This is a follow up question to this question. If this violates the idea of S.E, I will remove it.
In the last question I asked what free variables are and what sentences containing free variables mean. This time the question is concerned with examples from professional philosophy, in order to dodge problems of terminology and understanding.
Van Inwagen, in his 'Material Beings' starts by asking his Special Composition Question:
SCQ : When is it true that ∃y the xs compose y?
With 'the xs' being a plural variable and 'compose' a variably polyadic predicate. The first thing that comes to my mind is that it looks like van Inwagen thinks that it is possible for this sentence to be true, contradictory to most answers of the last question.
In van Inwagens view, if I understand it correctly, he treats free variables as belonging to an expression of a relation. A predicate and a free variable together can express a relation.
His answer to the Special Composition Question is this:
∃y the xs compose y iff the activity of the xs constitute a life (or there is only one of the xs)
Which brings me to the final part of the question. Ted Sider didn't like van Inwagens view and argued against it in his 'Van Inwagen and the Possibility of Gunk'. There he states van Inwagens view as follows:
For any material objects X, the Xs compose something iff the activity of the Xs consitutes a life, or there is only one of the Xs.
For me, and this brought my last question up, it looks like Siders version is different to van Inwagens version. In Siders version the plural variable is no longer free, but bound by the universal quantifier 'for any', and he completely got rid of van Inwagens 'y', which is replaced by a 'something'.
So the situation is this: One formulation has 'the xs' as a free variable, and one formulation has 'the xs' bound by a quantifier.
My question then is this: Are the two formulations expressing the same? What do the differences (as shown) mean, if they mean anything at all?
ps: If this question is answered by replies to my last question, I am sorry and just need a pointer in the right direction, because I am still puzzled.