An interesting approach—formally elegant and convenient, especially from the standpoint of set-theoretic formalization. It allows properties to be neatly defined through their instances and resolves the problem of uninstantiated properties via possible worlds.
However:
The set of all red things does not quite align with our intuitive grasp of “redness” as something unified and shared. A set is merely a collection of objects, whereas a property seems to be something more abstract—something that binds those objects together.
Identifying properties such as “equilateral triangle” and “equiangular triangle” as one and the same disregards their contextual distinctions. In geometric analysis, for example, whether emphasis is placed on sides or angles can carry significant implications, even if the extension is the same.
In the end, your approach requires a metaphysical commitment to the reality of possible worlds, which is itself a contested position.
I propose we step away from a substantialist approach to ontology and turn instead toward a processual one, which I am actively developing.
Rather than conceiving of properties as static characteristics inherent in objects—or as sets of such objects—we can understand them as dynamic events that emerge through acts of interaction (or, if you will, participation) between beings. A property is not something a thing has, but something that happens at the threshold where it encounters other beings.
Take “redness” as an example. Under the substantialist view, redness is either an inherent trait of the object or the set of all red things. In processual ontology, redness is an event that unfolds through the interaction of:
The thing (e.g., the apple, whose structure determines how it reflects light);
The light (photons of a particular wavelength);
The observer (a human or other creature interpreting that reflected light through their perceptual apparatus).
Redness, then, is not inside the apple. It is born from the interplay of all three participants. This makes the property contingent: for a different observer (say, someone with color blindness), or under different lighting conditions, redness may not manifest at all.
Another example: “a pleasant scent.” Scent is not a static attribute of a substance. It is an event arising from the interaction of molecular structures, olfactory receptors, and a brain interpreting those signals within the context of memory and experience. What smells pleasant to a human may signal danger to an insect.
But how should we treat uninstantiated properties—without lapsing into subjective idealism?
To relieve this tension, we can distinguish between modes and properties:
Mode (internal disposition): This is the objectively existing structure of the apple’s embodied being—its surface texture and chemical composition—that predisposes it to reflect light of a certain wavelength. This mode exists independently of both light and observer. It remains even in complete darkness. This aligns fully with realism.
Property (realized event): This is the event of redness, which only occurs when the apple’s mode enters into participation with light and an observer. This property does not exist in darkness.
The unactualized mode remains. The potential for redness (the mode) is always there, as long as the apple itself exists. But the redness (the property) is an event that may or may not come into being.