On properties

In the brief discussion here Bill makes the point that one can assert (a)  black(x)-->coloured(x) without asserting black(x), or coloured(x) or indeed the existence of x.  This suggests to me that black(x)-->coloured(x) is not so much an extensional statement about some or all xs, but rather an intensional statement about the properties of blackness and colouredness, viz, that blackness subsumes (or includes) colouredness.    First order logic doesn't contemplate relations between properties, unfortunately.  However,  we can express them by projecting them down into relations over objects, as in (a) above.  Here the name x is a kind of 'dummy variable'.  But this seems inadequate.  Surely blackness subsumes colouredness even when there are no objects at all?

If we are to be realists about properties then I think we must contemplate such relations between properties.  But I don't want to interpret properties as sets of objects, as I think happens in second order logic.  I want properties to be a distinct category of entity.  That blackness subsumes colouredness would then follow from the nature of those two properties, independently of any objects instantiating them.  Can this be made to work?  Frege says no, I think.

I am starting to think of properties in terms of  ways in which (bits of) the world might be.  Such ways seem quite real to me, in the same sense of ways in which hands of cards might be dealt.  Some motivation for this idea can be found here.  It's interesting that modality makes an appearance right at the start of this.   

Here are a couple of principles I'd like to keep to the forefront.

First, there is no uniform sense in which a property inheres in an object.  One has only to consider colour, roughness, shape, smell, to see that these are quite distinct categories.   In each of these categories, inherence requires a distinct explanation.  This goes some way to accounting for the superficiality of constituent ontologies, for example, which as far as I can see say nothing about inherence.

Second, a realist theory of properties conceived in terms of possible arrangements will presuppose atomism.

Third, the discussion here on reductio ad absurdam shows that a pair of properties may be non-co-instantiable, though this may not be obvious on superficial inspection.   This should be explicable in terms of the structure of properties.