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Trope theory holds that properties are not universals shared across objects, but are particular instances — this redness of this apple is numerically distinct from that redness of that tomato, even if they exactly resemble each other. Objects can then be analyzed as bundles of tropes, and resemblance classes of tropes replace universals. Trope theory aims to combine the explanatory virtues of realism (properties are real) with a nominalist-friendly ontology (everything that exists is particular). The key challenge is analyzing the resemblance relation between tropes without invoking a universal.
Read Williams's 'The Elements of Being' and Campbell's Abstract Particulars. Construct a trope-theoretic analysis of a specific object and test whether it handles the problem of exact resemblance without circularity.
From the debate about universals and particulars, you know the central puzzle: when two roses are both red, what accounts for their sharing that property? The realist about universals says there is one entity—redness—that both roses literally have in common, a universal that is wholly present in each instance. The sparse nominalist says there is no such shared entity; "red" is just a word we apply to similar things. Trope theory carves out a third position: properties are real, but they are not universals—they are particulars. This redness of this rose is a distinct entity from that redness of that tomato. They don't share a single universal; they are two numerically distinct but exactly resembling property-instances.
The term trope designates these property-instances: abstract particulars that are as individual as the objects that have them. When you see a red apple, the redness you observe is not an instance of some shared universal redness—it is *this very redness*, numerically unique, as particular as the apple itself. Two apples of the same shade have two distinct tropes of redness that exactly resemble each other. The resemblance is perfectly real, but it holds between two particulars, not between two instances of a single entity. This lets the trope theorist say: "Yes, properties are real and play genuine explanatory roles" (unlike sparse nominalism), while also saying "No, we don't need the mysterious machinery of universals—everything that exists is particular."
The trope-theoretic analysis of objects builds on bundle theory from your prerequisite. An ordinary object—an apple, a chair, a person—is a bundle of tropes: the collection of its particular property-instances. The apple is a bundle of *this-redness*, *this-roundness*, *this-sweetness*, *this-weight*, and so on. This contrasts with substratum theory, which posits a bare particular—a featureless "thin object"—that *has* properties but is distinct from them. Bundle theory eliminates the bare substratum; the object just is its properties bundled together. When trope theory adopts bundle theory, it gains a parsimonious ontology: the world is just tropes (particular property-instances) in various patterns of bundling and resemblance.
The most serious internal challenge for trope theory is the resemblance regress. If two redness tropes are exactly alike, what makes them similar? The trope theorist cannot say they share a universal of resemblance—that would reintroduce universals by the back door. The standard response is that the resemblance between exactly similar tropes is a primitive: it is just a basic, unanalyzable fact that these two tropes are exactly similar. Critics find this unsatisfying—it looks like swapping one primitive (universals) for another (primitive resemblances). Defenders argue that brute primitive resemblances between particulars are metaphysically less extravagant than universals, which require a separate ontological category. Whether this tradeoff favors trope theory over universals realism is one of the live debates in analytic metaphysics, and it turns largely on what kinds of primitives your overall theory of the world can tolerate.
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