Understanding a sentence

If you don't like centered propositions, drop the "centered" from the following. I am using the phrase "knowledgeably understand" to boost understanding to a level that requires the kinds of justification that knowledge does. Perhaps understanding already has that built-in, in which case "knowledgeably" can be dropped.

Now, consider the following inconsistent triad, each proposition of which is defensible:

1. To knowledgeably understand a sentence it suffices to know the language and to apply appropriate symbol recognition, symbol manipulation and logical skills to that sentence.
2. Necessarily, someone who knowledgeably understands a sentence knows what (centered) proposition that sentence expresses or else knows that the sentence does not express a (centered) proposition.
3. There are sentences s such that one cannot know whether s expresses a proposition simply by knowing the language, and by applying appropriate symbol recognition, symbol manipulation and logical skills to that sentence.

I think (1) and (2) are quite intuitively plausible, but (3) needs an argument. Here is a standard argument (Kripke came up with cases like this). I erase my board and write on it "No sentence on Jon's board expresses a true (centered) proposition." Let s be this sentence. Then I cannot know whether s express a (centered) proposition unless I know what Jon has on his board. For Jon, being a philosopher, might easily have written on his board "Every sentence on Alex's board expresses a true (centered) proposition." But if that is what is on his board, then the sentence on my board cannot express a (centered) proposition. (If it expresses a true (centered) proposition p, then plainly p is true if and only if p is not true.) But I cannot know what Jon has on his board simply by knowing the language and applying symbolic and logical skills to the sentence on my board. Hence, (3) is true.

Given the above really good argument for (3), we need to reject (1) or (2). I am inclined to reject (1), as (2) seems very, very plausible. Or, perhaps better yet, we might reject the notion of sentences that the paradox is predicated on.

Actually, everybody should reject (1) in the case of natural languages, simply because of the problems of homonymy, and that's not very interesting. But the argument against (1) (assuming the notion of sentences that the paradox is based on) continues to work even if we distinguish homonyms with subscripts, and similarly deal with other "standard" contextual ambiguities.