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What text in the GPL says that a GPL'd library really cannot be used in a non-GPL application?

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Obligatory disclaimer: I am not any sort of lawyer

Section 5 of the GPLv3 details how to "convey a work based on the Program". Among other requirements is the following:

c) You must license the entire work, as a whole, under this License to anyone who comes into possession of a copy. This License will therefore apply, along with any applicable section 7 additional terms, to the whole of the work, and all its parts, regardless of how they are packaged. This License gives no permission to license the work in any other way, but it does not invalidate such permission if you have separately received it.

I recommend reading the definitions section, which I will not copy here for the sake of brevity, as it does clarify the use of some terms.

So, the GPL(v3) does not say that "a GPL'd library really cannot be used in a non-GPL application". What it discusses is this idea of "a work based on the Program", which is typically called a derivative work. So, if a program P uses a library L, is P necessarily a derivative work of L? The Free Software Foundation would say yes. Unfortunately, the concept of a derivative work is not precisely defined. Although I am not aware of anyone who would argue that P is never a derivative work of L, some would argue that there are contexts in which P may not necessarely constitute a derivative work. I would not recommend testing that. The particulars may depend on the laws of a specific jurisdiction, but that is a matter for lawyers to discuss.

If nothing else, I can tell you with certainty that you would be in violation of the spirit of the license, and with that knowledge I hope you'll decide against trying to find some loophole (unless to aid in ensuring the license as written more accurately represents its intent in future).

You didn't really specify why you need to know, though, perhaps you want to distribute a library that can be incorporated into proprietary programs? If so, you may want to check out the LGPL.

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