More About Analyticity
Here comes the promised reply to Sam's reply to my previous posting. In that posting, I first suggested that some sentence S (in a given language) is analytic iff you can't understand it unless you believe it. Then I said that, "put slightly differently", S is analytic iff it is impossible to believe that not-S.
As Sam notes, the first definition implies that even very complicated analytic truths have to be believed in order to be understood, which might be somewhat unintuitive. I'm not sure how bad this is for lack of a clear example. Sam uses "the sum of the digits of the first prime number greater than 1 million is even", but this is not analytic, so here I can perfectly well admit that you may understand it without either believing or disbelieving it. He also mentions infinitely long sentences, but I don't believe there are any of those in ordinary languages.
At any rate, if this is a problem, I would have to weaken my first definition, making it equivalent to the second: S is analytic iff it is impossible to understand S and yet believe that not-S.
There is another problem with very complicated (not necessarily analytic) truths: If a sentence is so complicated that it just can't be understood at all, it will come out both analytically true and false on my definitions. I'm not sure if, given suitably broad possibilist quantifiers, there are any such sentences. For some reasons I think it's more likely that some sentences are in fact too complicated to be believed or disbelieved even by any possible being, so this would be trouble mainly for the second definition.
The problem I see with extremely trivial synthetic truths is not that some of these are understandable but not answerable. Even if it were impossible to know the weight of Sam's pen, it would still not be absurd to say of somebody that he believes that the pen weighs, say, 120 grams, nor that he believes that it doesn't. That these beliefs would be unjustified doesn't matter. (Not having to get into justification is one of the reasons why I prefer to talk about analyticity rather than apriority.)
The problem I had in mind was that there may be synthetic sentences that are so trivial that any competent speaker of our language must believe them. "There are people" or "some things are parts of other things" could be examples. But in both cases there are philosophers who deny them, so again I lack a clear example.
On second thought, I'm a bit worried that my explication of analyticity is mainly in terms of the possibility or impossibility of a corresponding belief. What worries me is that belief ascription is such a messy business, and not very directly related to sentences and their meaning. But I'm afraid this is as close as one can get if one wants to define analyticity independent of semantics. Given a semantic theory, it is possible to say much more directly that, for example, S is analytic iff it has a necessary 1-intension.