We consider lifting to be the process of inferring what is true in one context based on what is true in another context. Here is our favorite example: relating databases with different conventions [MB94]. Imagine that the Airforce and General Electric have databases both of which include prices for the jet engines that the company sells the Airforce. However, suppose the databases don't agree on what the price covers, e.g. spare parts. We use one context for the Air Force database, another for the GE database, and a third context c0 that needs to relate information from both. Lifting formulas in the context true in c0 relate information in the different databases to the context in which reasoning is done, e.g. they tell about the relation of the prices listed in and to the inclusion or not of spare parts. This example is worked out in detail in [MB94].
Translation is a certain type of lifting. In [BF95] we show how lifting can be used to formalize translation using a context logic. Implementations based on this formalization are currently being implemented at Stanford's KSL, and is proposed to serve as the basis for translation which will be built into their Ontolingua system.
However, often no direct translation between two contexts is possible, because the languages of the databases differ in meaning. Thus in the AF-GE jet engine example, in one case the price includes a spare parts kit, and in the other it doesn't. In this case it is necessary to use a higher level context usually with a richer language to lift formulas from the lower level databases making an appropriate translation. This translation will be possible if the higher language includes all the concepts of the lower languages.