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. 
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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/

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