| (*) | Rx1 … xn |
| (*P) | RxP(1) … xP(n). |
| (∇) | τQ = τQ∗ iff there is some S ∈ SYMR such that Q* = SQ, where for Q ∈ Sn, τQ is the property which (*) entails that xQ(1) has relative to xQ(2), …, relative to xQ(n). (Donnelly 2016: 94) |
| (*) | x1 is next to x2 |
| (*[2 1]) | x2 is next to x1 |
Figure 1
The fixed arity relation being next to2.
In this diagram and the ones to follow, a relative property instantiated by one thing relative to another is represented by an arrow going from the first thing to the second.
| (*) | x1loves x2 |
| (*[2 1]) | x2 loves x1 |
Figure 2
The fixed arity relation loving2.
| (*) | x1, x2, and x3 are arranged clockwise in that order |
| (*[2 3 1]) | x2, x3, and x1 are arranged clockwise in that order |
| (*[3 1 2]) | x3, x1, and x2 are arranged clockwise in that order |
| • | l has τ4 relative to c, relative to m | • | l has τ4 relative to m, relative to c |
| • | c has τ4 relative to m, relative to l | • | c has τ4 relative to l, relative to m |
| • | m has τ4 relative to l, relative to c | • | m has τ4 relative to c, relative to l |
| • | l has τ5 relative to m, relative to c | • | l has τ5 relative to c, relative to m |
| • | c has τ5 relative to l, relative to m | • | c has τ5 relative to m, relative to l |
| • | m has τ5 relative to c, relative to l | • | m has τ5 relative to l, relative to c.5 |
Figure 3
The fixed arity relation being arranged clockwise in that order3.
In this diagram and the ones to follow, a relative property instantiated by one thing relative to another, relative to another is represented by an arrow going from the first thing to the second, then to the third thing. In this diagram, red arrows depict assignments of τ4, while blue arrows depict assignments of τ5.
Figure 4
The variable arity relation meeting for lunch.
Figure 5
The variable arity relation being arranged clockwise in that order.
As before, red arrows depict assignments of τ4, while blue arrows depict assignments of τ5.
| (C1) | Any reasons we have for believing in the existence of variable arity relations must be defeasible. |
| (C2) | Relative positionalism must have advantages over its closest competitors, viz., antipositionalism and ostrich realism, that are sufficient to defeat the reasons we have for believing in variable arity relations, all else being equal (i.e., assuming that relative positionalism does not have any disadvantages relative to its competitors that would tip the balance back in their favor). |
| (Q1) | Why can each relation apply in the way(s) it can? |
| (Q2) | Why are the ways in which some relations can apply to their relata the same as one another, and those in which others can apply to their relata different from one another? |
| (Q2′) | Why are the ways in which some relations can apply to their relata more similar to one another than to the ways in which other relations can apply to their relata? |