M. Tabak's book Plato's Parmenides Reconsidered (Palgrave Macmillan, 2015) is a breath of fresh air in the ocean Platonic and specifically Parmenidean literature which is characterized not only by the insoluble difficulty of the subject matter, but by a certain propensity to a heavy philosophical hermeneutic bias which is ultimately more informative about the philosophical ideas of the author than about Plato's precise intention and method in writing this puzzling dialogue. The original nature of Tabak's thesis - that of the ironic, parody-like - even light-hearted - content of the second half of the dialogue, and the equation of Zeno's and Parmenides' argumentation to that of the sophists criticized in the earlier Platonic dialogues - as well as his according due importance to the briefer treatment of the same questions in the Sophist, certainly invites and a more neutral and lucid approach.
One of our interests in the second half of the dialogue involves the following question: is it possible to formalize in detail the arguments therein in a system of modern logic (including mereology) ? In particular, is this possible for the first part of the second of the eight arguments ?
Tabak makes some very interesting observations on this last matter. Basically he says that if A is a part of B then since A is a part then A participates of unity and since A is something it participates of being. We can state this in the general case as
\[\phi x \rightarrow (Ux \& Bx) \tag{1}\]
It also clear that Plato is assuming that if $\phi x$ then $x$ participates of something (the form corresponding to $\phi$) and if $x$ participates of something then it has a (proper ?) part ($Pyx$). Let us just state this very weakly as
\[ \phi x \rightarrow \exists y. PPyx \tag{2}\]
where $PP$ denotes proper part which excludes the cases $PPxx$.
Now the hypothesis of the second argument is : if the one is. Let $\odot$ denote the one and let us state this hypothesis as
\[ B\odot \tag{H2}\]
It seems we could use 1, 2 and a convenient supplementation principle to show that
\[ \exists uv. u\neq v \& PPu\odot\& PPv \odot \& \neg Au\& \neg Av \tag{T1} \]
and in particular that $\neg A\odot$. But the problem here is that in 2 we have no guarantee that the parts of $x$ corresponding to two different predicates (either logically or syntactically or intensionally) will be in turn different. Since $U$ and $O$ are both different and apparently co-extensional, this might be difficult to formulate. We note that passages in the Parmenides as well as others in Aristotle's Topics already anticipate some of Frege's later distinctions. One solution is to introduce a form-forming operation $[\phi]$ and an axiom scheme guaranteeing fine-grained distinction
\[ [\phi]\neq [\psi] \tag{Fg} \]
where $\phi$ and $\psi$ range over syntactically distinct formulas with one free variable. We replace 2 by
\[ \phi x \rightarrow PP[\phi]x \tag{2'}\]
Using H2 and 1,2' we can now derive T1 directly as well as stronger results closer to Plato's conclusion. The problem with this solution is that Fg is already postulating an infinite number of distinct entities so the Platonic conclusion of $\odot$ having an infinite branching tree of proper parts is no longer too surprising. An even more serious problem is that the same form will be a part of completely distinct entities so that there will be universal overlap, $\forall xy. Oxy$. One solution would be to introduce a function symbol $pxy$ which yields the participated mode of form $y$ for $x$. In other words, we replace 2' by
\[ \phi x \rightarrow PP(px[\phi])x \tag{2''}\]
and replace Fg by
\[ px[\phi]\neq py[\psi] \tag{Fg'} \]
where $\phi$ and $\psi$ range over syntactically distinct formulas with one free variable.
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