Some Type of Formulas and the Faithfulness of Flagg and Friedman’s Translation1

Flagg and Friedman [2] introduced a translation (·) Γ from EPC (epistemic first–order predicate calculus with equality, i.e. classical first–order predicate calculus based on Lewis’s modal calculus S4 with equality) to IPC (intuitionistic first–order predicate calculus with equality). In the same paper, by making use of the soundness of the translation, they gave in a uniform way an alternative proof of the faithfulness of the celebrated Gödel translation (·)2 from the intuitionistic formal system to the epistemic one. It is important to prove the faithfulness since it implies that the epistemic formal system is a conservative extention of the intuitionistic one. (For the topic, see e.g. [2], [4], [5], [14], [15], [16], [20], [21] and [28]. For a historical account of the topic, consult [2] and [28].) In [9], it was shown that Flagg and Friedman’s translation from EPC to IPC is not faithful: the converse of the following theorem does not hold: