Should expresses advisability. It ranges in strength from suggestion to responsibility (duty). Had better and ought to also express this meaning. "This is a good idea; this is my advice; this is an important responsibility." (Azar 9-7)
Modal auxiliaries generally express speakers attitudes. Modals can express whether a speaker feels something is:
- necessary (must, need to, have to)
- advisable (should, ought to, had better)
- permissible (may, can, could, might, would you mind)
- possible (may, might, could)
And a modal can express the strength of the attitude. Each modal has more than one meaning. (Azar 9-1, 10-1)
Should expresses obligation/necessity: He should leave. (intrinsic (184.108.40.206) [prediction (likelihood): He should leave soon. (extrinsic)]
Nine central modal auxiliaries: can, could, may, might, must, shall, should, will, would — They differ from other verbs both lexical verbs and primary auxiliaries, in that they have no nonfinite forms. (but have contracted forms I'd, can't, )
- permission/ possibility / ability: can, could, may, might
- obligation / necessity: must, should, had better, have (got) to, need to ought to, be supposed to
- volition / prediction: will, would, shall, be going to
Semi-modals → be going (to), ought (to), have (to), need (to), had better, have got (to), be supposed (to)
- obligation, duty (less strong than must)
- a reaction or unexpected occurrence (that clause) – It's surprising that she should appear now.
- conditional – Should you see her, call me. / If I had more time, I would / should / might call her.
Should is the preterit of shall, but is only used in that sense when "backshifting" tense in a subordinated clause. I shall do it. I thought that I should do it.
Should expresses "medium strength" deontic (advice) or epistemic (inference) modality—less than must, "strong".
- mandative should — He should be told. (morally right)
- emotive should — I was walking and who should I see but my ex-husband.
- conditional should – Should you need me, please call.
(Huddleston 3 §9.4)