Allen, Reginald Edgar. 1960. "Participation and Predication in Plato's Middle Dialogues." The Philosophical Review no. 69:147–164

Reprinted in: R. E. Allen (ed.), Studies in Plato's Metaphysics , London: Routledge & Kegan Paul 1965, pp. 43-60; N. D. Smith (ed.), Plato. Critical Assessments, Vol. II: Plato's Middle Period: Metaphysics and Epistemology , London: Routledge 1998, pp. 57-71.

"I propose in this paper to examine three closely related issues in the interpretation of Plato's middle dialogues: the nature of Forms, of participation, and of predication. The familiar problem of self-predication will serve as introduction to the inquiry." (p. 43 of the reprint)

(..-)

"The fundamental difficulty underlying the Third Man is ontological, not linguistic. Not only the regress arguments but all of the objections to participation in the Parmenides posit an identity of character between Forms and particulars; the Many and the One are to be unified, in effect, by a further One. These arguments demonstrate conclusively that this supposition is absurd.

Yet they point to a difficulty which is crucial in any exemplaristic ontology. For though there can be no identity between exemplars and exemplifications, there must be community of character; and how is this community to be explained? It can be explained by treating exemplifications not as substances in which qualities inhere but as relational entities, entities in which resemblance and dependence so combine as to destroy the possibility of substantiality.

Plato's use of the metaphors of imitation and reflection, and his characterisation of particulars and Forms, indirectly indicate that he accepted this solution." (pp. 59-60 of the reprint)
Bailey, Dominic T. J. 2009. "The Third Man Argument." Philosophy Compass no. 4:666–681

Abstract: "This paper is a brief discussion of the famous ‘Third Man Argument’ as it appears in Plato’s dialogue Parmenides. I mention, criticise and refine the most influential analytic approach to the argument; show that the actual conclusion of the argument is different from the one attributed to it by the majority of scholars; and elaborate two responses to the argument, both of which shed interesting light on the Theory of Forms."
Barford, Robert. 1978. "The Context of the Third Man Argument in Plato's Parmenides ." no. 16:1–11

"In this paper ! intend to explore the relation of the logic and context of the TMA [Third Man Argument] in the Parmenides . Thus we shall have what I think is Plato's own evaluation of the TMA, given in the Parmenides itself, from which it will follow that there is no need to assume any puzzlement on Plato's part. Specifically, I want to show that: (1) in the Parmenides Plato actually does expose the TMA as unsound, that is, he exposes it as a "sophistical refutation" of the theory of Forms; and (2) this exposition takes place in the "aporematic" context of that dialogue. It will follow from (1) and (2) that he was aware of all the necessary premises of the TMA as well as of the resources within the theory of Forms sufficient to expose the TMA as unsound.

Hence, there is no ground for a "puzzlement" hypothesis. What is new in my interpretation is not merely the claim that the TMA is unsound from the viewpoint of the theory of Forms. I want to show that in the Parmenides itself Plato is exposing the TMA as unsound and that he is doing so within the larger context of the aporematic character of the dialogue. In what follows I shall give my reasons for holding (1) and (2) above. We shall then have what I think is Plato's own evaluation of the TMA and perhaps a better understanding of what is going on in the first part of the Parmenides ." (p. 2)
Bestor, Thomas Wheaton. 1978. "Common properties and eponymy in Plato." The Philosophical Quarterly no. 28:189–207

"Dating right back to Proclus, there has long existed a minority opinion that Plato was aware that his notorious "Third Man" argument was not an effective objection to his theory of Forms after all. In recent times this opinion has been based on a distinctive insight: since for Plato words have both direct and also strictly second-hand referents, it is plain that though the (second-hand) application of a word 'F' to a particular inevitably involves regressing to a Form, the (direct) application of the same word to the Form itself does not. Usually this minority opinion is dismissed very casually. But recently Roger Shiner and Frank White(2) have subjected it to searching criticism: no matter in what other ways it might be F, Plato's theory of predication has to presuppose that the Form b is also F in the ruinous sense of possessing the property F-ness, or else it remains utterly mysterious why it is specifically the Form φ which is "responsible", as required, for the particulars possessing that property rather than some other. I am sure that this is the most damaging criticism which the minority opinion has to face. In what follows I want to do what I can to weaken it, concentrating primarily on White's recent article in this journal." (p. 189, a note omitted)

(2) Shiner, "Self-Predication and the "Third Man" Argument", Journal of the History of Philosophy , 8 (1970), pp. 373-5, 384-6; White, "Plato's Middle Dialogues and the Independence of Particulars", Philosophical Quarterly , 27 (1977), pp. 206-11.
Bluck, Richard Stanley. 1956. "The Parmenides and the 'Third Man'." Classical Quarterly no. 6:29–37

"In a recent article [The Third Man Argument in the Parmenides ] in the Philosophical Review (lxiii (1954), 319 ff.) Professor Gregory Vlastos has given an acute analysis of the 'Third Man' Argument as it appears in the Parmenides for which all Platonic scholars will be grateful. In view of the importance of the article and the interest that it has aroused, I should like to offer one or two criticisms of his conclusions." (p. 29)

(...)

"I wish to suggest that although Vlastos's analysis of the Third Man Argument is most valuable and helpful, his conclusions about the state of mind in which Plato wrote the early part of the Parmenides are open to objection. I wish to contend, in the first place, that the Self-Predication Assumption does not constitute the substance of any of the objections here brought against the Theory of Forms, and secondly that the Non-identity Assumption, which plays so vital a role in the Third Man Argument, is not in fact involved in Plato's Theory.

I shall then contend, thirdly, that in view of these two facts there is no special reason for supposing that Plato could not and did not recognize the nature of the weaknesses in his Theory, as hitherto presented, that give rise to these objections; and fourthly that one aim of the objections was probably to suggest, in a 'masked' form, Plato's comments on defects in the earlier presentation of his Theory-defects of the nature of which he was thoroughly aware." (p. 30)
———. 1957. "Forms as Standards." Phronesis no. 2:115–127

"Plato called his Forms πᾰρᾰ́δειγμᾰτα, and he is taken to mean that they were patterns or standards of which objects and acts in the sensible world are copies. But in what sense were they 'standards'?

My purpose is to offer a few remarks on this question." (p. 29)

(...)

"In seeking to establish that the use of the terms 'just' and 'good' and so on was not merely νόμος, and that these 'things' really existed φωνει and could really be known, Plato will have observed not only the constant change of sensible objects but also the relative (2) nature of things of this world. He obviously thought it at least as important that moral standards should be non-relative as it was that they should be exempt from constant change; and that is why in the Symposium (211a) the Form of the Beautiful is contrasted with that which is 'in one respect beautiful, in another ugly, or at one time beautiful, at another not, or in relation to one thing beautiful, in relation to another ugly, so as to be beautiful in the opinion of some and ugly in the opinion of others'." (pp. 115-116)

(2) By this I mean [I] relative to human judgement or perception - a thing may appear different to different people, and [II] varying according to situation. The Phaedo (102b-d) teaches that qualities even as they exist 'in us' do not change their nature, but they manifest themselves only in certain circumstances. (...)
Booth, N. B. 1958. "Assumptions involved in the Third man argument." Phronesis no. 3:146–149

"This article takes its start from G. Vlastos's excellent discussion in Philosophical Review lxiii (1954) pp. 3 19 ff. I am in general agreement with Vlastos, but I think that his ideas need further development before they can be fully accepted. It is hardly necessary for me to comment on the criticisms advanced by Geach and Sellars; Vlastos has already defended himself in Philosophical Review Vol. lxiv (1955) and Vol. lxv (1956). Possibly, however, there is need to refer to R. S. Bluck's work in Classical Quarterly N.S. vi (1956) and in Phronesis Vol. 2 no. 2 (1957)." (p. 146)

(...)

"My tendency, therefore, is to revert to Vlastos's theory of "unresolved puzzlement"; I do not believe that Plato fully realized the nature of his own assumptions. I would grant that Plato continued to have very strong faith in absolute Knowledge as opposed to shifting opinion, and accordingly in firm objects of Knowledge as opposed to shifting phenomena; it was also a matter of very great concern to him to prove the unity of moral qualities, especially of virtue itself (sec Laws 96 3), and he made considerable efforts to deal with the problem of how virtues (or any other Ideas) can be both one and many. But I see no evidence to show that he ever really came to grips with the fundamental difficulties involved in the theory of Ideas." (p. 149)
Brakas, Jurgis (George). 2011. "Plato, Aristotle, and the Third Man Argument." In Just the Arguments: 100 of the Most Important Arguments in Western Philosophy , edited by Bruce, Michael and Barbone, Steven, 106–110. Malden: Wiley-Blackwell

"Many scholars believe that the Third Man Argument (the TMA) is one of the most powerful arguments against the existence of Plato’s Forms, many going so far as to maintain that it is successful. It exists in two versions.

