Nilpotent Minimum Logic NM and Pretabularity

Authors

  • Eunsuk Yang Jeonbuk National University, Department of Philosophy & Institute of Critical Thinking and Writing

DOI:

https://doi.org/10.18778/0138-0680.2020.01

Keywords:

pretabularity, nilpotent minimum logic, algebraic semantics, fuzzy logic, finite model property

Abstract

This paper deals with pretabularity of fuzzy logics. For this, we first introduce two systems NMnfp and NM½, which are expansions of the fuzzy system NM (Nilpotent minimum logic), and examine the relationships between NMnfp and the another known extended system NM. Next, we show that NMnfp and NM½ are pretabular, whereas NM is not. We also discuss their algebraic completeness.

 

References

[1] L. Běhounek and P. Cintula, Fuzzy logics as the logics of chains, Fuzzy Sets and Systems, Vol. 157 (2006), pp. 604–610.
Google Scholar

[2] P. Cintula, Weakly Implicative (Fuzzy) Logics I: Basic properties, Archive for Mathematical Logic, Vol. 45 (2006), pp. 673–704.
Google Scholar

[3] J. M. Dunn, Algebraic completeness for R-mingle and its extensions, The Journal of Symbolic Logic, Vol. 35 (1970), pp. 1–13.
Google Scholar

[4] J. M. Dunn and G. Hardegree, Algebraic Methods in Philosophical Logic, Oxford University Press, Oxford, 2001.
Google Scholar

[5] J. M. Dunn and R. K. Meyer, Algebraic completeness results for Dummett's LC and its extensions, Mathematical Logic Quarterly, Vol. 17 (1971), pp. 225–230.
Google Scholar

[6] F. Esteva and L. Godo, Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems, Vol. 124 (2001), pp. 271–288.
Google Scholar

[7] D. Gabbay and V. B. Shetman, Undecidability of modal and intermediate first-order logics with two individual variables, The Journal of Symbolic Logic, Vol. 58 (1993), pp. 800–823.
Google Scholar

[8] L. Galminas and J. G. Mersch, A pretabular classical relevance logic, Studia Logica, Vol. 100 (2012), pp. 1211–1221.
Google Scholar

[9] J. Gispert, Axiomatic extensions of the nilpotent minimum logic, Reports on Mathematical Logic, Vol. 37 (2003), pp. 113–123.
Google Scholar

[10] C. Noguera, F. Esteva, and J. Gispert, On triangular norm based axiomatic extensions of the weak nilpotent minimum logic, Mathematical Logic Quarterly, Vol. 54 (2008), pp. 387–409.
Google Scholar

[11] V. Rybakov, V. Kiyatkin, and M. Terziler, Independent bases for rules admissible in pretabular logics, Logic Journal of the Interest Group in Pure and Applied Logics, Vol. 7 (1999), pp. 253–266.
Google Scholar

[12] T. Sugihara, Strict implication free from implicational paradoxes, Memoirs of the Faculty of Liberal Arts, Fukui University, Series 1, 1955, pp. 55–59.
Google Scholar

[13] K. Świrydowicz, 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 nite height, The Journal of Symbolic Logic, Vol. 73 (2008), pp. 1249–1270.
Google Scholar

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Published

2020-03-30

How to Cite

Yang, E. (2020). Nilpotent Minimum Logic NM and Pretabularity. Bulletin of the Section of Logic, 49(1), 1–11. https://doi.org/10.18778/0138-0680.2020.01

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Section

Research Article