Questions: Fundamental and Derivative Properties: Sparse and Abundant Ontologies

Question 1

A sparse theorist is asked whether 'being located within ten miles of the Eiffel Tower' is a genuine property. Which response best reflects the sparse view?

• A. Yes — the predicate is coherent and applies to many objects, so it corresponds to a property
• B. No — the predicate doesn't track a natural kind with causal efficacy; it's a gerrymandered classification that doesn't carve nature at its joints
• C. Yes — location is a physical fact, so any location-based predicate picks out a real property
• D. No — properties must be intrinsic, and this predicate is relational

Answer: (B)

The sparse theorist asks not 'is the predicate coherent?' but 'does it track something with causal power that explains resemblance and supports laws?' A location relative to an arbitrary landmark does none of this — two objects within ten miles of the Eiffel Tower may share nothing physically relevant. Option A is the abundant-theory answer, which accepts any coherent predicate. Option D confuses sparse/abundant with intrinsic/extrinsic — these are separate distinctions.

Question 2

Consider the predicate 'being-such-that-2+2=4,' which applies to every object since 2+2=4 is necessarily true. From this, an abundant theorist concludes that every object shares this property. What does this example reveal about the explanatory limits of abundant properties?

• A. It reveals the abundant theory is self-contradictory, because a property shared by everything cannot exist
• B. Abundant properties can be trivially shared without tracking any real similarity — they therefore cannot explain resemblance, causation, or why some generalizations are laws while others are accidents
• C. It shows that abundant theory only applies to contingent predicates, and necessary truths generate a different kind of property
• D. The example proves that abundant theory collapses into nominalism

Answer: (B)

The abundant theory's strength (logical tractability) is also its weakness: properties proliferate without doing explanatory work. Two objects sharing 'being-such-that-2+2=4' tells you nothing about their causal similarity. Sparse theorists argue that genuine properties must explain why similar objects behave similarly in law-governed ways. Option A misunderstands the abundant theory — it doesn't claim a universally shared property is incoherent, just that it exists. That's exactly the problem: it exists but explains nothing.

Question 3

On the sparse theory, whether 'redness' is a genuine property depends on whether there is a corresponding fundamental physical property with causal efficacy — not merely on whether 'red' is a coherent, commonly-used predicate.

• T. True
• F. False

Answer: True

This is the core commitment of sparse ontology. The sparse theory asks: does this predicate pick out something that does real causal work, supports laws of nature, and grounds resemblance? If 'redness' turns out to be fully reducible to surface reflectance properties that do the actual causal work, then 'redness' itself may be a useful shorthand rather than a sparse property. Coherence and common use are criteria for abundant properties, not sparse ones.

Question 4

The sparse/abundant distinction is primarily a terminological dispute with no real consequences for philosophy of science, since laws of nature can be formulated using any predicates as long as they are consistently applied.

• T. True
• F. False

Answer: False

The distinction has significant stakes for philosophy of science. Laws formulated using non-natural predicates — like Goodman's predicate 'grue' (green before a date, blue after) — fail to support counterfactuals and inductive projection. A sparse theorist explains this: 'grue' doesn't pick out a natural property, so regularities involving it aren't genuine laws. The abundant theorist has no such explanation. The distinction also matters for causation: if only sparse properties can be causes, then ordinary-language predicates that don't track natural kinds are causally otiose.

Question 5

Why can't abundant properties explain the difference between laws of nature and accidental generalizations? What does the sparse theory offer that the abundant theory cannot?

Model answer: Abundant properties exist for every coherent predicate, so both 'all electrons repel each other' and 'all things within ten miles of the Eiffel Tower expand when heated' have corresponding properties — abundant theory treats them symmetrically. It cannot explain why the first supports counterfactuals and the second doesn't, or why the first is a law and the second is an accident. The sparse theory resolves this by restricting genuine properties to natural kinds that carry causal efficacy. Laws are regularities among sparse properties, which ground necessity and support inductive projection. The geographic predicate doesn't pick out a sparse property, so the regularity it describes is an accident, not a law.

This is the philosophical payoff of the sparse/abundant distinction. The problem of distinguishing laws from accidents (the 'problem of induction' and 'problem of lawhood') runs through philosophy of science. Sparse ontology provides a principled basis: natural properties ground nomological necessity. Without this restriction, any accidental regularity becomes a candidate law — which is no theory at all.