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Towards the beginning of a course on knowledge (we were reading the beginning of Theaetetus) we discussed definitions and what a definition or rather a good definition is. I'd like to know more about definability. Is there a particular word/concept/notion that has been undefinable? I know that every language must have undefinable/primitive notions/concepts* but aside from within formal systems in logic, set theory, and mathematics have I seen one. Has anyone been able to prove that a particular word is undefinable? If so, how? References are appreciated too, thanks.

*To prove the existence of primitive notions, we notice that a good definition of a word or concept does not use itself. For example, a definition of orange juice as juice that comes from oranges is not a good one (of course orange juice is a juice, but what is a juice, namely orange juice?). Moreover, if we start to exhaustively define every notion in terms of other notions (assuming the class of words or concepts/notions are countable), surely there must be a word (towards the end of our list) in which we cannot define in terms of another notion (maybe because this notion precedes all other notions, i.e. every other notion is defined, possibly indirectly, in terms of this final notion). Defining this (last) concept in terms of another previously defined will make them circular. Therefore, there must exist a primitive notion undefinable in terms of others and unsatisfactorily defined if in terms of itself.

Thanks in advance!!

Edit:

The following comment clarified the sort of circular reasoning I refer to above:

I think this is the problem the OP has in mind: dictionary.reference.com/browse/hill: "a natural elevation of the earth's surface small than that of a mountain." dictionary.reference.come/browse/mountain: "a natural elevation of the earth's surface rising more or less abruptly to a summit, and attaining an altitude greater than that of a hill. - DBK

Here the dictionary defines a hill in terms of a mountain, and mountain in terms of a hill! This type of defining is circular... we understand the definitions because of our experience with hills and mountains. If hills and mountains were completely foreign to us, these definitions wouldn't make much sense (try explaining this to a child without using pictures). This type of phenomenon will must occur if we try to define all concepts exaustively. This shows that there are notions that we must take as primitive or undefinable (understanding a concepts is not the same as defining it here - to see this, we understand the conepts of good, just, knowldge, etc. and can certainly give examples of each, but defining it is much more difficult).

Many concepts have been analysed by Plato through Socrates' dialogues. Some of these include piety, justice, knowledge,... At the end of each dialogue, they conclude with what the concept being analysed isn't! I'm wondering whether any of these concepts (or any other) have been shown to be undefinable.

  • Your question is unclear: I can say this word: "jhgfutrht" is undefinable since it is just a jumble of letters ... – slashmais Mar 28 '12 at 7:33
  • By 'word' I mean a coherent term within a language, specifically English. If you don't like 'word' will replacing it with concept or notion help? For example I briefly heard about Tarski's undefinability of 'truth' but this is within a formal system and does not apply here. I am looking for a word, like knowledge or truth (or any of the words Socrates discusses for example), that is undefinable. That is, can we find a concept such that any definition of it won't capture it's essence or entire meaning? - and a justification or proof of this. – Alborz Yarahmadi Mar 28 '12 at 7:53
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    Is this not better suited in the English usage of stackexchange? I fail to see how this is relevant to philosophy. – Outlier Mar 28 '12 at 19:22
  • I added some qualifications to my answer. – DBK Mar 29 '12 at 1:12
  • Kant's analytic and synthetic distinction is relevant here as well. – stoicfury Mar 29 '12 at 2:29
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I know that every language must have undefinable/primitive notions/concepts* but aside from within formal systems in logic, set theory, and mathematics have I seen one. Has anyone been able to prove that a particular word is undefinable?

You are using the word "language" to encompass very different kinds of languages, i.e. you are unduly mixing together formal languages and natural/constructed languages.

  1. Formal languages are structured deductively and that's is why you have primitive notions.
  2. Natural/constructed languages are not structured deductively and thus do not have (stable) primitive notions. They have more of a network-like structure and the meaning of terms are not set only (or mainly) by intensional definitions, but also (and mainly) by partial extensional definitions, particularly ostensive definitions.

As your question regards the Theaetetus, I take it that you are interested in primitive notions in (2). The answer to your question

Is there a particular word/concept/notion that has been undefinable?

is therefore: no, there is not.

There are a few qualifications to this answer:

  • There are no particular undefinable words unless you come up with a restrictive criterion of meaning which discard certain words of an established language as meaningless.

  • Some philosophers argued that certain concepts are undefinable. The most famous instance is G. E. Moore's claim that the moral term "good" is undefinable. (See his so called open-question argument for more details).