One, preserved to us only in a commentary on Aristotle’s Metaphysics by Alexander of Aphrodisias, uses the Form Man as an example; the other – offered first, to his great credit, by Plato himself – uses the Form Large. The difference between the versions is significant, because the first uses Forms of entities or substances as examples whereas the second uses attributes or properties.

Both versions use just three major premises (in addition to five that most people would find uncontroversial) to generate a regress that is vicious. For any group of things to which the same “ name ” (word) may be truly applied, there exists a Form having the same “ name ” in virtue of which that “ name ” may be truly applied to them. (This may be called the “ Existence Assumption ” or “ One - over - many Assumption. ” ) This Form is not a member of the group of things of which it is the Form. (This is usually called the “ Non - identity Assumption. ” ) Finally, this Form may be predicated of itself.

(This is usually called the “ Self - predication Assumption. ” It should be pointed out that both the formulation of this premise and its name are misleading. It is not the very same Form that is predicated of itself but rather another Form having the same name as the first, with the same point applying as the regress proceeds.) Since an infinite regress is impossible (at least, so both Plato and Aristotle agree), one or more of the three major premises must be false, if we take the additional five to be uncontroversial.

The problem is that it is extremely difficult, if not impossible, to see how Plato could give up any of those premises and be left with anything that resembles his philosophy." (pp. 106-107)
Brinkley, Robert A. 1982. "Plato's Third Man and the Limits of Cognition." Australasian Journal of Philosophy no. 60:152–157

Abstract: "Discussions of Plato's Third Man Argument have tended to obscure its force within the context of "Parmenides". The TMA introduces a demonstration by Parmenides of the logic of dialectic. The argument does not refute the theory of forms: rather it illuminates particular difficulties involved in any attempt to conceive of what forms do. As a form, the large enables us to observe the same attribute in a number of objects. As such it is not an object of cognition. When we try to think of a form, however, we transform it into such an object. That object presupposes another form which enables us to conceive of it. "There will be no end to this emergence of fresh forms," Parmenides tells Socrates. "If the form is to be like the things which participate in it". What the regress of the TMA manifests is that a form of which we conceive can never be but will always presuppose a form "with" which we conceive."
Brownstein, Donald. 1986. "Parmenides dilemma and Aristotle's way out." The Southern Journal of Philosophy no. 24:1–7

"Parmenides poses so formidable a threat to Plato precisely because his dilemma seems to involve a clash between DNI and Plato’s other principles. Plato sees no easy way to give up either DNI or those other principles. If what I have suggested about Aristotle is right, he gives up DNI, but without providing a way of salvaging what DNI secured. Thus as a response to Parmenides’ challenge Aristotle’s theory seems inadequate. It is not at all clear how such a theory could be made adequate." (p. 7)

DNI = principle of the Discernibility of Non-identicals (the principle that neither individual things nor their qualities may differ in number alone.
Burnet, John. 1928. Greek Philosophy: Part I. Thales to Plato . London: Macmillan

Chaoter XIII: Criticism 234; The Theaetetus 237; The Parmenides 253-272.

"§ 192. The Parmenides is a criticism of the doctrine of forms as stated in the Phaedo and Republic , and the selection of Parmenides as the chief speaker points to the conclusion that the objections to the theory of participation contained in the first part of the dialogue are of Eleatic origin. We know from the Theaetetus that Plato was busy with Eukleides about this time. Besides that, we have a remarkable piece of external evidence to the same effect. The most telling argument against participation is that known as "the third man," which we shall come to presently. We have unimpeachable evidence that this argument was introduced in some work or other by the " Sophist " Polyxenos.(1) He had been a pupil of the " Sophist " Bryson, who had been an associate of Sokrates along with Eukleides, and with him had founded the " Eristic " of Megara. He also stood in close relations of some kind with the Academy.(2) Now the detractors of Plato asserted that he plagiarised the lectures (διατριβαι)(3) of Bryson, and that is most easily explained if we assume that Bryson was the original author of this argument.

(...)

"Plato was, of course, far too interested in the world of experience to accept the acosmism of Eukleides, but he was clearly impressed by the force of the arguments against " participation " as an account of the relation between the sensible and the intelligible. His own account of that is not, however, given in the Parmenides. " (pp. 253-255)

(...)

"The second part of the dialogue has shown once for all the impossibility of maintaining the isolation of the forms from one another. " The others" are just as hard to grasp as " the One." If we regard them abstractly, we can say nothing whatever about them ; while, if we regard them as being, we are compelled to ascribe contradictory predicates to them. In fact, the intelligible and incorporeal forms vanish under our hands just as the things of sense had done. It is clearly shown that we must now endeavour to understand in what sense the forms can participate in one another ; for all the difficulties of the Parmenides arise from the assumption that they cannot." (p. 272)

(1) Alexander on Ar. Met . 990 b, 17. He quotes Polyxenos from Phanias of Eresos, a disciple of Aristotle and friend of Theophrastos.

See Baümker in Rhein. Mus. xxxiv. pp. 64 sqq. The word εἰσάγειν used by Phanias does not necessarily imply that Polyxenos invented the argument. Cp. εἰσάγειν " to bring on the stage."

(2) This appears from the comic poet Ephippos, fr. 14 Kock. It is not clear whether Bryson was a member of the Academy, but he may have been. It makes no difference. What is important is that he was an associate of Sokrates.

(3) Theopompos, ap. Athen. 509 c.
Castañeda, Hector-Neri. 1974. "The Phaedo and the Third-Man arguments." ITA Humanidades no. 10:217–236.
Chailos, George. 2013. "Mathematical Interpretation of Plato's Third Man Argument." Far East Journal of Mathematical Sciences no. 83:185–222

Abstract: "The main aim of this article is to defend the thesis that Plato apprehended the structure of incommensurable magnitudes in a way that these magnitudes correspond in a unique and well defined manner to the modern concept of the Dedekind cut. Thus, the notion of convergence is consistent with Plato’s apprehension of mathematical concepts, and in particular these of density of magnitudes and the complete continuum in the sense that they include incommensurable cuts. For this purpose I discuss and interpret, in a new perspective, the mathematical framework and the logic of the Third Man Argument (TMA) that appears in Plato’s Parmenides as well as mathematical concepts from other Platonic dialogues. I claim that in this perspective the apparent infinite sequence of F-Forms, that it is generated by repetitive applications of the TMA, converges (in a mathematical sense) to a unique F-Form for the particular predicate. I also claim and prove that within this framework the logic of the TMA is consistent with that of the Third Bed Argument (TBA) as presented in Plato’s Republic . This supports Plato’s intention for assuming a unique Form per Predicate; that is, the Uniqueness thesis."
Chame, Santiago. 2024. "Bryson of Heraclea and Polyxenus, Megarian Philosophers." Phronesis no. 69:251–278

Abstract: "Bryson of Heraclea and Polyxenus have received little attention from scholars. Sources on these philosophers are few and difficult to interpret. However, they present interesting dialectical arguments that concern some of Plato’s and Aristotle’s most important theoretical elaborations: Bryson’s arguments on the issue of semantic ambiguity were explicitly discussed by Aristotle, and Polyxenus is credited with a particular version of the Third Man argument. My purpose in this paper is to reconstruct the historical background of these two philosophers and to analyze the philosophical implications of the arguments that the ancient tradition ascribes to them."
Cherniss, Harold. 1962. Aristotle's Criticism of Plato and the Academy . New York: Russell & Russell

Especially pp. 229-234, 284-376, and Appendix IV , pp. 500-505.

