Is there any word or some idiomatic expression that would mean "half believe, half doubt"?
Having listened to that weird story, children half believed it, half doubted.
Meaning that none of the children fully believed the story.
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As far as idioms are concerned, it's quite possible to express "half belief, half doubt" about something.
Do you think it will work out?
I'm on the fence about it.
I"m of two minds about it.
on the fence
: in a position of neutrality or indecision
of two minds
: not decided or certain about something : having two opinions or ideas about something
Or, in a different context:
How sure are you about the plan?
I think it has a fifty-fifty chance of success.
I think it could go either way.
Also from Merriam-Webster:
2 : half favorable and half unfavorable
// a fifty-fifty chance
go either way
—used to say that either of two possible results is likely to occur and that neither is more likely than the other // I don't know who's going to win. The game could go either way.
The children were dubious, uncertain, unsure or suspicious about the truthfulness or accuracy of the story.
Of course, one cannot half believe, half doubt something.
If you are not certain whether it is true, you simply doubt it.
I would say:
The children were skeptical about the story.