There is a long debate in philosophy if the Third Man Argument (=TMA) is a refutation of the theory of Ideas of Plato. The name of the argument comes from Aristotle, who used man as an example. It goes like this:
How do you recognize a man as a man? What Plato says is that there is an Idea of a man which can be used to recognize a man. That Idea can be used to identify another man as well. It is therefor the same and not the same as any man. And now comes the TMA question: how can you bridge the gap between partly different and partly the same? By making comparisons of course and show that the essential characteristics are the same. Aha, but at that point where the concrete man and the Idea of man come together have we actually created a third man. That third man is the set of characteristics of the concrete man by which it can be proven that the concrete man and the Idea of the man share enough with each other to say they are the same. (The same is not saying that they are equal to each other obviously.)
Well, if we have to create a third man to bridge the gap between a concrete man and the Idea, and this third man differs from both the concrete man and the Idea, how can we bridge the gap between this third man and the concrete man and how can we bridge the gap between the third man and the Idea of man? Showing that the concrete man and the third man, although different from each other are essentially the same? Showing that the Idea of man and the third man are essentially the same? But fort that do we have to create new men. And to bridge their respective gaps do we need new men again. This is an infinite regression and is the method how the TMA undermines each comparison.
Let's focus on the nature of the gap, because that is the real problem addressed by the argument. The gap is the problem that two different entities are said to be equivalent to each other, although they are defined to be different. The Idea of Man is different than any man or woman on earth, yet all men and women can be identified using that one and only same Idea of Man. What makes the Idea of Man better to identify concrete men and women than for instance the Ideas of Cars or Monkeys? Not the assertion that it is obvious that they are men, because that is exactly the case to be proven. It is equally impossible to use any concrete man or woman as an example, because individual cases can not be used as a proof of any general concept.
The Idea of Man can therefor be not an individual case. If it was an individual case, then could it not be used as a prove that a certain man or woman is a human being. When it is not individual, yet there exists one exactly, then must it be an abstract concept. The gap is by the TMA supposed to be unbridgeable from the abstract Idea to the concrete object and vice versa. Yet, that is what we do all day long. We bridge a gap, that is supposed to be unbridgeable from both sides of the comparison.
The nature of the gap is in the comparison and therefor in the alignment of two different objects. In any comparison are three characteristics present:
The nature of the gap must be found in those three characteristics, because they are needed to let an comparison make sense. The first characteristic is essential for the TMA to exist. There is no problem comparing two equal objects with each other. That is happening in mathematics all the time. The result of the addition 3 + 4 is 7. Therefor can we say that 3 + 4 equals 7. It is not the same as 7, because that can also be the result of 1 + 6 or 70 / 10. There is no gap possible here, hence no TMA. Two different objects means that it is without any doubt that the objects are not the same nor equal to each other yet it is possible that they are the same.
The second one might need some more explanation. Take a look at the sentences:
(1) 'the man is called Peter',
(2) 'Peter is the man',
(3) 'that is me' and
(4) 'I am that'.
In the first sentence can you perceive the ordering from concrete to abstract. The man is a particular man and it can be true or not that he is called Peter. Peter is more abstract than 'the man', because a lot of men can be called Peter, but there is only one man 'the man'. The second sentence makes therefor no sense as a comparison. The ordering is now from abstract to concrete and that is not a valid scheme. It still can be a good sentence, but it has then a slightly different meaning and is not a comparison anymore, but a sentence that is supposed to be true.
In the third and fourth sentence are two abstract objects compared to each other. Both sentences are correct comparisons. The ordering is now performed in time. First is there an object identified about which something will be said. Then is the comparison made and can it be said to be true or not. The first term is made concrete before the second one is introduced.
Please take a look at the next two sentences thinking about the third characteristic that different objects can be equal in a specific context: (5) 'Rik Smits is a big man',
(6) 'the president of the United States is a big man.'
In both sentences is the word big used, in both sentences has big another meaning. Which big is used depends on the context and that context is apparently delivered by the concrete object, because the context changes due to the concrete object. Being a big man is the abstract object and the perspective how one is asked to look at the concrete object.
It must be a valid way of comparison, which is another way of saying that the abstract object can be a feature of the concrete object. Not all perspectives of an object are valid. They can be false by nature and should then not be used in those sentences. It is allowed by language, but not by context. Now can we see how those three characteristics interact with each other:
There is no gap possible due to one of the objects. Although a lot could be said about problems with identifying objects, that is not important considering the TMA. The TMA is about the comparison between two different and correctly identified objects. (Yes, we could have a whole tree of arguments about the statement 'correctly identified objects', but that is out of scope of the perspective of this blog. Here is it an assumption supposed to be true.) The context of the sentence is based upon the most concrete object. That is shown in the sentences about Rik Smits and the president of the United States. The meaning of big changes immediately due to the context presented by the subject of the sentence, being the concrete object. The concrete object is always the subject of the sentence and therefor the provider of the context. The context is therefor connected with the first and correctly identified object and will therefor not be the cause of the gap noted in TMA.
