The violence in Iraq is widening the foreign-policy gap between Kentucky Sen. Rand Paul and other potential Republican candidates for the White House, and testing the strength of the pro-defense wing that has long dominated the party.

Among Republicans weighing 2016 bids, Mr. Paul is the most skeptical of foreign entanglements and the only one blaming the current crisis in Iraq on the Bush administration's 2003 invasion, as well as on President Barack Obama.

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