If we set aside the rookie error of thinking that 'self-evident' means 'automatically evident to everybody', we often find that we are in the position of having to argue for self-evident principles. It is impossible to give a demonstrative argument for a self-evident principle, because such principles are immediate and thus there is no middle term you could use for a demonstration. But demonstrative argument is far from being the only kind of argument, and you can in fact argue for self-evident principles in some of these other ways. Which raises the question: What kinds of legitimate argument are there for self-evident principles? Here's one possible, although very rough, classification.
ABSTRACT
Abstract arguments for self-evident principles address the principle itself and are either theoretical or practical. Each has a major kind that is positive (trying to show the necessity or reasonableness of holding the principle) or negative (trying to show the impossibility or unreasonableness of rejecting the principle).
(1) Oscillation (Theoretical Positive): One can argue for a self-evident principle on the ground of other evident principles. For instance, one could argue that the principle of noncontradiction is implied logically by some other logical principle or principles. Now, this would certainly fail as a demonstration, because it would be circular, but it has been noted (by Schlegel, if I'm remembering correctly) that this circular reasoning is not automatically wrong, and this has often been recognized. (Aristotle, for instance, explicitly notes that the problem with begging the question that makes it a fallacy is that it treats a non-evident conclusion as if it were a self-evident principle.) What this argument does is to show that a principle has a central role in a coherent web of principles.
(2) Retorsion (Theoretical Negative): This form of argument has been a standard one since Aristotle used it to argue for the principle of noncontradiction: argue that rejecting the principle is self-defeating: the rejection requires the truth of what it denies.
(3) Transcendental (Practical Positive): In a transcendental argument you show that it is inevitable that you will be using a principle anyway, and therefore that you have a practical right to use it. Although this kind of argument is most closely associated with Kant, he should really be seen as trying to build really rigorous and skepticism-resistant versions of a broader family of arguments that is quite common.
(4) Ad Hominem (Practical Negative): 'Ad hominem' is often used to name a fallacy, but as I have noted before, this is not its only use. Locke uses it to indicate a form of argument in which you appeal to your interlocutor's own concessions (either in argument or in action). Thus, you are showing that the objector has actually already granted the principle. Aristotle also uses this to argue for the principle of noncontradiction: the person who is contradicting you about the necessity of noncontradiction is already assuming the principle of noncontradiction in order to contradict you.
CONCRETE
Concrete arguments take a more indirect approach. There are at least two major kinds: casuistic, which focus on particular applications of the principle in question, and testimonial, which appeal to the fact that others directly or indirectly recognize it. Each of these has at least two common versions; for the casuistic version, one is theoretical, one is practical. The testimonial approach doesn't categorize easily; for convenience here, I will treat one approach as positive and one as negative.
(1) Example (Casuistic Theoretical): The example approach can be thought of as taking its inspiration from geometrical diagrams -- by looking at how a given principle works in a given principle, and taking into account that what applies to one example applies to any sufficiently like it, one can make it more obvious that a principle is reasonable to accept because it is true in cases that allow you to generalize widely.
(2) Utility (Casuistic Practical): Principles are things we actually use, and so it's possible to show that use of a principle gives you good and desirable results -- e.g., allows you to build bridges, avoid misleading arguments, simplify complex theories, and so forth. This may be direct, or it may be indirect. In either case, it aims to show that acceptance of the principle is practically reasonable, as seen in actual cases.
(3) Consensus Gentium (Testimonial Positive): Once we set aside the rookie mistake of thinking that consensus gentium requires absolute universality (I, II), we can see how it it would be used for arguing in this context: one appeals to the intellects of others, either in their verbal testimony or the witness of their behavior. The point of the argument is to establish that human beings generally are already treating the principle as self-evident, thus indicating that accepting it is a normal part of human functioning and thus is reasonable. This approach is used extensively by Thomas Reid.
(4) Consensus Sapientium (Testimonial Negative): As we already noted, 'self-evident' doesn't mean 'automatically obvious to everyone'; self-evident principles are evident when understood, but some self-evident principles are really hard to understand. As Aquinas often says, some enunciations are evident only to the wise. Thus sometimes you need to appeal to 'the wise', the people whom we have some reason to think actually know what they are talking about. Locke calls this kind of argument 'ad verecundiam', the appeal to shame or bashfulness; the point of the title being, I take it, that human beings are hesitant to contradict recognized authorities because doing so recklessly is itself evidence that you are a fool. It is in this sense that I think the appeal can be seen as a negative form of argument: it works by showing that there is reason to think only a fool would reject the principle.