"Alexander [in his Commentary on Metaphysics , ed. Hayduck, 1891] then proceeds to the reference concerning the τρίτος άνθρωπος , which he explains by giving first the demonstration that " introduces the third man " (pp. 83, 34-84, 2).—The Platonists say that the common predicates of substances are properly what such predications signify of their subjects and that they are ideas; furthermore they say that things which are similar to one another are so by participation in some identical thing which is essentially that element of similarity, and this is the idea(188). Hereupon there follows an exposition of four types of the " third man " argument, the last of which, Alexander says (Metaph ., p. 85, 9-12), Aristotle used in the Περὶ Ἰδεῶν and εν τούτω μετ ολίγον, by which he means Metaphysics 991 A 2-3 (cf. Alexander, Metaph. , p 93, 5-7), the first of which was employed by Eudemus, among others, in his book Περὶ λέξεων. This first form (that used by Eudemus) states that, if the common predicate, not being identical with any of the things of which it is predicated, is something else apart from the subjects (since " absolute man " is made a genus for this reason, because it is identical with none of the particulars of which it is predicated), there must be a "third man" apart from the particular man and the idea which is itself a single individual (p. 84, 2-7). The form of the argument ascribed to Aristotle runs as follows (pp. 84, 21-85, 3):—If what is truly predicated of a number of things also exists as something other than these subjects and separate from them (as those who posit ideas believe that they demonstrate when they say that " absolute man " is an existing thing because " man " is truly predicated of the multiplicity of particular men and is other than the particular men),—if this is so, there must be a " third man," for, if the predicate is other than his subjects and has independent subsistence and " man " is predicated both of the particulars and of the idea, there must be a third man apart from these. In like fashion there must be a fourth winch is predicated of this third, of the idea, and of the particulars; and so on indefinitely Alexander himself notes (p 85, 3-5) that the latter form of the argument is the same as the first, inasmuch as both take as their starting point the doctrine that similar things are similar by participation in a single identical thing and proceed by pointing out that the men and the ideas are similar He does not mention that in the form ascribed to Aristotle there is the explicit charge of an infinite regress which is lacking in the first form as here stated (see note 194 infra [end])."

None of these arguments to establish the existence of the ideas occurs in the Platonic dialogues in the form in which they are here reported by Alexander, and, even if it be admitted that his account of them is a substantially exact representation of the Περὶ Ἰδεῶν, allowance must be made for the possibility that the way in which they were there staled was due to Aristotle's penchant for casting into syllogistic form the more loosely worded Platonic arguments (cf. Aristotle, frag. 45 and Jaeger, Aristoteles , pp. 41, 44-45) It is possible, however, to find in Plato's own writings arguments which could easily have been developed into these demonstrations, whether such developments were the work of Plato himself or of one or another of his students in the Academy. It will, therefore, be well to consider the demonstrations one by one, such evidence as there is for them in the writings of Plato and in other parts of the Aristotelian corpus, their probable meaning for the Platonists, the way in which Aristotle uses them in his refutation, and the validity of the criticisms of them reported by Alexander and indicated elsewhere by Aristotle." (pp. 233-234)

(188) These two sentences (pp 83. 34-84, 2) give the Platonic argument from which Eudemus and Aristotle develop their refutations, but the passage is not to be taken as an integral part of the argument of Eudemus which follows it, for the Platonic hypothesis from which the " third man " follows is stated as a protasis again in the arguments of Eudemus (p. 84, 2-3) and Aristotle (p 84, 22-27). Cf also p 85, 3-5 and Robin, Idées et Nombres , note 51* (p 611).

(256) See Appendix IV.
Clarke, Timothy. 2024. "In Search of Aristotle’s Third Man." Phronesis no. 69:279–315

Abstract: "Aristotle thinks that the Platonic theory of Forms is vulnerable to the Third Man regress. According to Alexander of Aphrodisias, the regress arises from the conjunction of three Platonist claims, which I label ‘Exemplification’, ‘Similarity’, and ‘Distinctness’.

It is clear why, taken together, these three claims generate an infinite regress of Forms.

What is not clear is why Aristotle thinks that a Platonist should have to accept each of the claims. My answer begins from the fact that, in Metaphysics A and M, Aristotle mentions the Third Man as a consequence, not of positing Forms, but of certain ways of arguing for Forms. By working out what these arguments are likely to have been, we can understand why he took the Platonist to have been committed to all three claims, and thus why he saw the Third Man as a serious difficulty."
Clegg, Jerry S. 1973. "Self Predication and Linguistic Reference in Plato's Theory of the Forms." Phronesis no. 18:26–43

"In all these cases the Third Man is judged as showing that a Platonic Form cannot have, without disastrous paradox or serious qualification, the property it is the Form for.

There is ample reason to object to all these positions and to the judgment which links them. It is most natural, I wish to argue, to see Plato's Forms as self-predicating. I wish to argue further that the Third Man poses no unmet challenge to that feature of a Form.

If I can make these two theses plausible the discussion of an already extensively treated topic will be justified. I shall try providing that justification by proceeding in four stages. First, I shall offor a set of reasons for why a Form should always be considered self-predicating. I shall then list some objections to the views of those who have thought otherwise. Third, I shall point out Plato's way of escaping the paradoxes of the Third Man. In conclusion I shall consider some possible objections to the value of what I take his escape route to be." (p. 27)
Code, Alan. 1985. "Appendix on 'The Third Man Argument' " In How Things Are: Studies in Predication and the History of Philosophy and Science , edited by Bogen, James and McGuire, James E., 323–326. Dordrecht: Reidel

Appendix to: "On the Origins of Some Aristotelian Theses About Predication", pp. 101-131. (The material in the appendix is taken from unpublished work in progress by H. P. Grice and myself).
Cohen, S. Marc. 1971. "The Logic of the Third Man." The Philosophical Review no. 80:448–475

Reprinted in: Nicholas D. Smith (ed.), Plato. Critical assessments, New York: Routledge, 1998, vol. IV pp. 28-50 and in Gail Fine (ed.), Plato. 1, Metaphysics and Epistemology , New York: Ixford University Press 1999, pp. 275-297.

"The nmain problems facing the interpreter of the Third Man Argument (TMA) in Plato's Parmenides (132a1-b2) arise not so much from what Plato says as from what he does not say.

Gregory Vlastos, in his famous paper of 1954,(1) points out that the argument is formally a non sequitur and sets out to discover the suppressed premises of the argument. The literature dealing with the TMA, already large in 1954, has become enormous since then, and all of the authors I have read have followed Vlastos at least this far. But beyond a shared belief that the TMA as written is formally invalid and that in order to understand the argument we must identify its suppressed premises, there has been little agreement among the commentators. What are the suppressed premises? Is Plato committed to holding them? Is the argument, with the addition of such premises, valid? Did Plato think it was? What does it prove? What did Plato think it proves? Radically different answers have been offered to these questions, and I do not expect to offer definitive answers to any of them in this paper. What I hope to do is to show in what way the main lines of interpretation offered to date are inadequate, and to advance a formalization of the TMA which avoids these inadequacies and seems to me better to reveal the logical structure of the argument.

On the basis of my examination of the logic of the TMA I conclude that the philosophical point of the argument is different from what it has been generally supposed to be." (p. 448)

(1) "The Third Man Argument in the Parmenides ," Philosophical Review , LXIII (1954), 319-349; reprinted with an addendum in Studies in Plato's Metaphysics , ed. by R. E. Allen (London, 1965), pp. 231-263. Subsequent references will be to the reprinted version, which will be cited hereafter as "TMA I."
Cohen, S. Marc, and Keyt, David. 1992. "Analysing Plato's arguments: Plato and Platonism." Oxford Studies in Ancient Philosophy. Supplementary volume :173–200

§ 3: An illustration: The TMA , pp. 178-187.
Cornford, Francis Macdonald. 1939. Plato and Parmenides. Parmenides' Way of truth and Plato's Parmenides. Translated, with an Introduction and a Running Commentary . London: Kegan Paul, Trench, Trubner & Co

Reprinted by Routledge & Kegan Paul, 1980.

"The ' Third Man ' seems to have been the title of several different arguments, which fall under two heads: (I) not involving an infinite regress, (2) involving an infinite regress (as here).

(I) Alexander (Coomentary on Metaphysics ) mentions an argument used by 'the Sophists'. When we say' a man walks' we do not mean the Form Man (which cannot move) nor yet any particular man (for we do not know what man it is who is walking). It must, then, be some ' third man '. Alexander adds that those who separate the common predicate from the individuals, as do the assertors of Forms, give a handle to this sophistical argument. But, on the face of it, the argument seems to point out merely that, granted there is a Form Man, we can make statements not only about that Form and about one specified individual man, but also about 'a man ' or 'some man ' unspecified : ' some man stole my umbrella.' Stoic logic recognised this type of 'indefinite proposition ' (αξιωτων αξιωμα) : 'someone (τις;) is walking' or 'he (εχείνος) is moving' (Diog. L. vii, 70; Sext. adv. log . ii, 97). Evidently this is not the argument used by Parmenides. There is no infinite regress ; and the ' third man ' is not an extra Form, but an unspecified individual." (pp. 88-89)
Cresswell, Max J. 1975. "Participation in Plato's Parmenides ." Southern Journal of Philosophy no. 13:163–171

"This paper is a discussion of Parmenides 128e-133a. Its aim is to shew that the absurdities which Parmenides finds in Socrates’ theory of Forms arise because the Forms are conceived as both immanent and transcendent; and that the view being attacked is precisely the view presented in the ‘relative tallness’ passage of Phaedo 102.

