An overview of Interpretability Logic
نویسنده
چکیده
A miracle happens In one hand we have a class of marvelously complex theories in predicate logic theories with su cient coding potential like PA Peano Arithmetic or ZF Zermelo Fraenkel Set Theory In the other we have certain modal propositional theories of striking simplicity We translate the modal operators of the modal theories to certain speci c xed de ned predicates of the predicate logical theories These special predicates generally contain an astronomical number of symbols We interpret the propositional variables by arbitrary predicate logical sentences And see the modal theories are sound and complete for this interpretation They codify precisely the schematic principles in their scope Miracles do happen Our miracle as any good miracle involves transsubstantiation We trans late between languages of incomparable signature The modal languages do not contain quanti ers the predicate logical languages do not contain modal opera tors The modal operators can be translated to predicates because we transsub stantiate formulas occurring in the scope of a modal operator to closed terms numerals representing codes g odelnumbers of formulas of the target theory The miracle does not always work as is to be expected of true miracles we get no analogous result if we try to work with modal predicate logical languages See Provability Logic studies formal provability as a modality employing ideas and methods of modal logic Interpretability Logic extends Provability Logic by adding a binary modality This modality can be given several interpretations Relative interpretability and conservativity are the most salient We will
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