|1.||The population of New York is greater than 1 million|
|2.||Either the population of New York is greater than 1 million or it is less than 1,000,001.|
|3.||2 + 2 = 4|
|4.||p (q p)|
Proposition 1 is true in some possible worlds and false in others. It need not necessarily be true and hence it is not logically necessary and we say it is contingent.
Proposition 2 would probably be thought meaningless if New York did not exist, and so it might not be true. But it cannot be false. We may therefore prefer not to say it is contingent, and we may be willling to consider it necessary.
It is difficult to conceive any possibility that proposition 3 is false except that the sentence we use to express the proposition means something other than we normally mean by "2+2=4". It is this possibility that talk of propositions is intended to exclude, since, if the meaning of sentence were changed it would then express a different proposition. If changes in meaning are not ruled out, in this or some other way, then no sentence will be necessarily true. Any sentence will express a false proposition in some possible worlds.
Assuming then, that we are considering the truth in all possible worlds of the proposition expressed by the sentence in this world, then it is difficult to see how "2+2=4" can fail to be true. Hence we may consider it necessary.
Since the term "necessary" can be used to mean something like "entailed by the known laws of physics" the term logically necessary may be used to make clear the sense of necessity under consideration. If there could be a world in which the natural numbers did not exist then the statement "2+2=4" would be in such a world, meaningless. However, the natural numbers are not of this world or of any other world. We could therefore take the position that the existence or otherwise of the natural numbers is not something which is determined by the possible world under consideration, and that there is no world in which the proposition "2+2=4" fails to be true.
Proposition 4, where the symbol "" is intended as material implication and "p" and "q" are arbitrary propositions, is true in all possible worlds without our needing to make special pleas about the non-contingency of abstract objects.