Usually if we want to formalize a statement such as “the king is bald” we might be tempted to put it in the form B(k), k = the king. Here we have used the predicate B to designate the property of being bald, and k as the object that is the king. Russell however would argue that we should instead have formed the sentence as follows: There exists some x such that K(x) & B(x). Now instead of identifying some object as the king we have formulated the properties that make the king the king (and not someone else) as K. In this formulation the object or substance has become simply a placeholder, and it is the predicates (that represent properties) that give the statement its meaning. This development also fits nicely with Leibniz’s law (if two objects have all the same properties they are the same object). We let the predicate A pick out all the properties of the object in question. If we allow this then we can do away with equality, instead of stating that b = c now we can simply say that A(b) & A(c), which by Leibniz’s law ensures that they are the same object.
What does this way of formalizing sentences tell us? To me it seems to imply that it is only the properties of an object that are important. All of an object’s casual interaction with the world, including our perception of it, is dictated by its properties, which as I have argued before implies that objects are not part of the “real” world. At this point some will protest, and argue that the object or substance somehow binds the properties together. They might say that even though color and shape are different properties you can’t see the color of an object without perceiving a shape, and this connection comes about because of the underlying object or substance, and thus showing that objects or substances really do have a role to play. This statement however is itself questionable. Studies of the biology of vision have revealed that there are neurons that are specifically sensitive to the shape of objects. If these neurons were damaged it seems reasonable to assume that we would no longer perceive shapes. Since our sensitivity to color has not been damaged we would still perceive it, and thus we are in a situation where color has been separated from shape. From this I would argue that there is really no necessary connection between the properties we perceive that requires some extra object or substance to tie them together, and thus no reason to believe that a substance or object exists at all.
So then what are these objects that we think we know so much about? Well, like causation. or the identity of objects over time, I would say that they are a convenient abstraction that arises naturally from dealing with the world. It is an empirical fact that certain properties seem to be found together under most circumstances (for example shape and color). Thus without the abstraction of objects we would have to deal with collections of many properties instead of a single thing. Instead of typing this text on a computer I would be typing this text on a rectangular, somewhat heavy, gray object, which responds to my touch by displaying different images on its upper half. Clearly to think in such a way would be extremely cumbersome, and thus to simplify matters we naturally abstract the constant conjunction of certain properties as a single object. Normally this abstraction is a useful tool of thought, but there are times when it can mislead us, especially in philosophy (for example into believing a theory of metaphysical forms).