0231.02024
[5] Memoirs of the American Mathematical Society 77 (1989)
[6] DOI: 10.1007/BF00375898 · Zbl 0616.06003 · doi:10.1007/BF00375898
[7] DOI: 10.1007/BF00370315 · Zbl 0457.03018 · doi:10.1007/BF00370315
[8] Entailment 1 · Zbl 1090.08009
[9] DOI: 10.1007/BF00370150 · doi:10.1007/BF00370150
[10] Journal of Philosophical Logic 22 pp 449– (1993)
[11] There exist exactly two maximal strictly relevant extensions of the relevant logic R 64 pp 1125– (1999) · Zbl 0947.03029
[12] Reports on Mathematical Logic 29 pp 19– (1995)
[13] Truth, semantics and modality pp 194– (1973)
[14] Algebra i Logika 14 pp 28– (1974)
[15] Algebra i logika 12 pp 445– (1973)
[16] Algebra i Logika 11 pp 558– (1972)
[17] DOI: 10.1002/malq.19900360606 · Zbl 0696.03004 · doi:10.1002/malq.19900360606
[18] Handbook of philosophical logic 6 pp 1– (2002)
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