The modern version of Description Theory stems from John Searle's paper ``Proper Names.'' Starting from Frege's footnote in ``Über Sinn und Bedeutung,'' Searle takes account of the fact that different people attach different senses to a name, even in cases where these senses are uniquely identifying, they may differ substantially. Since Frege takes our epistemic attitude toward statements as significant (the puzzle involving the epistemic difference between `a = a' and `a = b,' where `a' and `b' designate the same object), the fact that these different senses lead to different attitudes toward sentences (`Aristotle taught Alexander the Great' is trivial for one person but not for another) is important. The difference here is that if this sentence isn't trivial, it seems to follow that being the teacher of Alexander the Great cannot be part of the sense of `Aristotle.' Yet, for some people it clearly is. What are we to do?
Searle says that what we associate to a name is not a single description (uniquely identifying) but a ``cluster'' or ``family'' of descriptions. Kripke then cites a passage in the Philosophical Investigations to clarify this.
In fact, I believe that this is not relevant in the context in which Kripke employs it, so I want to take a moment to look at it before we go on with the Cluster Theory of Names.