The strategy of the paper is as follows: Section I presents an analysis of Plato’s logical theory of predication. This is necessary to get the logic of the third man argument straight so that we can see how it fits in with the other arguments. Section II states the metaphysical problems which arise in connection with the logical theory of predication; and Sections III-V go through the Parmenides passage in the light of the theory of Forms of Phaedo 102." (p. 163)
Cubillo, Matía, and Óscar, Gerardo. 2021. "Suggestions on How to Combine the Platonic Forms to Overcome the Interpretative Difficulties of the Parmenides Dialogue." Revista de Filosofía de la Universidad de Costa Rica no. 60:157–171

Abstract: "This paper provides an original approach to research on the logical processes that determine how certain forms participate in others. By introducing the concept of relational participation, the problems of self-referentiality of the Platonic forms can be dealt with more effectively. Applying this to the forms of likeness and unlikeness in Parmenides 132d-133a reveals a possible way to resolve different versions of the Third Man Argument. The method of generating numbers from oddness and evenness may also be of interest; relational participation in these forms clarifies the interpretation of Parmenides 143e-144a."

This article was originally published in Spanish, with some changes, in the journal Éndoxa . Series Filosóficas 43 (2019), pp. 41-66.
Currry, D. C. K. 1992. "Owen's Proof in the Peri ldeon and the Indeterminacy of Sensibles in Plato." Ancient Philosophy no. 12:351–373

"In what follows I shall lay out two distinct. though, I will suggest, related trains of thought. I begin by making a case against. a certain prevalent interpretation of the motivation behind Plato's middle period postulation of Forms. I uncover the genesis of this reading in the writings of G.E.L. Owen and point out how this reading has become an interpretative dogma. This part of the paper, then, is an essay in the history of contemporary Platonic scholarship. Yet, this interpretation deserves attention not only on account of its prevalence, but also, I argue, because of its untenability. An understanding of the origin of this reading and its rather profound drawbacks will help us to see what other interpretative options are allowed by the text. The second part of this paper, then, offers a sketch of a reading of the relevant texts which does more justice to Plato's own words." (p. 351)
Cusmariu, Arnold. 1985. "Self-Predication and the "Third Man"." Grazer Philosophische Studien no. 23:105–118

Abstract: "Considerable effort has gone into clarifying the structure and the content of the Third Man Argument. Nevertheless, the argument is still an enthymeme. A premise crucial to it has yet to be stated openly. This premise holds the way out of the predicament which, happily, enables Plato to retain intact the foundations of the Theory of Forms. The solution proposed allows us in addition to look beyond the TMA and place this ancient argument in the context of an important modern insight."
d'Hoine, Pieter. 2019. "Proclus and Self-Predication." Epoché. A Journal for the History of Philosophy no. 24:461–470

Abstract: "In Proclus, like in Plato, we find statements about the Forms that at least appear to allow self-predication of Forms. In his discussion of the Parmenides’ s Third Man Argument (TMA), however, Proclus argues that Forms and their participants are not synonymous, which means that the property that the Form causes in its participants cannot be predicated of the Form itself. In this paper, I try to show how such seemingly self-predicative statements about the Forms are to be understood in the context of Proclus’ metaphysics. I will argue that, in Proclus, statements such as ‘(only) the Form of Large is truly Large’ should be considered what I will call ‘causal predications’. Causal predication does not attribute any property to a subject, but only concerns the subject’s causal efficacy in relation to that property."
Di Lascio, Ermelinda Valentina. 2004. "Third Men: The Logic of the Sophisms at Arist. SE 22, 178b36–179a10." Topoi no. 24:33–59

Abstract: "This article aims at elucidating the logic of Arist. SE 22, 178b36–179a10 and, in particular, of the sophism labelled “Third Man” discussed in it. I suggest that neither the sophistic Walking Man argument, proposed by ancient commentators, nor the Aristotelian Third Man of the Περ`ι ’ιδε ˆων, suggested by modern interpreters, can be identified with the fallacious argument Aristotle presents and solves in the passage. I propose an alternative reconstruction of the Third Man sophism and argue that an explanation of the lines regarding the identity of Coriscus and Coriscus the musician (178b39–179a3) is indispensable for its correct understanding, since they hint at another sophism in some important aspects analogous.

Finally, I show that two contradictions concerning ’´εκθεσις spotted by scholars in the passage are only apparent and can be dissolved once the assumption that the anti-Platonic Third Man argument is at stake here is discarded, and once the passage is read in the light of its agonistic context."
Diaz, M. Richard. 1978. "What is the third man argument?" Southern Journal of Philosophy no. 16:155–165.
Durrant, Michael. 1979. "Plato, the 'third man' and the nature of the forms." Southern Journal of Philosophy no. 17:287–304

"In this paper I shall (i) press the contention that Plato did not regard the‘Third Man argument’ (the T.M.A.) in either version presented in the Parmenides as ultimately valid against his theory of Forms; (ii) seek to offer an explanation as to why he presents us with no formal refutation of the T.M.A.; (iii) present an account of the role of the T.M.A. in the context of the Parmenides.

For convenience this paper is divided into four sections:

I A consideration of Professor Vlastos’s thesis that Plato was not in a position to reject the T.M.A. as invalid or accept it as valid on the grounds that he did not know or realise all of its necessary premisses or assumptions. My treatment of Vlastos will be relatively brief, though 1 trust not unfair. My justification for such treatment is the existence in print of a number of papers criticising (rightly, I hold) the position Vlastos advocates. 1 shall have recourse to some of these papers.

II A consideration of what evidence we have from the immediate context in which the T.M.A. is set in the Parmenides to indicate that Plato did not hold that the argument was ultimately valid against his theory.

Ill A consideration of whether there is a case for saying that in the actual texts of the T.M.A., Plato, through the mouth of Socrates, unrestriciedly accepts the argument as valid, and if not, how we may explain the fact that Plato offers us no formal refutation of the argument.

IV Consideration of an objection arising from Parmenides 135A-B3 and an assessment of Plato’s position at 135A-E. An account of the role of the T.M.A. in the context of the Parmenides." (p. 287)
———. 1997. "Meinwald's pros heauto analysis of Plato's apparently self-predicational sentences." Australasian Journal of Philosophy no. 75:383–395

"In her seminal work on Plato's Parmenides ,(1) Constance Meinwald argues that Plato clearly differentiates between two senses of predication. These two senses are indicated by her distinction between 'is' pros heauto (in relation to itself) and 'is' pros ta alia (in relation to other things); these two senses she designates pros heauto and pros ta alla predication respectively. This distinction, in her view, is the key to understanding the Dialectical Exercise in the second part of the Parmenides and to appreciating its function." (p. 383)

(...)

"Meinwald's scholarly and careful teasing out of the distinction between these two senses of 'predication' and their importance is indeed a valuable service to our understanding of Plato's thought in particular and to Philosophy in general, but I shall argue that there is a very serious difficulty for the general thesis she advocates ~ namely that such sentences as 'The Just is just'; 'The Large is large' are to be construed as making

pros heauto predications. Having produced reasons for holding that there is such a difficulty for such a general thesis, I shall then raise the question of whether Plato is committed to such a construction and, if he is, whether such a construction is Plato's final word on the matter. Insofar as my objection to the general thesis holds and insofar as Plato is committed to it then both Plato and Meinwald fall to the objection but I shall suggest that Plato can be saved in that it can be held that his advocacy of such a thesis (if indeed he does advocate it) is an interim measure waiting further development. I contend that the same cannot be said for Meinwald since (a) she does not suggest that the construction may only be an interim measure; (b) by construing the use of such sentences as expressing pros heauto predications, she explicitly claims that as a result of training in the Dialectical Exercise the mature 'Socrates' (the mature Plato) is able to indicate the fallacious nature of the 'Third Man' Argument as set out a Parmenides 132A ff.6 This argument has depended for its seeming validity on a confusion between the two senses of predication she has set out." (pp. 383-384)

(1) Plato's Parmenides (Oxford: Oxford University Press, 199l).
Dybikowski, James C. 1972. "Professor Owen, Aristotle, and the third man argument." Mind no. 81:445–447

"Professor G. E. L. Owen in his British Academy Lecture The Platonism of Aristotle (PBA, vol. lI, pp. 125-150) argues that Aristotle's theory of predication underwent considerable revision and that this revision is to be understood in terms of Aristotle's assessment of the several premises of the Third Man Argument (or TMA). The TMA, on Aristotle's interpretation at any rate (cf. Alexander in Met . 84.21-85.12), is a valid argument, but trivially so, since it depends on two incompatible premises, self-predication (or SP) and non-identity (or NI)." (p- 445)

(...)

