The Logic of The Third Man S. Marc Cohen

The Logic of The Third Man [Overview] A. Central Topic: The Third Man Argument [TMA] I. the argument at Parmenides 132a-b B. Purpose I. Formulate a version of the TMA which: 1. is logically consistent 2. accurately represents Plato’s argument in the text II. Show how previous formulations are inadequate 1. Vlastos’ argument 2. Sellar’s argument C. Format (eight sections) I. sections(1-3): examination and critique of Vlastos’ and Sellar’s TMA formulations II. sections(4-6): Cohen’s TMA argument III. sections(7-8) implications of Cohen’s TMA argument and conclusions

Parmenides 132a-b [Cornford's Translation] “I imagine your ground for believing in a single form in each case is this. When it seems to you that a number of things are large, there seems I suppose, to be a certain single character which is the same when you look at them all; hence you think that largeness is a single thing.... But now take largeness itself and the other things which are large. Suppose you look at all these in the same way in your mind's eye, will not yet another unity make its appearance-a largeness by virtue of which they all appear large? ... If so, a second form of largeness will present itself, over and above largeness itself and the things that share in it, and again, covering all these, yet another, which will make all of them large. So each of your forms will no longer be one, but an indefinite number.”

Concepts A. Defining [Rough Definitions] I. One-Over-Many: For any plurality of F things, there is a form of F-ness by virtue of partaking of which each member of that plurality is F. II. Self-Predication [SP]: Every form of f-ness is itself F III. Non-Identity [NI]: No form partakes of itself B. Applying B One-Over-Many: A, B and C are all Tall, therefore they participate in one form, Tallness, which is over them all A C Self-Predication: the form of tallness is tall Non-Identity: tallness doesn't participate in the form of the tallness

Vlastos I. Formulation of the TMA II. Suppressed Premises of the TMA A1- if a number of things are all F, there must be a single form, F-ness. A2- if a number of things and F-ness are all F, then there must be another form, F-ness 1. II. Suppressed Premises of the TMA 1. Argues that A2 doesn’t follow from A1, therefore there must be suppressed premises SP: any form can be predicated of itself NI: if anything has a certain character, it cannot be identical with the Form in virtue which we apprehend that character 2. Contends that NI and SP are formal contradictories SP: F-ness is an F thing NI: amounts to the assumption that F-ness is not an F-thing 3. Still insists that these were the premises Plato used [unknowingly] III. Scholarly Critiques 1. Cohen disagrees, recognizing that if SP and NI are required, then the conclusion must necessarily be logically inconsistent. Cohen maintains that there is no support in the text for logical inconsistency in Plato’s conclusion and points out that if the conclusion is logically inconsistent, then any other inconsistent set of premises would work as well, not only SP and NI. 2. Sellar argues that the SP and NI assumptions don’t have to be viewed as contradictories.

Sellar I. Formulation of the TMA II. Scholarly Critiques 1. Argument focuses on the treatment of the expression “F-ness” 2. Contends that “F-ness” represents a class of variables rather than a class of names 3. In Sellar’s version of the assumptions, quantifiers are added and two more assumptions are included, G and P, in order to eliminate logical inconsistencies G: if a number of things are all F, there must be an F-ness by virtue of which they are all F P: a,b,c and so forth, particulars, are F SP’: All F-nesses are F. NI’: If x is F, then x is not identical with the F-ness by virtue of which it is F II. Scholarly Critiques 1. Vlastos regards this argument as “incomparably better” than his own, but he contends that it is not supported by the text, Cohen agrees 2. Vlastos formulates a revision of G, G1 G1: if a number of entities are all F, there must be exactly one form corresponding to the character, F; and each of those entities is F by virtue of participating in that Form 3. Cohen argues that Vlastos’ G1 is logically inconsistent with Sellar’s SP and NI assumptions. Likewise, he believes that it is not the only alternative to Sellar’s G. Cohen identifies G2 G2: if a number of entities are all F, there must be exactly one form by virtue of which they are all F. 4. Cohen argues that G2 is better than the previous formulations, but through examination he realizes is ultimately insufficient. Cohen contends that in order to come up with an adequate formulation of G2, a shift to a version which quantifies over sets of F’s as well as over F’s will be necessary.

