Geodesic
Modified realizability and predicate logic
Matematičeskie zametki, Tome 61 (1997) no. 2, pp. 259-269

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Semantics of predicate formulas based on the notion of modified realizability for arithmetic formulas and interpretations of the language of arithmetic in all finite types are considered. For a number of natural constructive interpretations, the corresponding predicate logic of modified realizability is proved to be nonarithmetical. 
@article{MZM_1997_61_2_a7,
     author = {V. E. Plisko},
     title = {Modified realizability and predicate logic},
     journal = {Matemati\v{c}eskie zametki},
     pages = {259--269},
     publisher = {mathdoc},
     volume = {61},
     number = {2},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1997_61_2_a7/}
}
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V. E. Plisko. Modified realizability and predicate logic. Matematičeskie zametki, Tome 61 (1997) no. 2, pp. 259-269. http://geodesic.mathdoc.fr/item/MZM_1997_61_2_a7/