Substance and intrinsic properties

A standard necessary condition for a property P of a substance x to be intrinsic is the loneliness condition: x could have P without anything other than x existing. There is good reason to doubt the necessity of this condition. For if the condition holds, and if there is a contingent substance that has an intrinsic property, then (a) there are no necessarily existing substances (like God), and (b) the principle that, necessarily, every contingent substance is caused to exist by a substance is false. It is implausible that a mere account of intrinsicness should immediately have such consequences.

Here, however, is a simple replacement for the loneliness condition: a substance x has P (where "P" is a name of a property, not a definite description of a property) intrinsically only if the truthmaker of the proposition that x has P does not involve any substances other than x. To explicate this further, one needs to say something about what it is for a truthmaker to "involve" y. Since all I'm trying to give is a necessary condition on intrinsicness, all I need to do is give some sufficient conditions for involvement. Here are three:

1. If y is a part of the truthmaker of p, then the truthmaker of p involves y.
2. If y's having Q is a part of the truthmaker of p, then the truthmaker of p involves y.
3. If x's standing to y in R is a part of the truthmaker of p, then the truthmaker of p involves y.