A framework for the transfer of proofs, lemmas and strategies from classical to non classical logics
Résumé
There exist valuable methods for theorem proving in non classical logics based on translation from these logics into first-order classical logic (abbreviated henceforth FOL). The key notion in these approaches is a translation from a Source Logic (henceforth abbreviated SL) to a Target Logic (henceforth abbreviated TL). These methods are concerned with the problem of finding a proof in TL by translating a formula in SL, but they do not address the very important problem of presenting proofs in SL via a backward translation. We propose a framework for presenting proofs in SL based on a partial backward translation of proofs obtained in a familiar TL: Order-Sorted Predicate Logic. The proposed backward translation transfers some formulas in TL belonging to the proof in TL into formulas in SL , such that the formulas om SL either (a) belong to a corresponding deduction in SL (in the best case) or, (b) are semantically related in some precise way, to formulas in the corresponding deduction in SL (in the worst case). The formulas in TL and in SL can obviously be considered as lemmas of their respective proofs. Therefore the transfer of lemmas of TL gives at least a skeleton of the corresponding proof in SL. Since the formulas of a proof "keep trace" of the strategy used to obtain the proof, clearly the framework can also help in solving another fundamental and difficult problem:the transfer of strategies from classical to non classical logics. We show how to apply the proposed framework, at least to S5, S4(p), K, T, K4. Two conjectures are stated and we propose sufficient (and in general satisfactory) conditions in order to obtain formulas in the proof in SL. Two particular cases of the conjectures are proved to be theorems. Three examples are treated in full detail. The main lines of future research are given.