Third Man Argument Response: The Better Version

Hey guys, I know I posted something on this already, but this is the better, more scholarly essay that I had to write on it. As always, plagiarism is not cool.

What has come to be known as the Third Man Argument (TMA) has brought scholars of Plato and philosophers at large many headaches, due to the seeming end that it brings to the Platonic Theory of Forms. The most puzzling part is that it is included in the Platonic corpus itself, so it seems that there must be a solution to it, that it is a challenge from Plato beyond the grave to his philosophical successors. This paper will outline that one of the premises of the argument is not faithful to the Platonic corpus  and that there are at least two ways to solve the argument, thereby dissolving the infinite regress and keeping Plato’s theory of Forms intact.

The TMA (Parmenides 132a1-b2), relies on two basic principles. First, the Principle of Abstraction, i.e. that for every property F there must be a Form, F-ness, through which all objects with F get that property[1]. There are multiple places in the Platonic corpus where this is affirmed. The second principle, hereby to be referred as the Feedback Principle, asserts that the idea of F-ness and all the objects that it substantiates form a new class of things with the property F[2]. The Feedback Principle is drawn out of two hidden axioms, namely Non-Identity, which follows directly from Separation, and Self-Predication.

The Stanford Encyclopaedia of Philosophy cites four instances[3] in the Platonic corpus from which this principle can be concluded, however, in analysing them, the issue of pulling out the Principle of Separation is problematic. The first speaks about the doctrine of recollection, the central idea in the Meno, by which we have knowledge of the Forms, with the mention of “the Beautiful itself, the Good itself…[4]” The second quotation, also from the Phaedo makes the same mention, but its content does not relate to Separation. The third and fourth quotations are thematically very close, with the fourth one pointing out that there is a difference between those who love “beautiful sounds and colours” and those who love “the beautiful itself[5].” None of the four give sufficient evidence to postulate Separation. Rather, they indicate that there is a difference between an instance of a Form being substantiated in a sensible object and the Form itself. The last quotation makes the additional point that, even though, on the Metaphysical level, the Form is the first principle used to understand objects that instantiate it, on the Epistemic level, the formulation of it itself is the last step[6].

As if that were not enough to damn Separation (and with it Non-Identity[7]), there is the additional difficulty that, when taken together with the Principle of Self-Predication, they are mutually exclusive. If F-ness is an F, and yet is not one of the objects that begets F through F-ness, then it is obvious that there is an infinite regress, but to state the argument in such terms is to beg the question, because the strongest claim one can make about Separation, while still remaining faithful to the Platonic corpus is that “The F is itself by itself, at least in the sense of being separate from, and hence not identical with, the things that partake of it.[8]” It is obvious that F-ness is not identical with the (other) things it instantiates, but that does not mean that it is not an instance of instantiation itself. This points to the idea of degrees in instantiation, by which the Form is not perfectly instantiated in every object that has that property, but is instantiated perfectly in itself[9]. To deal with this concept at length would be to stray from the TMA, therefore, one must come back to it.

Be the issue with Separation as it may, there is an additional problem with the TMA, namely, the issue of the property of the ‘new Form’ formulated by the Feedback Principle. The expanded set with every object that has the property F and F-ness itself bring forth a ‘new Form’, F-ness1. However, it follows from this principle that F-ness is not an F, but rather an F1 and so on and so forth ad infinitum. In this case, if F-ness is, in fact, not an F but and F1, it cannot be included in the expanded set, because it is not one of the things which has the property F. To simply assume that F-ness gets the property of being F from F1 etc., would be to assume that one needs more than one Form for each property, but this is the conclusion of the TMA in proving that the Uniqueness Principle, i.e. that there is only one Form corresponding to each property F[10], is wrong. If so, to include it as an enthymic premise makes the argument invalid[11]. If the property which F-ness1 represents is any different than property of F-ness, then the argument makes no sense, if it is the same, then the argument is circular, because it assumes what it is trying to prove.

That being said, the reading that Self-Predication necessitates Self-Participation means that the Forms have to be understood (in terms of instilling the property F) as being a non-well-founded set. Though under Russell’s theory a set of the kind Ω={Ω}, would be an absurdity[12], Aczel introduces a variation of the Zermelo-Fraenkel plus the axiom of choice theory with the anti-foundation axiom[13] by using which one can keep the theory of the Forms consistent. Schweizer explains the contribution of this theory as follows:

This object induces an infinite descending chain of membership, but it is nonetheless hereditarily finite, since each member of the chain has only one element. ZFC, with the axiom of foundation replaced by the AFA, is provably consistent relative to the original system. Thus circularity is formally absolved … and the world of ‘hypersets’ is rendered just as axiomatically secure as the cumulative hierarchy[14].

By implementing this method, therefore, there is a consistent way to understand the Theory of Forms in terms of Self-Predication and Self-Participation, which leaves no room for the TMA.

That being said, it seems that Parmenides’ argument rests on one other shaky premise, namely the idea that being able to think about a scenario is suitable grounding for positing that such a thing exists. Even if the TMA were logically consistent, simply the fact that one can think of an infinite regression of Forms in not sufficient reasoning to postulate that this regress exists. However, to introduce this principle is to open the door to Nominalist criticism, because to implement a weak limit for the sets that correspond to Universals necessitates that it be defended from the stronger claim that no sets correspond to Universals as well as the idea that all sets do. The answer to both sides is functionality.

