Excuse my ignorance on this topic: I might be making either a stupid mistake or an obvious point – pointers to the literature are welcome!
Consider the relational expressions ‘is a child of’ and the relation ‘is a parent of’. It seems that we shouldn’t have to admit relations corresponding to both expressions into our ontology. If we believe in the is a child of relation then it is a constituent of the state of affairs a is b’s child, which is the truthmaker both for [a is a child of b] and [b is a parent of a] – we don’t need a separate relation is a parent of that is a constituent of the state of affairs b is a’s parent which makes the latter proposition true.
So we only need one relation here, not two. And it also seems crazy to think that one of the descriptions is privileged – i.e. to say that there is a relation corresponding to ‘is a child of’ but none corresponding to ‘is a parent of’. If that were the case, how we could know which expression picked out a real relation. Distinguishing between the world where ‘is a child of’ picks out a genuine relation and ‘is a parent of’ doesn’t and the world where ‘is a parent of’ picks out a genuine relation and ‘is a child of’ doesn’t seems like a bad case of distinguishing without a difference. There is just one relation that holds between a and b, and it is appropriate to call it both the ‘being a child of’ relation and the ‘being a parent of’ relation, depending on what order we list the terms.
But if that’s right we need to rethink how we state the properties of relations. Orthodoxy has it that the is a child of relation is non-symmetric: a can bear it to b without b bearing it to a. But if the above is right then this is false: b does bear it back to a – but in the other direction!
Two options then: either we could let symmetry etc apply to descriptions of the relations rather than the relations themselves, or we could ass directionality into our ideology and say that a relation R is symmetric iff if a bears R to b in a particular direction then b bears R to a in the same direction.
I prefer the second route. Can anyone foresee any problems? Or am I missing something in the first place?