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The comments from Mrs. Clinton, who is considering a 2016 presidential run and is clearly distancing herself from the administration’s foreign policy blunders, underscore the heated criticism of the president’s policy in regard to Iraq and Syria.

Mr. Obama has indicated that the U.S. will not make a broader commitment to take out the Islamic State.

Instead, based on the president’s words, American forces will act on humanitarian grounds or when terrorists directly threaten U.S. diplomats in Iraq. Responsibility to defeat the Islamic State apparently will lie primarily with the new Iraqi government and the Kurdish regional authority, which the U.S. and Britain are now arming directly.

“These have been difficult days in Iraq, a country that’s faced so many challenges in its recent history. And I’m sure that there will be difficult days ahead. But just as the United States will remain vigilant against the threat posed to our people by ISIL, we stand ready to partner with Iraq in its fight against these terrorist forces,” Mr. Obama said Monday, addressing the Iraq crisis while on a two-week vacation in Martha’s Vineyard.

Prominent Republicans have called on Mr. Obama to partner with Iraq and to launch a sustained campaign against the Islamic State.

“It is far past time for President Obama to develop a comprehensive strategy to combat the threat posed by ISIS. While the humanitarian aid efforts undertaken by the administration are an important first step, they should be accompanied by additional steps to degrade ISIS’s capabilities, including U.S. air strikes against ISIS positions in both Iraq and Syria and the immediate provision of military assistance to our partners who are fighting against ISIS,” Republican Sens. John McCain of Arizona and Lindsey Graham of South Carolina said in a joint statement Tuesday.