This is a complicated issue, and one that is still not fully understood by linguists, or so I believe. In short: there is a tendency in many languages for words to shift in meaning between probability and desirability. This tendency is apparently strongest in certain verbs that are used without specifying who the judge is, which includes passive verb forms.
I can make a simple statement that describes a fact:
Athens is more civilised than Rome.
I can modify this to describe my attitude towards the statement. This can be done in two ways: I can give my opinion on its truth value (probability), and on whether or not I want it to be so (desirability).
Athens is probably more civilised
than Rome. (weaker probability than
when stated as a simple fact)
I want Athens to be more civilised than Rome. (I have added my opinion
on the desirability of this
statement's being true)
The desirability and probability of statements as expressed by the speaker is collectively called modality. Modal verbs generally express some form of modality (surprise!):
She goes. (no modal value, except the "default" probability value of "fact, 100 % probable")
She will go. (probability, very
She might go. (weak probability)
She should go. (medium desirability)
She has to go. (strong desirability)
The problem is that probability and desirability are not always separate:
She may go.
Does this mean that it is possible that she goes? Or is she permitted to go? Permission is a weak kind of desirability: "I do not not want her to go, so she may go if she likes". Both kinds of modality are sometimes mixed up. Historically it can be seen that most if not all modal verbs have changed in their modal functions. Will once expressed only that someone desired something:
I will go.
This used to mean "I want to go"; but it has gradually acquired a sense of probability, in that it now means "it is very likely that I go" for most speakers of English, especially Americans. The other modal verbs have gone through and are still going through various other shifts.
The same can be seen in words that express some form of modality other than modal verbs. Consider the word impossible:
Now I'd like to go, Sir. — I'm afraid
that is impossible. You must finish
your work first.
The teacher says that it is impossible for the pupil to leave, because he does not want him to leave, and his will is an unchangeable fact as far as the pupil is concerned. But this amounts to the same as saying "I want you to stay".
Command and permission have to do with desirability: there is the thing that you desire but have no power over (plain desire), and the thing you desire and have some power over (command, permission). If you command something, you have a fairly strong desire and strong power; if you permit something, you need somewhat less power, because it will happen anyway if you do nothing, and you have a weak desire: you basically say, "you may go if you like; I don't care enough to forbid you".
The reason why command and permission are so well represented in statements of desirability is probably that it is little use stating your desires unless you are empowered to affect them to some degree, be it by commanding or begging. (Plain statements of desire do exist, though, of course.)
Now on to be supposed to.
The Carthaginian fleet was supposed to
comprise two hundred ships; Scipio had
hoped that there should be fewer, but
he had to organise his fleet based on
Here supposed to is used to indicate probability: the Romans considered two hundred a probable number.
The new ships were supposed to
dominate the Mediterranean. But Rome's fleet failed her on several occasions.
Rome thought her fleet would be dominant; and she had constructed it because she wanted it to be dominant.
From here it is clear how the sense of desirability came to dominate to be supposed to. Somehow this has not affected the past participle without to be as much, and the active verb even less so.
This shift in modality can be seen in other verbs as well:
The same type of shift can be observed in all other languages I know, in all ages. In Ancient Greek, dokeô means "to seem, appear" (cf. our words paradox and deixis); dokei moi can mean "it appears to me, that ...", but it can also mean "it seems like a good idea to ...", and hence even "I decide to ...". The personal pronoun can also be left out.
I suspect that this shift is most likely to rear its head where a verb has no agent, no person who actually does the supposing or expecting or meaning; that is why it appears first or most strongly in passive constructions. I am not sure why languages have this possibly universal tendency to shift modalities around. I don't think linguists are quite sure yet why it happens: they just describe how it happens.
On a side note, there is a general tendency for "neutral" words to acquire some sense of desirability. Consider the word shapely: one might expect it to mean "having a shape", but it now means "having a good, beautiful shape". Note that this might be emulating Latin forma, "shape, good shape", and formosus, "beautifully shaped".
But there is also high quality: originally, quality meant something like "how-ness, nature, property"; but somehow an intensified ("high") property became a good property. This process is still going on, as can be seen in modern quality shoes: without a modifier like "high" or "low", quality is now good by default. This phenomenon can be observed in many different words in many different languages. I can see why it might be efficient to use a positive sense by default if a neutral word is not modified one way or the other: it saves the speaker words, as long as it doesn't cause confusion.