In the Parmenides, Plato did what he knew would be done by someone else anyway: he refuted a central plank of his own philosophy, the Theory of Forms. When Aristotle came along to do what Plato had already foreseen and further the refutation, the argument was already old hat. Nevertheless, Aristotle had chosen a far better example to illustrate the point, an example which also lent itself to a snappy title by which the argument is now known: the Third Man argument.
The Third Man argument is a nifty delight that is often confusingly expounded. I reckon I can do better, so here now I explicate either triumphantly or to no avail.
The Simple Part
There’s a single Form for each recognisable object or quality in the real world, all of which are the imprecise and inferior copies of their respective ideal Forms. Thus, we can recognise all real-life rectangles as rectangles, for instance, because we have the Form or essence of rectangles in our heads. So when presented with soccer pitches, books and rulers, we can assign them to the group headed by the ideal rectangle that we have a mental picture of and dub them all rectangles.
The Confusing Part
Soccer pitches, books and rulers are very distinct yet are nevertheless rectangles. If these rectangles are so distinct from each other, the one ideal rectangle must be just as distinct from the variety of rectangles in the real world as the real-world rectangles are all distinct from each other. Thus, how can the one ideal rectangle be of use in categorising all real-world rectangles as rectangles? Alternatively put, if the one ideal rectangle is itself a rectangle that heads the group that includes soccer pitches, books and rulers, how is the ideal rectangle itself recognised as a rectangle let alone as the ideal rectangle?
The ideal rectangle must itself be recognised as an ideal rectangle just as a real-world rectangle is recognised as a real-world rectangle. A real-world rectangle is recognised as a real-world rectangle via the ideal rectangle. Thus, if we want to identify the ideal rectangle as the ideal rectangle, we can do so only via the ideal of the ideal rectangle.
The Easy Part Once You’ve Understood the Confusing Part
Of course, we’ve now got ourselves a reductio ad infinitum or an infinite regress: if the ideal of the ideal rectangle is needed to identify the ideal rectangle, then the ideal of the ideal of the ideal rectangle will be needed to identify the ideal of the ideal rectangle and so on to infinity. And if we’ve got an infinite regress, than the Theory of Forms doesn’t explain much at all.
The Third Man Argument in But Three of its Forms
Substitue man for rectangle in the explanation above and you have Aristotle’s Third Man argument (the first man is the real-world man, the second the ideal man, the third the ideal of the ideal man). Substitute large for rectangle in the explanation above and you have Plato’s own refutation of his Theory of Forms. (Large, though, is a confusing example because it’s so difficult to imagine an ideal of something that is a relative quality. Cold is the absence of heat, so small can be considered the absence of large, nevertheless it’s still difficult to conceive of the ideal of large). Leave rectangle as rectangle in the example above and you have my own somewhat simplified version of the Third Man argument.