when writing, can a claim be true or false depending on the context of the paper?

Yes, if that paper is the authority of truth for the claim.

It depends on what you're claiming. If you are writing an argument, the claim needs to be supportable on either side, and then your paper needs to completely support your claim and "blow away" the other side.

http://www.uncp.edu/home/canada/work/markport/best/writing/argument.htm
Scroll down for the section on claims.

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Read the first three paragraphs.

http://pubpages.unh.edu/~dml3/guidefor.htm
Scroll down and read about the thesis.

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