"I conclude, therefore, that Owen has not established his claim that Aristotle's attitude toward the various premises of the TMA provides the necessary metaphysical background for tracing a line of development in his theory of predication." (p. 447)
Evans, David. 1996. "Platonic Arguments." Proceedings of the Aristotelian Society, Supplementary Volumes :177–193

"The first version of the Third Man argument (Parmenides 132a-b) appears to be entirely about the numbers and kinds of things that there are, and not at all about our modes of awareness of such things. The argument, as is well known, concludes that if the existence of a plurality of things with a common attribute justifies us in positing a Form distinct from each and all of them, the existence of the original plurality plus the Form justifies us equally in positing a further, second Form; and the move from the first to the second step can be repeated indefinitely." (p. 187)

(...)

"The same strategy would help us both to understand and to make better philosophical sense of the suggestion that immediately follows this first version of the Third Man argument. This is the idea that Forms are thoughts (19) In the dialogue this idea is interpreted as meaning either that they are psychological activities or that they are the objects of such activities-and 'object' here is again construed only in the extensional sense. This bipartition of options ignores the possibility that a Form is indeed an object but an intentional one. That is, its conceptual value consists in registering a unity in the things which we think about, rather than a unit in the things about which we think." (p. 190)

(19 Parmenides 132b-c.
Fehr, René Ardell. 2021. "Aristotle, Thomas Aquinas, and The Third Man Argument." New Blackfriars no. 102:517–533

Abstract: "The Third Man argument, as it originated in Plato’s Parmenides , is unjustly read into Aristotle. The Parmenides argument is briefly examined, followed by an analysis of the relevant Aristotelian texts, with a special emphasis on the commentary of Thomas Aquinas. Three different versions of Aristotle’s Third Man argument are identified, of which none contain the essential infinite regress that characterizes the Parmenides argument. Finally, current scholarship on the Third Man argument, especially as it pertains to Aristotle, is reviewed. In this respect, I note that the overwhelming tendency has been to identify Aristotle’s Third Man argument with that of the Parmenides , in spite of the fact that Aristotle only once articulates his version of the Third Man argument, and that this articulation is vastly different from its Parmenides counterpart. I conclude that contemporary Third Man scholarship must take this into account."
Fielder, John. 1978. "Plotinus' Reply to the Argument of Parmenides 130a—131d." Apeiron no. 12:1–5

"In a well-known passage in the Parmenides (130a-132b) Plato presents a series of objections to the Theory of Forms. The youthful Socrates is unable to reply to these objections and nowhere else in the dialogues do we find a refutation of them. Some of these arguments are taken up by Plotinus.(1) His responses have considerable philosophical merit and provide an additional perspective on the philosophical possibilities of the Theory of Forms. In what follows I set out Plotinus' responses to the arguments of Parmenides 130a-13ld and show how they meet the objections raised there." (p. 1)

(...)

"Although Plotinus is clearly concerned to defend the Theory of Forms against these two objections in the Parmenides, he is silent about another famous argument found in the same dialogue, the "Third Man" argument (TMA). No reference is listed in the Henri and Schywzer text, 13 nor have I been able to discover any mention of the argument in the Enneads . This is unusual, given the repetition of the argument by critics of Platonism. Although I do not have a convincing reason why Plotinus ignored this argument, I believe that his philosophy possesses the philosophical resources to disarm TMA, just as he was able to meet the other objections posed by Parmenides.(14)" (p. 4)

(1) Plotinus does not have anything to say about the 'Third Man' argument (Parm . 130a - 132b), and consequently it is omitted from the present discussion. This point is discussed briefly at the end of the article.

(14) A paper on this topic is in preparation.
Fine, Gail. 1982. "Owen, Aristotle and the Third Man." Phronesis no. 27:13–33

"In his British Academy lecture, "The Platonism of Aristotle", Professor G.E.L. Owen provides an account of how Aristotle arrived at his theory of predication and the categories.(1) He argues that both developed as a result of Aristotle's criticism of Plato, in particular as a result of his reflection on the Third Man Argument (TMA). In this paper I present an alternative account of Aristotle's response to the TMA. If this account is correct, it is unlikely that Aristotle's theory of predication and the categories developed as a result of reflection on the TMA." (p. 13)

(...)

"Conclusion.

I have argued that Aristotle's "first short reply" to the TMA - the this-such distinction - is also his last. But that reply, as we have seen, does not discriminate between strong and weak predicables, however we construe that contrast. Rather, all predicables are treated alike, all escape the TMA in just the same way - by being suches rather than thises. If we want to uncover the roots of Aristotle's theory of predication and the categorief, we shall need to look elsewhere.(35)" (p. 28, a note omitted)

(1) G.E.L. "The Platonism of Proceedings of the British Academy ( 1965), pp. 125-50 (,,,)
———. 1982. "Aristotle and the more accurate arguments." In Language and Logos: Studies in ancient Greek philosophy presented to G. E. L. Owen , edited by Schofield, Malcolm and Nussbaum, Martha, 155–177. Cambridge: Cambridge University Press

"In Metaphysics I 9 Aristotle mentions several Platonic arguments for forms. These were set out in detail, along with Aristotle's criticisms, in his essay Peri Ideon , portions of which are preserved in Alexander's commentary on I 9.(1) In this chapter I explore the logic of some of these arguments and the interconnections between them."

(...)

"I consider OMA [Over Many Argument] in section II. In section III I examine accurate-OMA. I show how it fits our interpretation of 'more accurate', as OMA does not. In IV I show that accurate-OMA, but not OMA, is vulnerable to the third man regress. In V I briefly consider AR [Argument from Relatives], and show how it is an accurate argument. In VI I argue that AR, unlike accurate-OMA, is not vulnerable to the third man regress." (p. 159)

(1) I use M. Hayduck's text of Alexander's in Aristotelis Metaphysica Commentaria , in volume I of Commentaria in Aristotelem Graeca (Berlin 1891). The relevant portions of Alexander may also be found in W. D. Ross, Aristotelis Fragmenta Selecta (Oxford 1955). A new edition of the Peri Ideon , by D. Harlfinger, appears in W. Leszl, Il 'De Ideis' di Aristotele e la Teoria Platonica delle Idee (Florence 1975), 22-39.
———. 1993. On Ideas: Aristotle's Criticism of Plato's Theory of Forms . New York: Oxford University Press

Chapter 15: Third Man Arguments 203; Chapter 16: Is Plato vulnerable to the Third Man Argument? , 255-241.

"I conclude that here as elsewhere Aristotle neither misinterprets Plato nor offers a decisive objection to him; we can see this once we understand how complicated his argumentative strategy is. Part of his argument depends on intruding into Platonism claims Plato probably rejects but that Aristotle takes to be true (such as the claim that universals cannot exist uninstantiated). Part of it depends on disallowing Plato claims he accepts but that Aristotle takes to be false (for example, the claim that sensible men are imperfectly men). Part of it depends on disallowing Plato arguments he hints at but does not defend in sufficient detail (such as the version of the Imperfection Argument that applies to man no less than to large). Part of it depends on seeing what happens if one commits Plato to claims he does too little to rule out, even if he does not intend them (this might be Aristotle's strategy in interpreting Rep . 523-5 and the TBA). Aristotle presents his criticisms of Plato briefly, and one might think they are straightforward and simple. But, like Plato's arguments, they are highly compressed and presuppose a variety of complex and controversial moves. Once we uncover Plato's and Aristotle's background assumptions, we can see that once again it is not easy to decide how successful Aristotle's criticism is." (p. 241)

[TBA = (Third Bad Argument) (Rep . 597 c1-d3)
Frances, Bryan. 1996. "Plato’s Response to the Third Man Argument in the Paradoxical Exercise of the Parmenides ." Ancient Philosophy no. 16:47–64

Abstract: "In this article I am not seriously challenging the accuracy of the main parts of Meinwald's interpretation of the dialogue. Rather I am challenging her specific claim that with the distinction between the two forms of predication she (and Plato) can solve the problem of the third man in the manner presented in Meinwald 1991 and 1992. Her proposal suffers from at least two defects: her solution fails for the problematic pros ta alia claims as well as in the pros heauto cases. In light of these problems we should not yet say 'good-bye to the Third Man'. There is, however, ample reason to think that if Meinwald's exposition is accurate, then Plato did solve the problem of the third man in the Parmenides - in fact, in the very way suggested by some contemporary scholars."

References

Meinwald, Constance. 1991. Plato’s Parmenides . New York: Oxford University Press.

