Among true propositions, some are true independently of experience, and remain true however experience varies: these are the a priori truths. Others owe their truth to experience and might have been false had experience been different: these are a posteriori truths ... Kant argued that a priori truths are of two kinds which he called 'analytic' and 'synthetic' [A 6-10). An analytic truth is one like "All bachelors are unmarried" whose truth is guaranteed by the meaning, and discovered by the analysis, of the terms used to express it.That'll do to start with (there's also this, from Wikipedia: 'Galen Strawson wrote that an a priori argument is one in which "you can see that it is true just lying on your couch. You don't have to get up off your couch and go outside and examine the way things are in the physical world. You don't have to do any science."; a posteriori knowledge or justification is dependent on experience or empirical evidence (for example "Some bachelors are very happy"). A posteriori justification makes reference to experience; but the issue concerns how one knows the proposition or claim in question—what justifies or grounds one's belief in it.'
So far, so good, and so elementary. On what grounds, though, does Kant argue that a priori truths are true? Since one shortcut answer might be something like 'because a priori truths are tautological' I'm going to bracket a small section of a priori truths and call them 'tautological a priori truths'. These are statements of the 'A = A' sort, which are hard to argue against. But -- and this is my point -- it seems to me that any a priori truth that departs in any way from the simple tautology is possible to argue against, and since refuting such arguments always entails bringing in 'experience' (the very thing a priori truths are supposed to circumvent) I find myself wondering: are there any a priori truths at all? Is there any such thing?
Start with "All bachelors are unmarried". You say that to me. In reply I say: 'ah but Phil is a bachelor, but only because in a very real sense he's married to his job. Ergo he's a bachelor who is married.' You will then be compelled to point out that this is not the sense in which you meant 'married' in your original statement. But this can hardly help but tumble into a No True Scotsman fallacy. (As it might be: 'no, being "married" to your job is not actually being married. I meant: as in holding a marriage certificate.' And I reply: 'my friend Bob holds no such certificate, yet considers himself married to his civil partner Jim. Isn't he a married bachelor?' 'No,' you say. 'You're maliciously trying to pick holes in my initial statement.' To which I say: 'yes, yes I am.') Let's say you shift your ground and say, 'a better example would be: 5 plus 5 equals 10.' If I refute you by saying 'not in base 9 it doesn't' you are then faced with an infinite trail of supererogatory qualifications narrowing down the precise terms of your statement, any one of which can be refuted by a suitably ingenious and stubborn interlocutor. This last bit is the crucial thing. A clever enough antagonist will find ways in which the statement might not true. To avoid being driven back into the narrowest Tautolgical form of the a priori truth, you will be compelled to say something like: 'but you're not arguing in good faith! You know perfectly well in what sense I used the term "bachelor".' This is probably true, but it is not true a a prori -- it is, in fact, only true if we import experience.
This, then, is my claim: not that a priori truths are necessarily untrue, but that it would be possible for a suitable inventive individual to think of ways in which they might be untrue. It's not that I necessarily think any given example of an a priori truth is untrue; that's not my argument, and doesn't need to be. It's that I'm suggesting a suitably ingenious individual can find grounds for falsifying any a priori 'truth'. Or to put it another way: a priori statements depend upon 'good faith' at some level, and good faith is not a priori.