An Oct. 9 article incorrectly suggested that Defense Department officials leaked the name of Gen. Eric K. Shinseki's successor as Army chief of staff in revenge for his public comments that more troops were necessary in Iraq. Shinseki made those comments after the name of his successor was leaked. (Published 10/11/04)

President Bush and Sen. John F. Kerry disagreed vigorously last night as they tossed out plenty of numbers, and both demonstrated a talent for relying on facts and assertions of questionable origin.

Kerry said the administration retired Gen. Eric K. Shinseki, the Army chief of staff, because he said that more troops were necessary in Iraq, which is technically incorrect but close to the mark. Shinseki was permitted to retire on schedule, but in revenge for his comments, Defense Department officials leaked the name of his replacement 14 months early, effectively undercutting his authority.

Bush was skating close to the line when he said that he spoke to generals in the White House, asked if they had enough troops, and "they looked me in the eye and said, 'Yes, sir, Mr. President.' " In that 2002 White House meeting, Gen. Tommy R. Franks, whom Bush mentioned, said there were enough troops, but Shinseki told the president there were not. Other senior members of the Joint Chiefs of Staff told Defense Secretary Donald H. Rumsfeld that they were concerned about troop levels.

As in the previous debate and in his stump speech, Bush repeated a number of assertions about Kerry's voting record on taxes, intelligence spending and budgets that are out of context and misleading.

Bush, hitting Kerry for alleged inconsistency, also asserted: "He said he thought Saddam Hussein was a grave threat, and now he said it was a mistake to remove Saddam Hussein from power."

Kerry has never said that. This attack is derived from a Kerry statement that "the satisfaction that we take in [Hussein's] downfall does not hide this fact: We have traded a dictator for a chaos that has left America less secure." Kerry prefaced that statement, however, by saying that although Hussein was "a brutal dictator who deserved his own special place in hell," that by itself was not a reason to go to war.

In last night's debate, Bush also asserted often that Kerry was rated the "most liberal" senator, citing a study by the magazine National Journal. This is correct for the past year -- Kerry earned the rating in part because he missed a number of votes while campaigning -- but in a recent issue, National Journal said "the shorthand used to describe our ratings of Kerry and Edwards is sometimes misleading -- or just plain wrong."

Bush said Kerry's tax-cut rollback would raise taxes on 900,000 small businesses. This is misleading. Under Bush's definition, a small business is any taxpayer who reports some income from investments, partnerships or trusts. By that definition, every partner at a huge accounting firm or at the largest law firm would represent a small business.

Although Bush expressed surprise at Kerry's assertion that the president earned $84 from his investment in a timber company and, thus, qualified as a small business -- "I own a timber company? That's news to me. Need some wood?" -- the Web site backed up Kerry's assertion.

"President Bush himself would have qualified as a 'small business owner' under the Republican definition, based on his 2001 federal income tax returns," the Web site's analysis said. "He reported $84 of business income from his part ownership of a timber-growing enterprise. However, 99.99% of Bush's total income came from other sources that year."

On health care, Bush continued the specious accusation that Kerry is proposing a "government takeover" of the U.S. health system, saying Kerry's position is: "Let me incent you to go on the government." Kerry's plan builds on both private sector and government programs. Kerry does propose broad expansions of Medicaid and the State Children's Health Insurance Program. A report by the Commonwealth Fund, a private social research institute, estimated that under Kerry's proposal an additional 18 million to 21 million people would be covered. Like Kerry, Bush has been a supporter of SCHIP and said he intends to start an "aggressive" outreach effort to sign up a few million more children.

The president suggested he is on the verge of supporting the legal importation of lower-cost prescription drugs from countries such as Canada, which would be a major reversal. Despite overwhelming support in the House and Senate, the White House has blocked legislation opening the borders to the reimportation of U.S.-made pharmaceuticals.

Bush was correct in noting that during the Clinton administration, Health and Human Services Secretary Donna E. Shalala concluded that she could not guarantee a safe system for drug imports. In a tense back-and-forth over Medicare, Bush took credit for achieving much more in his four years in Washington than Kerry has done in his 20 years in the Senate.

"Show me one thing on Medicare he accomplished," Bush said. Kerry responded that he was "involved in" 1997 legislation that extended the solvency of the program, adding: "We balanced the budget and paid down the debt of the nation." In fact, his role was limited to voting for the Balanced Budget Amendment that was designed to keep Medicare afloat through 2030.

On the hot-button subject of medical malpractice, both men skipped over details that did not suit their cases. As he often does, Bush suggested that limiting non-economic damages would sharply reduce health care costs for most Americans. Analysis by the Congressional Budget Office found that legislation capping damage awards to $250,000 would lower physician malpractice premiums by 25 percent to 30 percent. But that reduction "would lower health care costs by only about .4 percent to .5 percent, and the likely effect on health insurance premiums would be comparably small," the CBO said.

Kerry glossed over his opposition to that bill, saying only: "I think we should look at the punitive [damages] and we should have some limitations."

Kerry stretched the truth when he said that the Bush limits on federal funding for embryonic stem cell research "makes it impossible for our scientists" to pursue an array of medical cures. Overwhelmingly, the scientific community has complained that the administration's policy has slowed research efforts but not curtailed them entirely. But Bush's assertions that the work "requires the destruction of life" is subject to debate. The research does involve destroying 5-day-old embryos, but people differ on whether that is life.

Kerry at one point said that "the president has presided over an economy where we've lost 1.6 million jobs." Kerry misspoke. He meant to qualify that statistic by referring to "private sector" jobs. The net number of jobs lost since Bush became president is about 800,000, because of growth in the public sector. It is the first time in 72 years, as Kerry correctly noted, that a president has presided over a net loss of jobs.

Bush asserted that he had tripled spending on homeland security, which depending on the numbers chosen could be an exaggeration. The budget authority has essentially doubled, from $20 billion in 2001 to $40 billion in 2004, according to the Congressional Budget Office. The White House budget office says that Bush was including his proposal for fiscal 2005, and that he was not including non-homeland security activities under the Department of Homeland Security.

Staff writer Thomas E. Ricks contributed to this report.