I have problem with the meaning of following sentence and its grammatical structure.
If they are to be believed
How is this different from
If they are believed
When do we use am/is/are + passive infinitive?
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The verb "be" is among modal verbs (like "must", "should", "ought", etc.) and used to express a pre-existing agreement or common sense perception of behavior.
We are to arrive to New York terminal at 6:15pm.
meaning: if all goes well, the train/bus will be there at that time, according to the schedule or timetable.
John was to give a lecture on the 16th.
meaning: John was put in charge of giving the lecture (and agreed to do it).
If they are to be believed...
means that "If common sense suggests that we/you should believe them..." or that "if, according to some predefined plan or agreement, we/you accept what they are saying as truth..."
If they are believed...
is a supposition of acceptance of "their" statements as true, simple passive voice.