Monday, September 16, 2019

Arity-increase and heavy-weight Platonism

Here is a curious problem. To give a heavy-weight Platonist analysis of an n-ary predication requires an (n + 1)-ary predication:

  1. Alice is green [unary]: Alice instantiates greenness [binary].

  2. Alice and Bob are friends [binary]: Alice and Bob instantiate friendship [ternary].

But higher arity predication is more puzzling than lower arity predication. Hence, heavy-weight Platonism explains the obscure in terms of the more obscure.

What got me to thinking about this was exploring the idea that Platonists can curry higher arity relations into lower arity ones. But doing so requires a multigrade “instantiates” predicate, and the curried expression of an n-ary predication seems to require an n-ary use of “instantiates”.

On a function- rather than relation-based Platonism, the issue comes up as follows. To say that the value of an n-ary function f at x1, ..., xn is y is (n + 2)-ary predication which gets Platonically grounded by the application of the (n + 1)-ary function applyn such that applyn(f,x_1,…,x_n) = f(x1, ..., xn).

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