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WI an amendment in the terms of th e 22nd had been passed earlier, perhaps out of fear of TR.

The first big effect of this I think would be paradoxical. In OTL Cooleridge did not run because of the 2 term tradition. However the 22nd would actually have allowed him to run in 1928 (he got less than 2 years of Harding's term.

I wonder how different FDR would have behaved if he were not allowed to run in 1940.
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