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In a series of papers beginning in 1944, the Dutch mathematician and philosopher George Francois Cornelis Griss proposed that constructive mathematics should be developed without the use of the intuitionistic negation 1 and, moreover, without any use of a null predicate. In the present work, we give formalized versions of intuitionistic arithmetic, analysis, and higher-order arithmetic in the spirit of Griss' negationless intuitionistic mathematics and then consider their relation to the current formalizations of these theories.