The bundle theory, a theory in metaphysics and ontology about objecthood, is the view that an object is a bundle, or collection, of properties. Hence, one cannot even conceive of a propertyless object: the object just is its properties, and so if you take away its properties you take away the object itself.
To illustrate, think of an apple, but do not think of its color, of its shape, of the fact that it is a kind of fruit, of the cells that its made of, of its taste, and so on. Think of the apple, but do not think of any of its properties. Is that possible? The bundle theorist says it is not possible. So the apple is no more than a collection of properties. There isn't any more to the apple than that. In particular, there is no "substance" that these properties inhere in.
Arguments for the bundle theory
The most common sort of argument for the bundle theory capitalizes on the above insights. The basic idea is that however we might choose to conceive of, or describe a thing, it will be a conception, or a description, of a property of a thing. In other words, there is nothing that can be described, or even conceived, about a thing, which is not a property of the thing. (Or a relation of the thing, but we can ignore relations for simplicity's sake.) We cannot have the slightest idea of any aspect of the thing that is not a property of it.
This implies, the bundle theorist maintains, that we cannot have any conception whatsoever of a "bare particular." As the English philosopher John Locke said, a substance by itself, apart from its properties, is "something, I know not what." The only way that we can conceive of an object is by conceiving of its properties.
Therefore, the bundle theorist concludes, to conceive of an object just is to conceive of a bundle of properties. The only conception we can have of an object is as a bundle of properties. The bundle theory wins out, on this view, simply because it is the only game in town: the only way we can conceive of things is as bundles of properties. We could not imagine a bare particular, a propertyless substance, even if we tried. So the bundle theory is correct and the substance theory is wrong.
Many philosophers in the Anglo-American tradition today think this argument, or one much like it, is a very powerful argument. How can the substance theorist reply? How could one hold onto the substance theory if one had no conception of substance as distinct from properties and relations?
It seems that the best that we can do is to say we have a very basic sort of idea of substance; namely, substance is the subject in which properties inhere. There, it seems, is a basic idea we can have. So the substance theorist maintains: conceded, we conceive of substances by conceiving of their properties, but neverthleless, the thing we are conceiving of is the thing that has those properties. That thing is different from the properties. So we can have a notion of a substance as "the subject in which properties inhere." The substance theorist would maintain that we should just content ourselves that we are surrounded by lots of examples of subjects in which properties inhere. All the bodies around us are subjects in which properties inhere. The bundle theorist is not at all be satisfied with that view.
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Objections to the bundle theory
Several objects to the bundle theory have been raised. On first glance, the very notion of a bundle of properties sounds rather mysterious. An apple is supposed to be a bundle of properties, redness, being four inches wide, juicyness, and so on. But if there is no substance that underlies all these properties, then what exactly is the relation supposed to be between the properties of a thing? More importantly, how does it happen that any two given properties are determined to be properties of the same object, if there is no substance in which they both inhere. For example, the apple is both red and juicy. According to the bundle theorist, there is no substance that is the apple. There is just a "bundle" of properties that includes redness and juiciness. So in virtue of what are we to say that, indeed, the bundle-that-is-the-apple includes redness and juiciness? The bundle theorist says they are "bundled" together, that they are a collection. But how are they tied together? What collects them together? Why should it be that redness and juicyness just happen to be found together on top of the table?
The substance theory seems to be able to explain this, but the bundle theory--without further theorizing--seem unable to. The substance theory says: redness and juiciness are two properties that are found together on top of the table, because it is the apple that is red and juicy. The apple is a substance in which those two properties inhere.