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According to formal sementics propositions (semantic term for "sentences", "clauses") have truth value. The truth value shows whether sentence is true or false and it is denoted as 1 or 0. What about sentences that have failed in presupposition? Do they have truth value? Ex: The king of USA lives in NY. Here, the interlocutor presupposes that the USA is ruled by the king, and s/he fails in his/her presupposition. So what be the truth value of this proposition? I think that it is impossible for proposition that has failed in presupposition to have a truth value but I am not sure.

Note: here I am talking about the direct meaning of the word "king", not possible metaphoric.

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First, proposition is a term for Statement (i.e, not a question, order, promise, etc.), not for sentence or clause. Proposition is a logical term; sentence and clause are syntactic terms. See the logic guide for terminology. –  jlawler Feb 19 '13 at 21:35
Second, there is no single solution for the problem you ask about, which goes back at least to Russell's famous The present King of France is bald. Russell's solution was to say they're all F. Another is to posit a ternary logic (usually T, F, and # are the symbols, though numeric ones like -1, 0, 1 or 0, ½, 1 also occur), with a third value for failed presuppositions. Ternary truth tables can get complex, and there are several different kinds of ternary extensions. –  jlawler Feb 19 '13 at 21:41
The usual answer is, "it depends on what you mean". If I read the conversational implicature of the sentence as "the USA have a king and he lives in NY", then it would be false. This reading makes the most sense in examples such as yours. Alternatively, you could read it as "if there is a king of the USA, he lives in NY". Logically, this is true, because we evaluate an implication whose condition is unfulfilled as true. But this interpretation most people find unnatural. I would stick with the conversational implicature; language is not logic—it is far more comprehensive and complex. –  Cerberus Feb 19 '13 at 21:48
Thank you for you comments. –  Dariya Feb 19 '13 at 21:58
@jlawler Why not post your comment as an answer? –  jyc23 Feb 20 '13 at 14:26
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