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tim-kt
2 days ago
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Finding a bug in Dummit and Foote's Abstract Algeb...
It's true precisely because it's vacuous. If you quantify over the empty set, anything is true.
In other words, the statement "for every x in {} it holds that <anything>" is always true.
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layer8
2 days ago
[–]
What can be confusing is that the statement "for every x in {}, it
doesn’t
hold that <anything>" is always true as well.
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tim-kt
2 days ago
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[–]
I mean, yes. But "it doesn't hold that <anything>" is equivalent to "it holds that <not anything>" and since not anything is also anything... Ah, I see.
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In other words, the statement "for every x in {} it holds that <anything>" is always true.