"I claim x" is ALWAYS true, at least at the moment it's uttered, in a self referential way. If we consider saying "I claim x" as a claim of x, then "I claim x" must be true.
"I believe X" may not be true - people can lie about their beliefs. Thus, someone can say "I claim x" while not believing x, but despite not believing it, they have still made the claim.
