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
It also clear that Plato is assuming that if
where
Now the hypothesis of the second argument is : if the one is. Let
It seems we could use 1, 2 and a convenient supplementation principle to show that
and in particular that
where
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
and replace Fg by
where
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