Meinwald, Constance. 1992. ‘Goodbye to the Third Man’ 365-396 in The Cambridge Companion to Plato . Richard Kraut ed. Cambridge: Cambridge University Press.
Gazziero, Leone. 2012. "The Latin "Third Man": A Survey and Edition of Texts from the 13th Century." Cahiers de l'institut du Moyen-Âge Grec et Latin no. 81:11–93

"Introduction

0. Prolegomena

Readers of the Cahiers de l’Institut du Moyen Age Grec et Latin are long familiar with editions of medieval texts which bear on similar issues or belong to the same topic. Those gathered here for the first time all dwell on the Latin interpretation of the « Third Man » argument as it occurs in chapter twenty-two of Aristotle’s Sophistici elenchi.

The introduction will give the general outlines of the subject, which is to be dealt with in more detail in a forthcoming book whose scope will be the argument’s history as a whole.[*] The edition serves a twofold purpose: primarily, it is meant to provide a new, substantial contribution to the medieval corpus of the « Third Man » argument, rather incomplete to this day. Secondarily, it provides a background survey which ought to enable readers to restate the case, should they disagree with the thesis argued for in the forthcoming monography ]*] and summarized hereafter." (pp. 11-12, a note omitted)

[*] L. Gazziero, Rationes ex machina. La micrologie à l’âge de l’industrie de l’argument , Paris: Vrin, 2008.
Geach, Peter. 1956. "The Third Man Again." The Philosophical Review no. 65:72–82

"I am very much indebted to Professor Vlastos for helping me to interpret the Third Man Argument (TMA) - indebted both to his recent article in The Philosophica Review , and to his elucidations in private correspondence."

(...)

"There are, however, some important disagreements between Vlastos and myself; and so I have thought it worth while to set out my own interpretation of the TMA. First: I cannot believe Vlastos has rightly located the inconsistency in the premises of the TMA. On his view, Plato reaches the conclusion of the TMA by using two tacit assump tions: Self-Predication ("F-ness is itself an F") and Nonidentity ("no F is identical with F-ness"). Now these premises are not merely inconsistent; they are formally contradictory, related as "p" and "not p." For "no F is (identical with) F-ness" is equivalent to "F-ness is no F," which is the direct contradiction of "F-ness is itself an F." The con clusion of the TMA would therefore "follow," as Vlastos himself points out, only in the trivial way that absolutely anything "follows" when you use "p" and "not p" together as premises. Such a thorough muddle is not lightly to be imputed to Plato; it will be more satisfactory if we can find a set of assumptions each of which Plato may well have made, and interpret the TMA as bringing out a buried and unobvious inconsistency of the set.

Second: in formulating Plato's arguments Vlastos continually uses abstract nouns like "largeness" and "F-ness" as designations of Forms. Now, if our aim is to bring out Plato's implicit assumptions about Forms, we had better not call Forms by names that embody a lot of implicit assumptions of our own. Terms like "F-ness" have just this disadvantage. The way "F-ness" is formed from "F" embodies a tacit assumption that something or other stands in a special relation (expressed by the suffix " ness") to the things that are F; and to use "F-ness" (as Vlastos does) with the grammar of a singular term embodies a further tacit assumption that there are not several bearers of this relation to "the many" Fs. So far, indeed, no assumptions have been brought out that Plato would not have accepted, since he held that there is one and only one Form answering to a term "F" and standing in a peculiar relation to "the many" Fs. But the conclusion of the TMA calls in question this very view of his; and we can no more formulate that conclusion properly in terms of "F-ness" than we could formulate polytheism by saying "Allah is many," when the mere use of "Allah" embodies monotheistic assumptions, (pp. 72-73)
Gerson, Lloyd. 2011. "Proclus and the Third man." Études platoniciennes 105–118

"The so-called Third Man Argument (hereafter TMA) in Plato’s Parmenides (132A-B) continues to rivet the attention of Plato’s friends and foes alike.

In this paper, I focuson Proclus’ treatment of the TMA in his Commentary on Plato’s Parmenides . I do so to repay a debt of piety, offered in a spirit I would like to think Proclus would understand perfectly. For though prior to having read Proclus, I arrived at an analysis of this argument essentially identical to his, it was my encounter with the Proclean text along with my reading of Plotinus that completely reoriented my approach to ancient Greek philosophy. I came to appreciate that the pejorative and anachronistic term ‘Neoplatonism’ was actually an impediment to understanding Plato and even Aristotles well. I also became acutely aware that philosophical sophistication in the interpretation of Platonic texts does not date from the appearance of Anglo-American scholarship in the latter half of the 20th century." (p. 105, a note omitted)
Goldstein, Laurence, and Mannick, Paul. 1978. "The Form of the Third Man Argument." Apeiron no. 12:6–13

"In the light of our treatment, not only does the concluding line of Parmenides' argument make sense, but its force becomes apparent. Though it is true that the Form of Largeness as reappraised at the conclusion of the T.M.A. turns out to be both one and not one, this by itself does not represent a problem if the term One' is equivocal. But further, Teloh and Louzecky deny the "oneness" of the Form in the very sense in which, in our view, it is retained, i.e., oneness in the sense of uniqueness. What is denied of the Forms, on our view, is oneness in the sense of completeness. This results from the indefinite plurality which a Form turns out to be. The crunch is that Socrates has provided us with Forms whose comprehension is as problematic as that of the particular instances, the comprehension of which the Forms were meant to facilitate. Given this treatment of the thrust of the T.M.A. it is no surprise to find Plato in the Sophist working on philosophical problems involving the relations among Forms and to find him later in the Philebus

sketching the relations of monadic Forms to the indefinite pluralities comprehended through them." (pp. 11-12)

References

H. Teloh and D.J. Louzecky: 'Plato's Third Man Argument', Phronesis , 1972
González-Varela, José Edgar. 2020. "The Platonist Absurd Accumulation of Geometrical Objects: Metaphysics Μ.2 " Phronesis no. 65:76–115

Abstract: "In the first argument of Metaphysics Μ.2 against the Platonist introduction of separate mathematical objects, Aristotle purports to show that positing separate geometrical objects to explain geometrical facts generates an ‘absurd accumulation’ of geometrical objects. Interpretations of the argument have varied widely. I distinguish between two types of interpretation, corrective and non-corrective interpretations. Here I defend a new, and more systematic, non-corrective interpretation that takes the argument as a serious and very interesting challenge to the Platonist."
Hales, Steven D. 1991. "The Recurring Problem of the Third Man." Auslegung. A Journal of Philosophy no. 17:67–80

"Pretty much everyone agrees that Plato must have relied on three assumptions in his regress, which have become known as the One Over Many, Self Predication, and Non Identity assumptions. Most writers take Non Identity to be a largely uncontroversial matter of stipulation. The main virtue of this paper is that it argues that Non Identity is consistent with Plato's explicit premises only if Plato accepted a view in the theory of reference which I believe he would have denied. If NI is rejected, the third man argument is not a legitimate regress and Plato's middle period theory of forms has nothing to fear from it. It is also argued that the third man argument is a purely an exercise in ontology, and is not concerned (primarily) with reference or language. The paper is divided into three main sections. The first section is a reading of what happens in the Parmenide s before the first third man argument at 132a. The second section presents two interpretations of the third man. The third section is a discussion of reference necessary for evaluation of the two versions of the third man argument." (p. 67(
Hand, Michael. 1993. "Mathematical Structuralism and the Third Man." Canadian Journal of Philosophy no. 23:179–192

"Plato himself would be pleased at the recent emergence of a certain highly Platonic variety of platonism concerning mathematics, viz., the structuralism of Michael Resnik(1) and Stewart Shapiro.(2) In fact, this

species of platonism is so Platonic that it is susceptible to an objection closely related to one raised against Plato by Parmenides in the dialogue of that name. This is the Third Man Argument (TMA) against a view about the relation of Forms to particulars. My objection is not a TMA against structuralism; the position avoids that objection, but is vulnerable to a different one precisely at the point where it avoids the TMA.

The way structuralism avoids the TMA has in fact been considered, as one of Plato's options, by at least one commentator on the Parmenides, Colin Strang(3), who explicitly rejects it on logical grounds. In the course of the discussion, I shall clarify the reason that I believe led Strang to reject this option, and shall modify his own statement of that reason."

(1) See, in particular, his 'Mathematics as a Science of Patterns: Ontology and Reference,' Nous 15 (1981) 529-50; 'Mathematics as a Science of Patterns: Epistemology,' Nous 16 (1982) 95-105; 'Mathematics from a Structuralist Point of View,' Revue Internationale de Philosophie 42 (1988) 400-24.

