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**Linear Kripke frames and Gödel logics.**
*(English)*
Zbl 1118.03016

First-order Gödel logics are determined by closed subsets of the unit interval [0,1], containing 0 and 1, as their truth-value sets, which are called Gödel sets in this paper. Recently, the logics have received increasing attention particularly in relation to fuzzy logic and, in fact, several interesting results have been obtained in the last decades. Since Gödel logics are intermediate predicate logics enjoying both the so-called constant domain axiom and Dummett’s linear axiom, Kripke frames for them should be those linear with constant domains.

In this paper, the authors establish a correspondence between Gödel sets and countable linear Kripke frames so as to determine the same logics. That is, for any such Kripke frame there is a Gödel logic which coincides with the logic defined by the Kripke frame on constant domains and vice versa. The correspondence allows us to transfer results on one side to those of the other, respectively, among which two remarkable applications from observations on Gödel logics are mentioned here: one is a complete characterization of axiomatizability of logics based on countable linear Kripke frames with constant domains, and the other that the total number of logics defined by countable linear Kripke frames on constant domains is countable.

In this paper, the authors establish a correspondence between Gödel sets and countable linear Kripke frames so as to determine the same logics. That is, for any such Kripke frame there is a Gödel logic which coincides with the logic defined by the Kripke frame on constant domains and vice versa. The correspondence allows us to transfer results on one side to those of the other, respectively, among which two remarkable applications from observations on Gödel logics are mentioned here: one is a complete characterization of axiomatizability of logics based on countable linear Kripke frames with constant domains, and the other that the total number of logics defined by countable linear Kripke frames on constant domains is countable.

Reviewer: Osamu Sonobe (Follonica)

### Keywords:

linear Kripke frame; Gödel logic; intermediate predicate logic; constant domain; Dummett’s LC; axiomatizability
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\textit{A. Beckmann} and \textit{N. Preining}, J. Symb. Log. 72, No. 1, 26--44 (2007; Zbl 1118.03016)

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### References:

[1] | Annals of Pure and Applied Logic (2006) |

[2] | Intermediate predicate logics determined by ordinals 55 pp 1099–1124– (1990) |

[3] | Computer science logic pp 1–15– (1996) |

[4] | Tsukuba Journal of Mathematics 11 pp 101–105– (1987) |

[5] | Theoretical Computer Science 160 pp 241–270– (1996) |

[6] | Algebraic and Topological Methods in Non-classical Logics II, Barcelona, 15–18 June, 2005 pp 73–74– (2005) |

[7] | Proceedings Gödel ’96, Logic foundations of mathematics, computer science and physics–Kurt Gödel’s legacy 6 pp 23–33– (1996) |

[8] | Die Nichtaxiomatisierbarkeit des unendlichwertigen Prädikatenkalkülus von Łukasiewicz 27 pp 159–170– (1962) |

[9] | Proceedings of the American Mathematical Society 3 pp 677–680– (1952) |

[10] | Logic with truth values in a linearly ordered Heyting algebra 34 pp 395–409– (1969) |

[11] | Metamathematics of fuzzy logic (1998) · Zbl 0937.03030 |

[12] | Journal of Applied Logic 1 pp 309–392– (2003) |

[13] | Ergebnisse eines mathematischen Kolloquiums 4 pp 34–38– (1933) |

[14] | Semantical investigations in Heyting’s intuitionistic logic 148 (1981) · Zbl 0453.03001 |

[15] | Comptes Rendus de l’Académie des Sciences Paris 226 pp 1330–1331– (1948) |

[16] | Intuitionistic logic, model theory and forcing (1969) |

[17] | Heyting algebra, I, Duality theory (1985) |

[18] | Intuitionistic fuzzy logic and intuitionistic fuzzy set theory 49 pp 851–866– (1984) |

[19] | Formal systems and recursive functions (Proceedings of the Eighth Logic Colloquium, Oxford, 1963) pp 92–130– (1965) |

[20] | On the complexity of propositional quantification in intuitionistic logic 62 pp 529–544– (1997) · Zbl 0887.03002 |

[21] | Classical descriptive set theory (1995) |

[22] | Modal logic 35 (1997) · Zbl 0871.03007 |

[23] | Lattice theory (1967) |

[24] | Computer Science Logic, Proceedings of the CSL 2000 pp 178–201– (2000) |

[25] | Handbook of mathematical logic pp 973–1052– (1977) |

[26] | 28th International Symposium on Multiple-valued Logic. May 1998, Fukuoka, Japan. Proceedings pp 108–113– (1998) |

[27] | A propositional logic with denumerable matrix 24 pp 96–107– (1959) |

[28] | Houston Journal of Mathematics 5 pp 183–192– (1979) |

[29] | Proceedings of LPAR’2002 pp 327–336– (2002) |

[30] | Studia Logica 47 pp 391–399– (1988) |

[31] | Publications. Research Institute of Mathematical Sciences, Kyoto University 8 pp 619–649– (1972) |

[32] | Publications. Research Institute of Mathematical Sciences, Kyoto University 6 pp 461–476– (1971) |

[33] | Descriptive set theory 100 (1980) · Zbl 0433.03025 |

[34] | Studia Logica 46 pp 137–148– (1987) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.