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I was talking with my friend about logical connectives and he noticed that OR in informal speech is basically used only as exclusive and in other cases we add "or both" to it.

So, it's the reverse of what we do in formal logic.(we add "and not both" if we mean exclusive)

I need to find some counterexamples to his point of view, in which it's crystally clear that OR is being used inclusively.

I think i have found one already: "There are many examples that confirm given phenomenon, such as ... or such as ...".

famesyasd
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"If your mother or father had high blood pressure then there's a good chance you might also."

"If you call me nasty names or stick your tongue out at me, I'll slap you silly."

"Reading Shakespeare or attending the West Coast Number Theory Conference will make you smarter."

"Driving drunk or high is almost as bad as driving while texting."

"You could ask your instructor or post your question on MathStack."