  • Further, you might be interested in the so called paradox of analysis, which is related to your initial worry.


Bonus-answer ;)

Now, if you read the Theaetetus, I'll see that Theaetetus gives partial extensional definitions in response to Socrates question. Socrates in turn tries to delegitimize extensional definitions by implying that they are no definitions at all. Instead, he requests intensional definitions, implying that they are the only (relevant?) type of definitions:

Soc. Do you hear, Theaetetus, what Theodorus says? The philosopher, whom you would not like to disobey, and whose word ought to be a command to a young man, bids me interrogate you. Take courage, then, and nobly say what you think that knowledge is.

Theaet. Well, Socrates, I will answer as you and he bid me; and if make a mistake, you will doubtless correct me.

Soc. We will, if we can.

Theaet. Then, I think that the sciences which I learn from Theodorus-geometry, and those which you just now mentioned-are knowledge; and I would include the art of the cobbler and other craftsmen; these, each and all of, them, are knowledge.

Soc. Too much, Theaetetus, too much; the nobility and liberality of your nature make you give many and diverse things, when I am asking for one simple thing.

Theaet. What do you mean, Socrates?

Soc. Perhaps nothing. I will endeavour, however, to explain what I believe to be my meaning: When you speak of cobbling, you mean the art or science of making shoes?

Theaet. Just so.

Soc. And when you speak of carpentering, you mean the art of making wooden implements?

Theaet. I do.

Soc. In both cases you define the subject matter of each of the two arts?

Theaet. True.

Soc. But that, Theaetetus, was not the point of my question: we wanted to know not the subjects, nor yet the number of the arts or sciences, for we were not going to count them, but we wanted to know the nature of knowledge in the abstract. Am I not right?

(http://classics.mit.edu/Plato/theatu.html)

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There are notions that precede all other notions: It is called axioms: assumptions made without proof. Every 'ontology' will have at its start some notions and assumptions that is unprovable and accepted as true. An attempt to avoid/bypass this is to have a circular definition i.e. a definition that references itself, but this in effect assumes itself to be that axiomatic 'truth'.

[edit]

Maybe this will help: small-number. It is interesting by itself, but also serves as an example of creating concepts or notions that may be undefinable.

  • Yes I agree, in fact I tried to give an informal proof of this after asking my question, intitiated with a star - *. What I am looking for is a particular notion (or more than one) that is undefinable and justification that it is undefinable, if not proof. I appreciate your effort, thank you. – Alborz Yarahmadi Mar 28 '12 at 7:28
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Your questions is fairly contradictory.

If a word is undefinable then why would it even be created as a part of the English language? How can we possibly develop a word of whose definition we are incapable of grasping?

If you mean concepts such as love, the infinite, or god which cannot be shown through material means, then that's a different story.

Or even if you mean things such as 'the' or 'be/are/is' (bringing forth the argument of "How can we define existence), then you must be out of your mind to think that they are undefinable. We know (more or less) what we try to convey by existence: merely our presence in our physical form.

Forgive me for being sidetracked. All I wanted to say is that, no, words cannot be undefinable. Think logically.

My source: Type any valid English word on Dictionary.com.

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    I think this is the problem the OP's has in mind: dictionary.reference.com/browse/hill: "a natural elevation of the earth's surface, smaller than a mountain." dictionary.reference.com/browse/mountain: "a natural elevation of the earth's surface rising more or less abruptly to a summit, and attaining an altitude greater than that of a hill." – DBK Mar 28 '12 at 23:28
  • Care to elaborate please? – Outlier Mar 29 '12 at 0:19
  • @DBK Yes! This is exactly what I'm talking about. This phenomenon must occur when we try to exhaustively define concepts. This shows that there must exist undefinable notions. Many concepts have been analyses by Plato through Socrates. Some of these include piety, justice, knowledge,... At the end of each dialogue, they conclude with what the concept being analysed isn't! I'm wondering whether any of these concepts (or any other) have been shown to be undefinable. DBK, I am going to include your comment in editing my question, thanks. – Alborz Yarahmadi Mar 29 '12 at 0:26
  • It's an example of an actual (narrow) circular definition in a dictionary. See WP Circular definition. @Alborz However, taking this lexicographic entries as examples won't do. Please see my answer for why I think that your intution that circular definitions imply undefinable terms is mistaken. – DBK Mar 29 '12 at 0:35

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