Cohen I. Formulation of the TMA [OM-Axiom] A. Terms object: any F thing which ‘F’ can be predicated. particular: an object in which nothing participates. form: an object that is not a particular. maximal set: contains every object on every level equal to or less than the level of its highest-level member one-over-many: For any maximal set there is exactly one Form in which all and only members of that set participate. B. Levels level 0 objects: particulars level 1 objects: objects composed of only particulars and every particular level (n) objects: all participants are of level n-1 or lower, and all objects ` of n-1 or lower participate in it C. Critique 1. Cohen concludes that the idea of maximal sets doesn’t align with the text and implements the one-over- one relationship rather than a one-over-many principle.

OM-Axiom 1. Let [a] be the set of all particulars. Proofs: 2. [a] is a maximal set (level 0) (1) (D4) (D7) (D8) 3. There is exactly one Form over [a], call it "F-ness I." (2) (OM-axiom) 4. F-ness I is of level one (1) (3) (D5) 5. F-ness I is not a member of [a] (2) (4) (T1) (D7) 6. [a] U F-ness I is maximal (level one). (2) (4) (T2) (D7) (D8) 7. There is exactly one Form over [a] U F-ness I, call it F-ness II (6) (OM-axiom) 8. F-ness II is of level 2. (6) (7) (D6) 9. F-ness II is not a member of [a] and F-ness. (6) (8) (T1) (D7) 10. F-ness II NOT = F-ness I (9) ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ (OM-axiom) For any maximal set there is exactly one Form in which all and only members of that set participate. (D4) I will also speak of a particular as an object of level 0 (D5) An object is an object of level one if (a) All of its participants are particulars, and (b) all particulars participate in it. (D6) In general, an object is an object of level n (n > I) if (a) All of its participants are of level n - 1 or lower, and (b) all objects of level n - 1 or lower participate in it. (D7) A set of objects is a set of level n if it contains an object of level n and no higher-level object. (D8) A set of level n will be said to be a maximal set if it contains every object of level m for every m < n. In other words, a maximal set contains every object on every level equal to or less that the level of its highest-level member. (T1) No object is on more than one level. (T2) There is exactly one object on each level (greater than 0).

Cohen II. Formulation of the TMA [IOM-Axiom] A. Terms one-over-many: for any set of F’s, there is exactly one Form immediately over that set Plato’s over relation: x is over y= if y is a set, every member of y, participates in x [many-over-many] immediately over: x is immediately over y=x is over y and x is over all and only those sets whose level is equal to or less than that of y [one-over-many] Plato’s uniqueness thesis: : when you consider a set of large things, exactly one Form of Largeness will come into view, immediately over that set; so there is exactly one Form of Largeness B. Conclusions 1. Cohen contends that the argument of one over many, thought to be a safe route to the uniqueness thesis, has been shown to be defective 2. apparent problem with the theory of the forms lies in the one-over-many principle

IOM-Axiom I. Let [a] be any set of F's (of level n). 2. There is exactly one Form immediately over [a], call it "F- ness I.” (1) (IOM-axiom) 3. F-ness I is of level n + 1 (1) (2) (T3) 4. F-ness I is not a member of [a] (1) (3) (T1) (D7) 5. [a] U F-ness I is of level n + I (1) (3) (D7) 6. There is exactly one Form immediately over [a] U F-ness I, call it "F-ness I I.” (IOM-axiom) (5) 7. F-ness II is of level n + 2 (5) (6) (T3) 8. F-ness II is not a member of [a] U F-ness I (5) (7) (T1) (D7) 9. F-ness II NOT= F-ness I (8) (D7) A set of objects is a set of level n if it contains an object of level n and no higher-level object (T1) No object is on more than one level (T3) If x is immediately over y, then the level of x is one greater than the level y (IOM-axiom) For any set of F's, there is exactly one Form immediately over that set.

IOM-Axiom(Many-Over-Many Relation) “I imagine your ground for believing in a single form in each case is this. When it seems to you that a number of things are large, there seems I suppose, to be a certain single character which is the same when you look at them all; hence you think that largeness is a single thing.... But now take largeness itself and the other things which are large. Suppose you look at all these in the same way in your mind's eye, will not yet another unity make its appearance-a largeness by virtue of which they all appear large? ... If so, a second form of largeness will present itself, over and above largeness itself and the things that share in it, and again, covering all these, yet another, which will make all of them large. So each of your forms will no longer be one, but an indefinite number.” F-ness 2 Particular S Form F-ness 1 A B C A B C Set 0 Set 1 Set 2