Endless Extrinsic & Intrinsic Properties
The thing about extrinsic properties is that they appear to be indefinite - or even infinite - in number. One could argue, then, that if extrinsic properties are indeed indefinite in number, then what's the point of the classification? Wouldn't you need to make a somewhat arbitrary (or random) choice as to which ones to include and exclude?
For example, Mary's being a friend of John is an extrinsic property of Mary. Then again, Mary's being near a sewage works is also an extrinsic feature of Mary.
Similarly, if having a mass of 200kg is an intrinsic feature of object a, then having a mass of 201kg can be an intrinsic feature of object b. What's more, object a may have a intrinsic mass of 200kg at time t and an intrinsic mass of 2001kg at t2. Thus a and b may change positions when it comes to their intrinsic masses.
Take this other problem.
Why would Mary's extrinsic property of being related to John be deemed more important than Mary's having a daily causal relations with professors? In fact, in order to decipher which extrinsic properties are important/fundamental and which aren't, we'd surely be raising the former to the category of intrinsic properties. Thus the closer extrinsic properties come to being important/fundamental, the more they resemble intrinsic properties.
Can we invert this argument by doing the same with intrinsic properties? Are there levels of intrinsicality (as it were) between intrinsic properties?
In other words, are some intrinsic properties more fundamental/important than others? And if that's the case, then perhaps some intrinsic properties are closer to fundamental/important extrinsic properties than to other intrinsic properties. So much so that they may as well be classified as extrinsic properties.
Similarly (as already said) with fundamental/important extrinsic properties, they too may as well be classified as intrinsic properties.
Or, better still, why not jettison the intrinsic/extrinsic distinction altogether and stick with the fundamental/not-fundamental (or important/not-important) distinction instead (as Quine once did in relation to essential properties)?
Endless Relational Properties
You have to also wonder what's the point of some relational properties; just as we've just wondered about the nature of some extrinsic properties.
For example, take the relational human property having more cells than fingers. Now that can be seen not only as a relational property, but also as an intrinsic property in that any all human beings will have the property having more cells than fingers. (That's because there must be more cells in one finger than there are fingers.) However, if it's a unprofitable relational property, then surely it's even more unprofitable as a intrinsic relational property.
Despite saying that, we can deny the property having more cells than fingers its intrinsic nature by saying that both fingers (of whatever number) and cells (of whatever number) aren't themselves intrinsic in the first place. Thus the relation between the two can't be deemed intrinsic either. And here, yet again, is another good reason to reject the intrinsic-extrinsic (or, in this case, intrinsic/relational) distinction entirely.
Here we can add that having X as a “proper part” is also seen as being intrinsic by some philosophers. Thus both fingers and cells can be deemed to be proper parts of human beings (or homo sapiens). If they are, then perhaps they're also intrinsic in nature.
Are they relational?
Well, a human being certainly has some kind of relation to his or her own fingers or cells. Though is it right to call this an intrinsically relational property? Isn't it best to say that fingers or cells are part of the package which constitute a human being? What does it mean to say that a human - or even a person - has a (intrinsic) relation to his cells or fingers? Sure, there's some kinds of causal relations that can be established between a human being and his/her cells and fingers; though is there an intrinsic (essential) relation?
In that case, what is related to what?
If we take away fingers, cells and similar properties, what would be left of a human being so that it could have a relation to what remains? What I mean by that is if one takes away all these kinds of properties, perhaps there's no human being left. And if that's the case, then it may not make sense to speak of a human being having a relation to such things as his cells or fingers. It would certainly make the notion of an intrinsic relation between such proper parts and a human being problematic.