Here is a little puzzle, inspired by the radio programme Desert Island Discs, on which people are required to choose eight pieces of music to take to a desert island. (There is no actual island: the programme was created long before reality television came to pass.)
Most of us would find it impossible to choose eight pieces of music directly. If one aims for eight, one ends up with a list of 20. The obvious thing to do would be to aim for three and end up with eight. But would one then have the top eight, as distinct from candidates to be in the top three?
I suspect that one would not have the top eight, because there would be no such thing as the top eight (for a given person, at a given time). “One of the top eight” would be a term with a perfectly clear intension, but no extension. And “candidate to be in the top three” would be the best approximation to “one of the top eight” which did have an extension.
The same type of argument could be conducted even if the point was not to pick eight favourite pieces of music, but to pick pieces of music to reflect eight important things in one’s life (such as one’s job or members of one’s family). A scale of importance of things in one’s life would replace a scale of liking for pieces of music, and there would be the same kind of competition for the eight places, and the same temptation to aim to select the top three in order to end up with eight.
Likewise, the argument would run if the task were to measure objective importance (in one’s own subjective view), rather than personal preference. One might for example be asked to identify the eight most important philosophers, or inventions. Aiming for three would be a sensible way to approach the task.
In similar vein, I am happy say things like “Descartes would have to feature on any defensible list of the ten most important western philosophers of all time”, but I would never try to list the ten most important western philosophers. It would not, however, be sensible for any one person to say, of more than four or five candidates, that they would have to feature on any defensible list of the top ten. “One of the ten most important” has no extension, but we can still identify some who would have to be within its extension if it had one, at least under the condition that the list would have to be defensible (before some reasonable judges – another source of vagueness).
There is another type of failure to have an extension, on which Simon Blackburn has remarked. It is rather different, because the problem is the existence of irresoluble disagreements between people. An example of a term without an extension that Blackburn has given (when giving his Presidential Address to the Aristotelian Society in October 2009) is “pig-headed”. If X claims, but Y denies, that Z is pig-headed, it is not possible for an arbiter to check whether or not Z is in the extension of the term “pig-headed” and to determine whether X or Y is correct. This is an attractive line of thought, and I would be very tempted to apply it to terms like “beautiful” and even “is a work of art” (in the sense that does not imply special praise). But I think that there are two types of limit to how far one could take this. First, although there may be no way of deciding borderline cases, there must be widespread agreement on a good number of cases. Some people would have to be regarded by most people who knew them as pig-headed, and some people would have to be regarded by most people who knew them as not pig-headed. Without that level of agreement, the term would not have an agreed intension. Second, if X and Y debated whether or not Z was pig-headed, one would expect them to make their cases by reference to a largely common range of factors, and by reference to examples of Z’s conduct which they would both approach with reference to those factors. Thus, while “pig-headed” might not have an extension, “appropriate way to test for pig-headedness” might well have an extension. That extension would do much to support agreement on the intension of “pig-headed”.