(2) See his 'Mathematics and Reality,' Philosophy and Science 50 (1983) 523-48; 'Structure and Ontology,' Philosophical Topics 17 (1989) 145-71.

(3) 'Plato and the Third Man,' in Gregory Vlastos, ed., Plato I: Metaphysics and Epistemology (Garden City, NY: Doubleday 1971) 184-200.
Hansen, Michael J. 2010. "Plato’s Parmenides : Interpretations and Solutions to the Third Man." Aporia no. 20:65–76

Abstract: "Plato’s Parmenides fulfills a special role in understanding Plato’s theory of Forms. Throughout his dialogues, Plato is pleased to invoke the theory intermittently, often as ancillary support for related pieces of his philosophy. But the theory itself remains concealed, even mysterious, never giving up a broad exposure for the reader’s immediate analysis. Hence, what we know of Plato’s Forms comes from piecemeal assembly.(1) Given this situation, the attacks levied against the Forms in the Parmenides may reveal vital aspects of the theory and perhaps even betray Plato’s own misgivings about the Forms. In this paper I will not consider whether Plato maintained his theory in the face of these critiques. Rather, I will present the famous Third-Man Argument from the Parmenides , review some competing perspectives on the argument among Plato’s interpreters, and defend Gregory Vlastos’ interpretation due to its historical plausibility."

(1) For a comprehensive listing of Plato’s comments regarding the Forms, see “Chapter 28: Form” of The Great Ideas: A Synopticon of Great Books of the Western World (Vol. II). Encyclopaedia Britannica (1952), 536–41.
Hathaway, R. F. 1973. "The Second 'Third Man'." In Patterns in Plato's Thought: Papers Arising Out of the 1971 West Coast Greek Philosophy Conference , edited by Moravcsik, J. M. E., 78–100. Dordrecht: Reidel

"Finally, the present analysis has shown that it is possible to give up the predicational model that has controlled our understanding of the Theory of Forms since Aristotle. Cohen, for instance, argues that "The one-over-many principle is supposed to provide an answer to questions like 'What makes it correct for many things all to be called 'F'? "(44). But it is obvious that the OM [One Over Many] assumption is powerless to explain the correctness of predications, whence Cohen's view that the TF [Theory of Forms] is vacuous as a theory of predication should follow without effort. According to whom, however, is OM supposed to explain predication, much less the correctness of predicates?

OM itself needs explaining. The genuine strength of the Theory of Forms must be recovered before we accuse it of vacuousness." (pp. 95-96)

(44) Cohen, ['The Logic of the Third Man', The Philosophical Review 80 (1971), 448-476.], p. 474. Perhaps what is intended is rather, "What makes it necessary for many things all to be called 'F'?"
Hunt, David. 1997. "How (not) to exempt Platonic Forms from Parmenides' Third Man." Phronesis no. 42:1–20

"One response invited by the Third Man Argument (or "TMA") of Plato's Parmenides is that it simply fails against the middle-period theory of Forms, not so much because its premises radically misconstrue that theory, but because there are good reasons endemic to the theory itself (and not simply introduced ad hoc) for exempting the Forms from at least someof the principles which drive the argument. Call this the "Exemption Defense".

(...)

If the Exemption Defense is accepted, it can have far-reaching consequences for the resolution of other issues, such as the development of Platoìs own thoughts on the Forms, his purposes in the Parmenides, and the relation of this dialogue to others. My focus in this paper is not on these further questions, however, but on the prior question of whether the Exemption Defense is in fact available. Following a standard formulation of the TMA, I develop what I take to be the most plausible version of the Exemption Defense. I then argue that, while the Exemption Defense is in fact successful in repelling an important attack on the theory of Forms, it's not clear that this is the attack being made in the Parmenides . I conclude with some thoughts on the actual attack and its legacy for the Platonic project." (p. 1)
Hunt, David 1997. "Plotinus meet the Third man." In The Perennial Tradition of Neoplatonism , edited by Cleary, John J., 119–132. Leuven: Leuven University Press

"The “Third Man Argument” (or “TMA”) was thought by Aristotle and other early critics of Platonism to pose a significant challenge to Plato’s theory of Forms, and recent commentators have evidently concurred in this judgment, lavishing more attention on the TMA than on any of the other problems that Plato raises for the Forms in the first part of the Parmenides . This is in marked contrast to the most acute of Plato’s ancient commentators, Plotinus, who ignores the TMA completely in his own discussion of these problems. Plotinus’s silence here has not, however, detened speculation on how he would have responded to the TMA had he deigned to consider it. Such speculation is surely no less legitimate in Plotinus’s case than it is in the case of Plato himself, who also avoided giving any direct answer to the TMA in his writings. Insofar as the TMA is valid and begins with premises which are at least not wildly misrepresentative of Plato’s views, a satisfactory' response to the argument must be regarded as a key test of the adequacy of (neo) Platonic theorizing.

I propose in this paper to review some of the resources at Plotinus’s disposal for blunting the force of the TMA. The discussion which follows is largely critical, inasmuch as I focus on mistaken directions in which these resources have generally been sought. But I also mention at the end some directions that I believe to be more promising." (p. 119)
Kung, Joan. 1981. "Aristotle on Thises, Suches and the Third Man Argument." Phronesis no. 26:207–247

"After mentioning some type distinctions in recent thought, I shall argue that understanding the this-such distinction as a distinction of logical type that is allied with these enables us to give a plausible account of Aristotle's response to the Third Man Argument, one which avoids the difficulties of some other important interpretations. I shall then indicate, more briefly, how reading the distinction in this way also helps to illuminate a number of otherwise perplexing passages. Such considerations constitute evidence for the correctness of so understanding it." (p. 208)
Lee, Edward N. 1973. "The Second 'Third Man': An Interpretation." In Patterns in Plato's Thought: Papers Arising Out of the 1971 West Coast Greek Philosophy Conference , edited by Moravcsik, J. M. E., 101–122. Dordrecht: Reidel

"My concern in this paper is not with the formal structure of the second TMA but with the philosophical strategies behind it. I shall begin by sketching an interpretation of these, though in a fairly rough-and-ready way, so as to have some main issues of this discussion fairly clearly set before us from the start (Section I). Then, in order to define some of the finer structure of these issues, I shall draw a few distinctions between various aspects of the concepts of resemblance and of imitation, and shall illustrate these distinctions in application to a few important texts in Plato (Sections II and III). To conclude (IV), I will return to a discussion of the second TMA, spelling out what will perhaps by then be obvious enough, the way in which all these distinctions bear upon the metaphysical issues implicated in the second TMA." (p. 101)
Lewis, Frank A. 1985. "Plato's Third Man Argument and the 'Platonism' of Aristotle." In How Things Are: Studies in Predication and the History of Philosophy and Science , edited by Bogen, James and McGuire, James E., 133–174. Dordrecht: Reidel

"In his 'The Platonism of Aristotle', G. E. L. Owen attempts to trace some key theses from Aristotle's Categories and Metaphysics back to what he argues are their origins in Aristotle's reflections on the Third Man Argument (TMA), first advanced by Plato in the Parmenides ." (p. 133, a note omitted)

(...)

"In Section I, I sketch the details of Owen's account, and suggest a reason for doubting the accuracy of his transcription of the assumptions Alexander takes to be at work in the TMA. In Section 2, accordingly, I consider the revised account of the argument in Alexander given by Code, and add some further refinements of my own. Code also states three semantical principles that bear on Aristotle's controversy with Plato: these will be the subject of Section 3.

(...)

In Section 4, I return to Owen's historical thesis. On a broader conception of self-predication than Owen's SP [Self-predication] allows, it appears that the TMA and the Categon·es are after all concerned with elements in the notion of self-predication to a considerable degree. Contra Owen, however, there is virtually nothing in common between his NI [Non-identity] ( or its denial) and Aristotle's thesis in Metaphysics Z6 that each thing is (essentially) the same as its essence. This last point is developed further in Section 5. I argue that Aristotle's sameness thesis in Z6 is unintelligible outside the new metaphysics of the rest of Book Zeta. Aristotle's notion of the substance of a thing, first, allows him to construct a variety of anti-Platonism that cannot even be stated within the conceptual framework of the Categories . Finally, I discuss the argument Aristotle gives at Z6, 103lb28-1032a2, for the conclusion that each thing is (essentially) the same as its essence. The convolutions of this argument dramatize vividly the way in which the Z6 thesis is linked irretrievably to Aristotle's mature theory of essence, definition, and substance." (pp. 133-134)

References

Alan Code,'On the Origins of Some Aristotelian Theses About Predication' same volume, pp. 101-131.