It is also not the verb 'to be', that is connecting both objects. If that would cause the gap, then would a sentence like '1 is 1' also have that problem. Those type of sentences are not influenced by the TMA, hence is the verb 'to be' not the culprit. Then is there only one possibility left for the gap to arise: it is the validity of comparison. Can it be proven that the comparison is valid? Take a look at the next two statements:
(7) the comparison is valid, and
(8) the comparison is wild.
The first comparison will be read without any notice. The concept of validness belongs to the context of comparison. But wild? Yes, it can, when wild has the meaning of farfetched or unexpected. But wild in the meaning of a wild animal or a wild dance? No, that is not possible. Comparisons are made with knowledge coming from the context. As shown in sentences 5 to 8 is the context provided for the comparison done automatically. It is required to understand any sentence, not just sentences with comparisons that could be influenced by the TMA. It is therefor clear that the gap exists not within the context itself. In other words: the gap of the TMA is not in the sentence.
The context is provided by the persons that are writing down or reading the sentence. Writers create texts with a meaning. Words are put in any order on purpose. Writers create the context and expect their readers to share more or less the same context. So, lets return to Aristotle, who introduced the name of the argument and review what he was doing. Aristotle asked himself how it is possible that a comparison can be made between an Idea of Man and a concrete man. To make the connection, there should be a third man in between the Idea and the concrete man. That is where the name comes from.
But is his line of argumentation valid when we now know that the gap comes from the context of the writer or reader and not from the comparison itself? Words can only be understood when their meaning (=context) is known. A sentence in a language that one can not understand for instance is meaningless in the eye of the reader. When the subject and language is known, one can read and understand the words of a sentence. It is possible to put all kind of question marks to the assertion that words can be understood, but that is not relevant for the discussion of the TMA. The TMA can only be a problem in situations in which words can be understood clearly. How can you otherwise measure that there is an unbridgeable gap?
Aristotle asserted that the third man was required to bridge the gap. How did he do that? We know that the context changes immediately when the concrete object is changed. We know immediately when comparisons are valid or not. How could he puzzle his readers introducing a third man, that no one has seen or thought of before? If you take a good look at the process of creation of the third man, then can you read that Aristotle is adding new steps to the comparison. He is freezing the results of the concrete object, next freezing the results of the abstract object and then is he comparing the two newly created results with one another. That implies that he is isolating both objects from each other. He is pretending that two independent objects are compared with each other, each having its own context.
But that is not what is happening when a comparison is made. There is only one object that is providing the context and that is the concrete object, i.e. the subject of the sentence. The context of a concrete object is a set of associations. For the word 'man' for instance have we built those association schemes while learning the language. It is not like when we read a sentence that we see the concepts for the very first time. The first time we are able to read a sentence, or to hear and comprehend it, we have already a lot of associations connected with each word of that sentence. Making a comparison is in fact checking if the predicative belongs to the set of associations we have stored with the subject while interpreting the sentence. If it belongs to it, then might we conclude that the comparison results in a true. If it does not, then will we conclude that it is false, or we might experience a problem in understanding the sentence. Every comparison is made one way, it is not reciprocal. Aristotle pretended it a comparison is reciprocal, like in a math equation, but it is not. The TMA does not exist in the comparison, it exists solely in the eye of the beholder.
Aristotle was the first to describe and categorize fallacies. He described 13 different fallacies. The TMA is the 14th. The process of the TMA is to first separating the two objects to be compared and then trying to equalize them. That is a sound form of reasoning in mathematics. It looks therefor rational thinking, but alas, it ignores the essential difference between formal languages and natural languages. The argumentation used is too formal for natural languages, because it overlooks the fact that comparisons go only from the subject to the predicate. Equalization in mathematics is between two subjects. There is no predicate. That is why equalization is about the check if two things are really equal and comparing is about the check if things can be said to be the same, despite they are inherently different. The TMA is a fallacy, not because it is an unsound method of reasoning, but because the method of reasoning is too strict for normal languages. It requires that the two words could be the same, yet two words being equal does not exist within normal languages. Even synonyms are most of the time not exactly the same, let alone homonyms :-).