On Logical Relations and Conservativity

Having developed a Montague-like interpretation of a fragment of natural language, one may want to modify it in order to accomodate further linguistic phenomena. This may require radical changes in the semantic interpretation of syntactic categories. For instance, one may want to raise the semantic type assigned to the category of adverbial modifiers in order to account for quantified adverbial modifiers, i.e., to interpret adverbial modifiers as terms of type (((e → t) → e → t) → t) → t rather than (e → t) → e → t. Such a modification must then be propagated throughout the current interpretation, which might be cumbersome if it cannot be done in a systematic way. Ideally, one would like to propagate the change by applying some automatic procedure that would guarantee that the new obtained interpretation is faithful to the original one. The intensionalization procedure introduced in [5] is an example of such a systematic procedure. It allows the extensional interpretation of a language to be transformed into an intensionalized interpretation that offers room for accommodating truly intensional phenomena. Moreover, this procedure is conservative in the sense that it preserves the truth conditions of sentences. Another instance of such a systematic procedure is provided by the dynamization procedure described in [6], which allows a static interpretation to be turned into a dynamic one capable of accommodating phenomena related to discourse dynamics. This procedure, which is also conservative, presents several similarities with the intentionalization procedure. This raises the following question: is there some abstract general scheme of which both the intensionalization and dynamization procedures would be instances? This paper reports some ongoing work along this research direction. It attempts to develop an abstract construct that generalizes the intensionalization and dynamization procedures, and for which conservativity results may be established using the notion of logical relation.