Necessity and Truth
It is generally assumed that necessity implies truth: from “Necessarily p” we can infer “It is true that p”. In possible worlds terms, if a sentence is true in every world, it is true in the actual world. This is taken to be self-evident and to my knowledge has never been questioned. We can certainly never infer falsehood from necessity! But have we considered the full range of cases? Take the sentence “Necessarily the king of France is a monarch”: is that sentence true or false or neither true nor false? It is a false sentence according to Russell’s theory of descriptions and neither true nor false according to Strawson’s theory. But it seems to me to be true; it might be said in reply to someone who claims that the king of France could fail to be a monarch. At least it has a reading on which it is true (“Being king of France logically implies being a monarch”). Yet the sentence “The king of France is a monarch” might not be true: it might be false or neither true nor false, depending on which theorist you agree with. Asserting that sentence either entails or presupposes that there is a king of France, but asserting the necessitation of it does not entail or presuppose that (on one reading). Intuitively, the necessitation says merely that if any object answers to the description “the king of France” it must also fall under the predicate “is a monarch”. The necessity operator cancels the implication of existence carried by the embedded sentence (the same might be said of the belief operator). So the modal sentence can be true without the embedded sentence being true.
Or consider ethical sentences, such as “Genocide is wrong”. Can we not assert, “Necessarily genocide is wrong” as a way to emphasize the universality of the moral proscription, while maintaining that ethical sentences are not strictly true or false? It may or may not be plausible to maintain that ethical sentences lack truth-value, but surely it is consistent to hold that they suffer that lack while being ready to assert the necessitation of an ethical sentence. To say that the wrongness of genocide couldn’t fail to be so in any world is not to commit oneself to the meta-ethical position that ethical sentences have truth-value. There is a logical gap here. If we take the modal sentence to mean something like, “It is undeniable that genocide is wrong”, that could be true without it following that the sentence that is undeniable is itself true—it might just be morally unquestionable. An emotivist might be happy with assigning the modal sentence a truth-value but deny that the non-modal ethical sentence is thereby a bearer of truth. There may be no circumstances in which someone can legitimately assert that genocide is right and yet “Genocide is wrong” might not be a true sentence.
If we choose indisputably true sentences to put after “necessarily” we will generate the impression that necessity implies truth, but if we choose other types of sentence that appears not to be the case. We cannot therefore affirm that truth follows from necessity across the board; we need some restrictions if we want to maintain the logical truth of “If necessarily p, then it is true that p”, to the effect that p has to be a truth-bearer. The mere fact of necessity does not entail the truth of what is necessary. 
 We can say that Sherlock Holmes is necessarily a man, but can we say that it is true that he is a man? Fictional sentences can express necessities without expressing truths. We might hold that mathematical sentences have no truth-value while also holding that they express necessities. The concept of necessity does not itself imply truth. We could dispense with the whole idea of truth but retain the concept of necessity.