In the Eudemian Ethics, Aristotle has an interesting passage on people engaging in fake philosophy:
...because to say nothing at random but use reasoned argument seems to mark a philosopher, some people often without being detected advance arguments that are not germane to the subject under treatment and that have nothing in them (and they do this sometimes through ignorance and sometimes from charlatanry), which bring it about that even men of experience and practical capacity are taken in by these people, who neither possess nor are capable of constructive or practical thought. And this befalls them owing to lack of education—for in respect of each subject inability to distinguish arguments germane to the subject from those foreign to it is lack of education. (1217a)
The word translated here as 'lack of education' is ἀπαιδευσία; one can indeed translate it as 'lack of education' or 'lack of training', but this often makes it sound as if it were just a claim that those who have it are ignorant, when in reality something deeper is certainly meant. Paideia was the cultivation of those qualities that made one suitable for civilized life. To lack paideia is not mere ignorance, but the kind of ignorance we might call barbarism. The person Aristotle is criticizing in the above passage is someone who has learned the philosophical knack for argument-giving, but uses it in what we might call a cargo-cult way; they have no real sense of good and bad argument, or of relevance in argument, because this is something that can only be had by cultivation -- and, indeed, can only be had fully by cultivation that involves participation in a community. They do not put forward arguments as participants in a community of inquirers; they put them forward because they are ignorant of how such a community works or because they are trying to ape the effects of being in such a community, without the actual work required to be so. Their imitation may be very good; but they are imitating the outward appearance, and not the internal character, of real rational argument.
Aristotle uses the term elsewhere. It shows up in the Rhetoric (1391a), where Aristotle talks about nouveaux riches, and how there vices with regard to money are often greater than those of old money because they are apaideutic with respect to wealth. I think this is arguably more than just a happenstance of using the same word; new-money people in Aristotle's example haven't learned to restrain themselves so as to be respectable in society, whereas the old-money aristocrats have. Likewise, those doing the imitation philosophy, having come into the wealth of reasoned argument, have not cultivated the habits of restraint that make one part of the community of inquirers.
The most important use of the term, however, is in the Metaphysics. Aristotle notes that order requires paideia (1005), and then goes on to give his most famous example of the kind of person who exhibits apaideusia: the person who demands that one prove the principle of noncontradiction:
But we have just assumed that it is impossible at once to be and not to be, and by this means we have proved that this is the most certain of all principles.Some, indeed, demand to have the law proved, but this is because they lack education; for it shows lack of education not to know of what we should require proof, and of what we should not. For it is quite impossible that everything should have a proof; the process would go on to infinity, so that even so there would be no proof....And I say that proof by refutation differs from simple proof in that he who attempts to prove might seem to beg the fundamental question, whereas if the discussion is provoked thus by someone else, refutation and not proof will result.The starting-point for all such discussions is not the claim that he should state that something is or is not so (because this might be supposed to be a begging of the question), but that he should say something significant both to himself and to another (this is essential if any argument is to follow; for otherwise such a person cannot reason either with himself or with another);and if this is granted, demonstration will be possible, for there will be something already defined. (1006a)
The person who demands that the principle of noncontradiction be proven does not understand how proof actually works. If he says nothing in support of his claim, Aristotle tells us that it's absurd to argue against him, because for all that he's actually contributed to reasoned discussion, you might as well be arguing with a plant; on the other hand, if he does say something in support of the claim, then, as quoted above, one can give a proof by refutation based on the fact that he says "something significant both to himself and to another", without which no one can reason. I think both prongs here flow directly from Aristotle's diagnosis of such people as apaideutic: they are either not reasoning (and thus might as well be a vegetable) or, in reasoning, they are not really familiar with what reasoning requires -- the importance of things like meaningfulness and relevance and communicability for it, and the things that follow from these. They have not developed the habits required to participate in the community of reasoners. They are barbarians in the land of reasoning.
There are also a number of passages in which he does not use the word apaideusia, but does talk about paideia in ways that are clearly relevant -- such as the famous passage in the Nicomachean Ethics (1094b) in which he says that it belongs to the trained to recognize the right amount of exactness to seek in inquiring into different topics. And he goes on almost immediately with an illuminating explanation:
To criticize a particular subject, therefore, a man must have been trained in that subject: to be a good critic generally, he must have had an all-round education. Hence the young are not fit to be students of Political Science. For they have no experience of life and conduct, and it is these that supply the premises and subject matter of this branch of philosophy. And moreover they are led by their feelings; so that they will study the subject to no purpose or advantage, since the end of this science is not knowledge but action. And it makes no difference whether they are young in years or immature in character: the defect is not a question of time, it is because their life and its various aims are guided by feeling; for to such persons their knowledge is of no use, any more than it is to persons of defective self-restraint. But Moral Science may be of great value to those who guide their desires and actions by principle. (1095a)
This passage puts together the bits we've already seen. To make good judgments in a particular field requires paideia in that field, to make good judgments generally requires good general training. You must, in particular, be familiar with the materials that "supply the premises and subject matter" and must have a sort of discipline, a self-restraint, that makes study advantageous by letting it be guided by principle rather than the passions.
Various Links of Interest
* Kenneth Pearce, George Berkeley and the power of words, at the OUP blog
* Miriam Burstein, Mill's Inaugural Address and the Contemporary University (or Not), at "The Little Professor"
* Richard Marshall interviews Michail Peramatzis about the notions of dependence and priority in Aristotelian metaphysics
* Geoffrey K. Pullum, Fear and Loathing of the English Passive, discusses the many wrong and misleading things said about passive voice in English
* A. C. Thompson, Sikhs in America
* Pius XI, Mit brennender Sorge
Murasaki Shikibu, The Tale of Genji
Martha Nussbaum, The Fragility of Goodness
Gaven Kerr, Aquinas's Way to God
Philip Jenkins, The Lost History of Christianity
George R. R. Martin, A Feast for Crows
George R. R. Martin, A Dance with Dragons