I know how to use this phrase - it means not to automatically accept something as true -- but I don't know where it comes from. Can anyone help? Thanks!
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Brewer explains the Latin equivalent thus:

'*** grano salis With great limitation; with its grain of salt or truth. As salt is sparingly used in condiments, so is truth in the remark just made.'

But to my mind, that doesn't seem quite to fit, since the 'grain of salt' is added by the listener, when he 'swallows' the remark; whereas the 'truth' is added by the speaker, when he makes it.

I'm more used to hearing 'with a pinch of salt'. Is that just BrE?

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I'm used to pinch of salt as well.
Maybe it's BrE, because I'm used to "pinch". However, (and I'm showing off now) '*** grano salis' comes from Pliny the Elder's 'Historia Naturalis' - one of my very favourite bed-time stories.

It refers to a recipe which was an antidote to poison, so the listener takes "a pinch of salt" to avoid being poisoned by the speaker's words. Emotion: geeked
I feel the AmE option is healthier.

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"With a grain of potassium chloride?"
Scientists would say, "Salt is not healthy for you, but please take it With 1 g. of potassium chloride."

Emotion: smile