In 1958 Gödel introduced the Dialectica interpretation  to prove the (relative) consistency of intuitionistic arithmetic and, over the years, this in- terpretation has been generalized from a categorical perspective by several authors, leading up to the notion of Dialectica category  (or more gener- ally fibration ). The most important clause of the Dialectica interpretation is the defini- tion of the translation of the implication connective. It is well-explained in [1, 3] that the crucial point is that this translation validates two logical prin- ciples which are usually not acceptable from a constructive point of view, namely a variant of the principle of independence of premises (IP) and a variant of Markov’s principle (MP). The main purpose of this talk is to provide a categorical explanation of the validity of (IP) and (MP) in the Dialectica interpretation by using the language of Lawvere’s doctrines. To achieve our purpose we employ the tool of existential (and univer- sal) completion introduced in  and further developed in  to define a proof-irrelevant categorification of the Dialectica interpretation given in  in terms of doctrines that we call Gödel doctrines. Then, we show that the categorical notions of existential-free elements introduced in  and universal-free elements developed in  provide a cat- egorical presentation of quantifier-free formulas and are the key ingredient to validate (IP) and (MP) underlying Gödel’s Dialectica interpretation. Finally, showing that every Gödel doctrine validates also the so-called principle of Skolemization, we can conclude the proof that the notion of Gödel doctrine provides a complete categorical account of the Dialectica interpretation and of the logical principles there involved.
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