First, it is quite clear that Universals are necessary as a means of language. Let us take ‘red’ as an example. The Nominalist would argue that there is no such one thing as ‘red’ that one can pick out. We can talk about a chair being ‘red,’ but that definition of ‘red’ would have to be loose, because even a chair seemingly identical to it would be a slightly different kind of red. This, however, is explained within the Platonic corpus as Impurity-S, namely that, “sensible things are impure inasmuch as they can (and, in fact, often do) have contrary properties.[15]” The reason why most of the red things witnessed on Earth are different from most others is because their pigment is some mixture between red and other colours. In trying to produce colours digitally, the RGB model uses red, blue, and green in varying degrees in order to represent all colours. One has to abstract, out of the idea that there are many things that one would classify as red (that is, colours made up mostly by red), that there is a perfect or pure red that is not mixed with any other colour. Unless there is such a kind of red, it would not make sense to posit that we can mix red with other colours to produce mixtures. In fact, in recreating images in television or computers, one would have to otherwise log an infinity of colours as primitive, whereas this ‘flowing’ chart is simpler and works better.

In addition, there are cases where one can only speak in abstractions, thereby necessitating that there be Forms in order to be able to communicate about a scenario. The scenario B proposes in Max Black’s “The Identity of Indiscernibles” goes a long way in illustrating this case. In this possible world, there are two spheres, both made of chemical iron, both the same temperature and colour, both having one mile diameters, etc.[16] There would be no way to talk about them except by abstractions, which is to say, except by appealing to the Forms. As Black illustrates, one cannot pick one sphere and name it, because there is no reference to which sphere the name applies as opposed to the other. Nonetheless, one can say that they are both spheres, that they are both made out of iron, etc. However, in order for those statements to bear any meaning, there needs to be a Form of the sphere, the perfect sphere, by which one can discern that these two objects are spheres. Therefore, if the Nominalist wants to keep intact his ideology that there are no Universals, he would either have to say that the postulation of such a world is impossible, because the string of words “there are two spheres” has no meaning, or otherwise accept that words such as ‘sphere’ are necessary out of the convention of language, but that they have no intrinsic meaning, at which point we would be back to square one. Because the Forms have a function, i.e. of picking out the perfect or pure red that is mixed in order to make colours in nature, or the perfect sphere from which one can discern spheres, they have a function, whereas the expanded sets of the Universals do not derive any function and are, therefore, unnecessary. If they are unnecessary, one need not postulate them.

In conclusion, the TMA breaks down because the Principle of Non-Identity is drawn from a Principle of Separation that does not faithfully follow the Platonic corpus, which allows and indeed requires for the Forms to be Self-Participating if they have are instances of Self-Predicament. If this is so, the expanded sets, and with them the infinite regression of Forms, are dissolved. Forms, then, are instances of non-well-founded sets as defined by Aczel. Nonetheless, Parmenides’ argument against the Forms is problematic in that simply being able to think of the expanded sets does not give sufficient reason to believe they exist, since the expanded sets have no function. In saying this, it may seem to open the road for a Nominalist criticism, but the Forms are necessary as reference points to reality, therefore, one cannot extract them without damaging both language and philosophy.

Works Cited

Aczel, P., Non-Well-Founded Sets, Center for the Study of Language and Information, Stanford University, Lecture Notes Number 14, 1988.

Black, Max. “The Identity of Indiscernibles.” Mind 61.242 (1952): 153-64. JSTOR. Web. <http://www.jstor.org/stable/2252291&gt;.

McInerny, Ralph.  “Are There Moral Truths that Everyone Knows?” in E. McLean (Ed.),  Common Truths. (Wilmington: Intercollegiate Studies Institute Books, 1999), pp. 1-15.

Plato. Plato: Complete Works. Ed. John M. Cooper. Indianapolis, Indiana: Hacket, 1997. Print.

Rickless, Samuel, “Plato’s Parmenides“, The Stanford Encyclopedia of Philosophy (Winter 2012 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/win2012/entries/plato-parmenides/&gt;.

Schweizer, Peter. “Self-Predication and the Third Man.” Erkenntnis 40.1 (1994): 21-42. JSTOR. Web. <http://www.jstor.org/stable/20012526&gt;.


[1] P. Schweizer, “Self-Predication and the Third Man”, pg. 23

[2] ibid.

[3] Phaedo 75c11–d2, 100b6–7; Republic 476b10, 480a11

[4] Phaedo, 75c11-d2

[5] Republic, 480a11

[6] R. McInerny,, “Are there Moral Truths That Everyone Knows?” pg. 14. The article has no relation to TMA, however, it outlines why it must be that, though the principle has to come first and be used to distinguish instances of it, the formulation of the principle, abstracted from particular instances, comes last.

[7] SEP, Parmenides

[8] Ibid.

[9] P. Schweizer, “Self-Predication and the Third Man”, pg. 34

[10] SEP, Parmenides

11 P. Schweizer, “Self-Predication and the Third Man”, pp. 27-28

[12] Ibid. pg. 38

[13]P. Aczel, Non-Well-Founded Sets, Lecture Notes Nr. 14

[14]P. Schweizer, “Self-Predication and the Third Man”, pg. 38

[15] SEP, Parmenides

[16] M. Black, “The Identity of Indiscernibles” pg. 156

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