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The current mainland China gov never had any real claims to it. The argument they use is the same as for why they believe they have the right to enforce Chinese law on Chinese people abroad, including having their own secret police in other countries, etc, they simply don't accept being anything less than the sole authority and sole representative for everybody they consider to belong to any ethnicity which is "theirs". The claims on the island doesn't really have much to do with the island, but that it's populated with people they consider theirs.