v.aux. (3rd sing. should) past of SHALL, used esp.: 1 in reported speech, esp. with the reported element in the 1st person (I said I should be home by evening). ° Cf. WILL(1), WOULD, now more common in this sense, esp. to avoid implications of sense 2. 2 a to express a duty, obligation, or likelihood; = OUGHT(1) (I should tell you; you should have been more careful; they should have arrived by now). b (in the 1st person) to express a tentative suggestion (I should like to say something). 3 a expressing the conditional mood in the 1st person (cf. WOULD) (I should have been killed if I had gone). b forming a conditional protasis or indefinite clause (if you should see him; should they arrive, tell them where to go). 4 expressing purpose = MAY, MIGHT(1) (in order that we should not worry).
Meaning of SHOULD in English
Concise Oxford English dictionary. Краткий оксфордский словарь английского языка. 2012