#### Metaphysics : Inductive Resolution to Bradley’s Regress

**Metaphysics : Inductive Resolution to Bradley’s Regress** When it comes to dealing with the abstract theory of properties, two prominent theories arise.

The two theories are universalism and trope theory; each of these theories have their variants, namely the bundle and substratum of each theory (however, specifics about these subtheories are beyond the scope of this paper, as I will be discussing a broader approach to property theory).

These two theories are very similar in their mechanics, and at first glance, trope theory can seem to be a subset of universalism; however, upon closer examination. We can see that these two theories function differently.

To illustrate this difference, I will be using Bradley’s Regress and the responses that each theory has to Bradley’s Regress.

First, let’s discuss each of the theories, starting with universalism. There are two building blocks to this theory: the universal and the particular. A universal is an entity that is repeatable. A particular is a thing that has a universal, and unlike universals, particulars are not repeatable. In this theory, no universal can exist without being instantiated by a particular. What this means is that every universal must “act” on a particular; it cannot just exist by itself.

#### In contrast to the repeatable nature of universalism, trope theory posits the existence of tropes.

Tropes are particularized properties. Meaning that they are not repeatable. Rather, these tropes are “tied” to the object that they are describing. This behavior is different from universals. As universals “act” on particulars, but are not constrained to be unique to that particular. Two different objects can have the exact same universals, but by definition, two different objects cannot have a common trope.

Now that we have explicitly defined each theory and its mechanics, I will introduce Bradley’s Regress. Suppose that there is a particular, *a, *and a property, P. Now, for P to be a property of *a*, we need to ensure that P *instantiates a*. However, instantiation is an external relation, and so in being a property for *a*, P requires another external relation, called P*, to tie it to *a*. Furthermore, P* needs another external relation called P** to tie it to P, and this cycle continues until we are left with the conclusion that nothing can ever be instantiated.

The trope theorist’s response to Bradley’s Regress is quite straightforward. A trope theorist can work around this argument by saying that each trope is localized to the particular that it is acting upon, so it lacks the necessity for instantiation. Another way to view this is that each particular has a unique identification code, and each trope associated with that particular also carries that code, making it so it automatically points at the particular. So, in the formalization presented above, P would immediately be related to *a*, without the need for a P* (and so on).

#### The universalist’s response to Bradley’s Regress is more involved, and there are many ways to go about it.

I will first start by presenting two canonical approaches, and then offer an original approach. The first defense against Bradley’s Regress is that somehow, relations relate things together in an “essential” manner. However, I feel as if this argument is the argumentative version of deus ex machina.

And as a result, is a weak argument. The other defense of Bradley’s Regress is to take facts of P being a property of *a *to be fundamental, which allows us to ground the facts about *a *and P. I think that this argument is logically sound, but I do not think that it is optimal, as it makes another requirement about the way that properties function (the requirement being that properties are fundamental).

#### I posit the following argument:

Assume that there is a particular, *a*, and a property, P^{0}, describing *a*. In the argument put forth by Bradley’s Regress. We would get an infinite cycle of external relations that would be required. Index these external relations, with the *i*-th relation (zero-indexed) being denoted P^{i}.

Then, we can define an infinite set of Ps. From here, we can apply induction. We first notice that for a given number *i*. P^{i}is instantiated by all of the P^{j}, where *j *> *i*. If you think in the framework of an ordered set. This equates to each relation being instantiated by all of the elements that come after it in the set. This induction hypothesis combines with the concrete case of the first iteration of induction. (namely that P^{0}is instantiated by all of the relations P^{j }with *j *> 0) we get that all external relations are instantiated, and the property is valid in its application to *a*.

#### One potential counter to this argument is that it is not clear how much “work” each external relation is doing with respect to the entire problem.

In Zeno’s paradox (whose inductive steps are similar to my arguments). Each iteration brings us closer to our goal of reaching 1. However, upon first glance, it may seem unclear how we are getting closer to our goal of instantiating our property. To see how we are actually making progress, first suppose that we only required P^{0}. In this case, one thing is doing 100% of the work.

Now, suppose that P^{0 }and P^{1} we require. In this case, each of the two things are doing 50% of the work.

As we expand this out to infinity, we get that each external relation is doing an infinitesimal amount of work for our goal; however, since there are an infinite number of them, we accomplish our task. (In mathematics, this is equivalent of taking the limit as *n *goes to infinity of �� * ~~,~~ which turns out ^{1}�� to be 1). So, we can see that each step is doing a meaningful amount of work with respect to the goal of instantiating our property. Thus, induction holds.

###### Metaphysics : Inductive Resolution to Bradley’s Regress

I think that this inductive argument is stronger than the two previous responses that universalism has to Bradley’s Regress. I also think that this argument has a new feature: it allows for the construction of minimal bundles (with respect to bundle universalism). To see what I mean, take a random integer, x. Now, this number is greater than (x – 1). It is also greater than (x – 2), and so on. Each of these is a property, and so in bundle universalism, we would have an infinite set of properties just dedicated to this “greater than” relationship.

#### Instead, we can use the inductive process.

And only include the least upper bound (smallest element larger than x) and the greatest lower bound (largest element smaller than x); now, if we want to get that (x – 2) is smaller than x, we would go to (x – 1) first, and then inductively make our way to (x – 2). This would reduce the size of the “smaller than” (or equivalently, “greater than”). Set to a single property. And if we repeat this process on every set of universals that have an ordering, we would get the minimal bundle for a particular.

In conclusion, trope theory and universalism are two theories about properties. Each handles the specific mechanics of properties differently, but there are similarities between the two. Each of these theories handles Bradley’s Regress differently, with trope theory having an easier time than universalism. This is not to say that universalism struggles with Bradley’s Regress, as it has many options available. Inductive argument is the strongest, as it does not require anything more about the mechanics of properties; rather, it relies only on inductive reasoning. Also, I think that the inductive argument allows us to define minimal bundles, which is an added feature.

In conclusion, the next steps in this line of reasoning would be to investigate an inductive argument for bundle trope theory. Using notions of similarity between tropes.