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I'm working on a paper concerning the third man argument.

I have my own solution in mind, but I need other arguments to refute so that I can then move onto my argument. I found another thread on this topic, but there were not many answers because the person admitted that he or she was asking to help another person to cheat. The one answer provided Gregory Vlastos, The Third Man Argument in the Parmenides (1954); P.T. Geach, The Third Man Again (1956); and S. Marc Cohen, The Logic of the Third Man (1971).

If the arguments provided are very confusing I would a appreciate a brief explanation, but that is not required as having the information is my top priority.

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I have ways wondered why infinite regression would not be allowed. In series calculus you can argue that given an infinite number of steps a variable will approach a certain value.

Given "beauty" as an example, the third man argument falls apart, as Aristotle is arguing in our favor. If a beautiful thing partakes of the form of beauty there must be a form of beauty even higher than this, ad infinitum. If infinite regression is allowed, then Aristotle is giving us a formula to understand the "ultimate" form of beauty; as we can understand this as the most essential form of beauty with all particulars removed.

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    FWIW in calculus the argument is that after a finite number of steps a variable will get as close as we like to a particular value. That's the revolutionary insight of calculus: to banish vague thinking about infinitely many steps, and to replace it with clear thinking about arbitrary closeness after finitely many steps.
    – user4894
    May 30, 2022 at 4:27
  • Interesting point here, an infinite regress isn’t inherently bad, so long as it converges onto a determinate value (in calculus one may integrate over an infinite interval, so long as the final answer is determinate and finite, such as f(x)=1/x^2 from x=1 to x=infinity: the area under the curve is exactly 1 in this example, not infinite).
    – Hokon
    Oct 23, 2023 at 20:44
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Cogito ....

Instances: {I1, I2}

We extract the form {F1}.

Now we have a new set {I1, I2, F1}, F1 is the 3rd man

But {I1, I2} = F1

So {I1, I2, F1} = {F1, F1}

But {F1, F1} = {F1} since F1 = F1

There is no 3rd man.

😁

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