Geodesic
Level logics
Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part IX, Tome 220 (1995), pp. 72-82

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This paper is devoted to description of level logics whose logical values are rational numbers. All positive logical values are true, negative logical values are false, and zero logical values are paradoxes. Level logic using all rational numbers from the segment $[-1/2,1/2]$ is similar to the fuzzy logic of Zadeh. A sequent calculus for such logics is proposed. Convertible logical rules of inference are defined. Note that the rules of predicate calculus for nonbinary logics with inequalities were not proposed previously. It is proved that the algorithm of checking the deducibility in this calculus belongs to the class EXP-LIN-TIME. Bibliography: 9 titles. 
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     author = {N. K. Kossovski},
     title = {Level logics},
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     year = {1995},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_1995_220_a4/}
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N. K. Kossovski. Level logics. Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part IX, Tome 220 (1995), pp. 72-82. http://geodesic.mathdoc.fr/item/ZNSL_1995_220_a4/