As we have seen, Rickless take the Third Man Argument (TMA) of the Parmenides to be arguing that each Form is inherently multiple. For him to claim this, he has to say that an argument such as the following is implicit in the passage:
(i) There is an infinite regress of Forms.The Standard Interpretation (as I've called it) can rest content with (i). Rickless has to say that all of (ii)-(v) is additionally present and implicit. This is already, I have argued, a large inconvenience with his interpretation.
(ii) Each Form earlier in the sequence participates in all the Forms later in the sequence.
(iii) Thus an infinite number of predicates can be asserted of each Form in the sequence.
(iv) But that of which many predicates can be asserted is itself manifold.
(v) And that of which an infinity of predicates can be asserted is itself infinitely divided.
(vi) Thus, each Form in the sequence is infinitely divided.
My concern in this post is with the key premise of this proposed implicit argument, premise (iv): That of which many predicates can be asserted is itself manifold. Where does this come from? Why should Plato have accepted it? Why should Plato have represented his interlocutors as all accepting this? Why should Plato have thought that readers of the dialogue might have seen that this principle was operative?
Rickless locates it in the Philebus, but there are many difficulties in this, which I'll raise in a subsequent post.