Often claims are based on beliefs. But not always. My claim may not be supported by my belief but I might use it to justify myself or to make my views more attractive.
A scientist may write a book saying there is no God because it is sure to become a best seller and you can mint money. If you write a book saying there is God few will buy it. Often for material, worldly gains people make claims against their own beliefs. So claims and beliefs are not the same. There is as distinction.
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