Geodesic
Extended fuzzy constructive logic
Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part XII, Tome 407 (2012), pp. 35-76

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A logical system is introduced which is similar to the “fuzzy constructive logic” earlier developed by the author, however this new system gives larger possibilities for establishing the truth of predicate formulas and logical deductions in the framework of this logic. The notions of strong and weak FCL$^*$-validity of predicate formulas are defined. It is proved that every formula deducible in the constructive (intuitionistic) predicate calculus is strongly FCL$^*$-valid. From other hand it is proved that some formulas not deducible in the mentioned calculus are not weakly FCL$^*$-valid. A definition is given for the semantics of the traditional constructive logic on the base of the developed logical apparatus. Theorems are proved showing differences between the extended fuzzy constructive logic and the traditional constructive logic. 
@article{ZNSL_2012_407_a2,
     author = {I. D. Zaslavsky},
     title = {Extended fuzzy constructive logic},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {35--76},
     publisher = {mathdoc},
     volume = {407},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_407_a2/}
}
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I. D. Zaslavsky. Extended fuzzy constructive logic. Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part XII, Tome 407 (2012), pp. 35-76. http://geodesic.mathdoc.fr/item/ZNSL_2012_407_a2/