Slot theory
The leading realist account of what an argument-place is: a universal has three places exactly when it contains three slots. Its own axioms allow one object to occupy more than one slot at the same time — so the theory built to take places seriously says, as an axiom, that counting places does not count occupants.
Slot theory is the view that argument-places are real entities — slots — and that a universal is three-placed exactly when there are three slots in it. It is, in other words, a theory built to take the "shape" of a relation seriously, which makes what it says next surprising.
Among its founding axioms is one stating that a single object may occupy more than one slot in a given completion. Its illustrating examples are "a is exactly as tall as a" and "a is taller than a". Slot theory's main rival, pocket theory, drops one of its axioms and keeps this one. The two leading accounts of argument-places therefore agree, as a starting assumption rather than as a concession, on exactly the point a slot-counting argument needs denied.
Where this comes up
Sources
- Cody Gilmore, “Slots in Universals”, Oxford Studies in Metaphysics 8 (2013)
- T. Scott Dixon, “Plural Slot Theory”, Oxford Studies in Metaphysics 11 (2018), 193–223