Owen, G. E. L., 'The Platonism of Aristotle'. Reprinted in Articles on Aristotle , Vol. 1 (edited by J. Barnes, M. Schofield, and R. Sorabji), Duckworth, London, 1975
Malcolm, John. 1981. "Semantics and Self-Predication in Plato." Phronesis. A Journal for Ancient Philosophy no. 26:286–294

"It is crucial to the notorious Third Man argument (Parmenides 132a 1 -b2, 132d 1- I 33a6), where an infinite regress is alleged to ensue from the adoption of Plato's Theory of Forms, that the Form and its instances be grouped together in the same class and this by virtue of the so-called Self-Predication of the Forms. In other words, a given Form, e.g., Beauty, or the Beautiful, is itself beautiful and this fact allows this Form to be included along with the beautiful particulars in the class of beautiful things. In general, not only is a certain predicate "F" true of a group of particulars, but the Form, F-ness, in virtue of which this is so, is F as well - assuring the truth of the formulation "F-ness is F."

Some scholars have convinced themselves that an examination of Plato's concepts of eponymy and homonymy is sufficient to disallow self-predication' and so free Plato from the questionable clutches of the Third Man. I propose to challenge this contention and shall maintain that Plato's use of eponymy and homonymy is inconclusive as regards self-predication and hence the evidence for or against the latter must be sought elsewhere. I hope, in passing, to cast new light on an interesting passage in Plato (Parmenides 133d) and, finally, to offer some considerations on the relation between predicate adjectives and Platonic Forms." (p. 286)
———. 1985. "Vlastos on Pauline Predication." Phronesis no. 30:79–91

"In the period following his epoch-making attention to the Third Man Argument' Gregory Vlastos was converted to the cause of what he has termed Pauline Predication (.2) Sentences such as "Justice is pious" or "Fire is hot" seem, on the surface, to be attributing a characteristic to a Form - a situation labelled by Vlastos "ordinary predication" (AS,(3( p. 273). But, Vlastos suggests, such cases are really to be read as assigning this property to instances of the Form. As applied to putative cases of self-predication,4 "F-ness is F" becomes, from the Pauline perspective, equivalent to "Necessarily, for all x, if x partakes of F-ness, x is F" (AS, p. 273; UVP, p. 235)." (p. 79)

(...)

"I share Vlastos' dominant motivation which is to reveal Plato in the most favorable light possible and it may come down to a question as to which flaws one finds least disfiguring. I see his earlier Plato, despite the unacceptable identification of paradigm- case with universal, a more challenging and stimulating thinker than the lacklustre, yet still blemished, creature of his later exegesis." (p. 91)

(2) This label is attributed to Sandra Peterson. Her version of Pauline Predication, appeared in "A Reasonable Self-Predication Premise for the Third Man Argument," Phil. Rev ., 82 (1973), 451-470, is different from that of Vlastos. In this paper I shall be concerned only with the Vlastos formulation.

(3) All references in this article, unless otherwise specified, are to Platonic Studies (Princeton, 1973; 1981 ). They are abbreviated as follows: "UVP" for "The Unity of the Virtues in the Protagoras"; "AS" for "An Ambiguity in the Sophist "; "TLPA" for "The 'Two-Level Paradoxes' in Aristotle."
———. 1991. Plato on the Self-Predication of Forms: Early and Middle Dialogues . Oxford: Clarendon Press

Chapter Three: The Third Man , pp. 47-53.

"I have just (p. 45) affirmed that self-exemplification is extended to cover all middle-dialogue' Forms. The justification for this claim will occupy subsequent sections, where it will be argued that all such entities are both universals (Chapter 4) and paradigm cases (Chapters s—7). But even one of the few explicit examples of self-exemplification in the middle dialogues, Equality as equal at Phaedo 74d, takes us beyond the unproblematic status of those of the earlier writings, for the universal, Equality, cannot in itself be regarded as an equal thing. The middle-dialogue Forms, therefore, rest very much in the shadow of the Third Man. I have already had several occasions to refer to this argument and, rather than continue to postpone its presentation, I shall outline it forthwith. Although anyone who presumes to add something substantive to the treatment of a passage which has proved so stimulating to contemporary commentators must approach the topic with trepidation, I believe-I can offer a few informal, yet useful observations." (p. 47)

(...)

"My ultimate goal for the remainder of this treatise will be to offer a suggestion as to why the Plato of the middle dialogues did not distinguish the universal from the paradigm instance in cases where their identification caused the embarrassing consequence presented in the TMA of the Parmenides , but made the same entity, the Form, fulfil both functions. My treatment will follow a four-step sequence. In Chapter 4 I will try to establish that the middle-dialogue Forms are, indeed, to be seen as universals. In Chapters 5-7 I will undertake to confirm that these Forms are also paradigm cases, to show that in these dialogues F-ness, or the F itself, is an F thing. In Chapters 8—9 I consider explanations as to why Plato postulates ideal paradigm instances, and finally in Chapter 10 I attempt to explain why these ideal paradigm instances, the Forms as paradigm cases, were not distinguished from the Forms as universals: a situation which resulted in the diffuse and detrimental self-exemplification of the middle dialogues." (pp. 52-53)
Mann, William E. 1979. "The Third Man = the Man Who Never Was." American Philosophical Quarterly no. 16:167–176

"My purpose in this paper is to present and defend a rather heterodox interpretation of the argument which appears at 132A-B of the Parmenides .

The argument is commonly known as the "Third Man Argument," but for reasons which will become apparent, I shall refrain from calling it that. I do not relish the prospect of being heterodox on this issue, and I would not bother if it were simply a matter of fussing about a particular Platonic passage. I believe, however, that all the various interpretations of the argument which have been offered share in a fundamental mistake, a mistake which in tum distorts our understanding of Plato's later metaphysics.

These interpretations suppose that the argument is intended to apply against all of Plato's Forms. The supposition immediately creates a quandary. For nowhere in the Platonic corpus is there even an attempt to rebut the argument that Plato places in Parmenides' mouth. Moreover, in dialogues which clearly were written after the Parmenides , the Theory of Forms is still very much in evidence. If e suppose that Plato thought that Parmenides' argument told equally against all the Forms, then we are forced to picture Plato as (a) having presented an objection which is potentially fatal to all the Forms, (b) having failed to respond to that objection, and (c) having continued nevertheless to espouse the Theory of Forms." (p, 167)
Marmodoro, Anna. 2008. "Is Being One Only One? The Uniqueness of Platonic Forms." Apeiron no. 4:211–227

Abstrac t: "Each Form is unique in number; no two numerically distinct Forms can share the same nature. Plato argues for this claim in Republic X. I identify the metaphysical principles Plato presupposes in the premises of the argument, by examining the reasoning behind them, and offer a reconstruction of the argument showing the principles in use. I argue that the metaphysical significance of the argument’s conclusion is to establish that if a Form F were not unique, if there were many Forms F, their nature would alter along with their number: a Form cannot recur without change in its constitution. This is why there can be only one Form for each character in the world."
Mates, Benson. 1979. "Identity and Predication in Plato." Phronesis. A Journal for Ancient Philosophy no. 24:211–229

Reprinted in: S. Knuuttila and J. Hintikka (eds.), The Logic of Being, Dordrecht: Reidel 1986, pp. 29-47.

"Among the Platonic statements that have most agitated his commentators, from Aristotle's time down to the present, are those in which he seems to be saying (and with great confidence, too, as though there were no question about it) that beauty itself is beautiful, justice itself is just, largeness is large, piety is pious, and the like. On the one hand, these statements are considered by many to involve some sort of category-mistake or serious ambiguity: beauty itself, they say, is not the sort of thing that can be beautiful, at least not in the same sense in which people, statues, paintings, or pieces of music are beautiful. And likewise with justice itself, largeness itself, and the other Ideas. On the other hand, though, there is the awkward fact that these so-called "self-predications" cannot be lightly dismissed as mere lapsus linguae on the part of our author, for they seem essentially related to his doctrine that each Idea is a paradigm or perfect exemplar for the particulars that fall under it; beauty itself is said not only to be beautiful, but to be the most beautiful thing of all.

In recent times this situation has been analyzed on the basis of the assumption that the verb "to be" has at least two senses, viz., the predicative sense, as in "Socrates is human", and the identity sense, as in "Socrates is the husband of Xanthippe". Plato's critics castigate him for being unaware of the distinction, while his defenders believe that he was perfectly well aware of it and that the allegedly self-predicative statements are to be understood as assertions of identity. In this paper I wish to investigate the possibility that the assumption is false, and that consequently neither the attacks nor the defenses that are based upon it are well-founded." (p. 211, a note omitted)

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