In the 3 p.m. hour on Fox News on Tuesday, Shepard Smith rejected the notion of a quid pro quo involving Hillary Clinton and the U.S. government's 2010 approval of a deal in which Russia's atomic energy agency bought a company with uranium-mining interests in the American West.

Smith argued that the theory makes no sense because “the Clinton State Department had no power to veto or approve that transaction” — which is true.

Yet in the 9 p.m. hour on the same network, Sean Hannity told viewers the opposite.

“Where I come from, that sounds like a quid pro quo,” he said of the deal's approval. “That's what we call pay to play.”

These are not differences of opinion. These are true-or-false statements that cannot be reconciled.

As I have written before, it is reasonable to second-guess the wisdom of allowing Russia to acquire Uranium One, the company in question. It is also fair to contend that the Clinton Foundation should not have accepted money from foreign donors while Hillary Clinton was secretary of state.

But, as Smith pointed out, it is illogical to claim that Clinton signed off on the Uranium One deal in exchange for donations to her family's foundation — because she didn't even possess sign-off authority. The State Department was just one of nine agencies that reviewed the deal and gave it a green light.

Smith and Hannity live in different realities, and that is a problem for Fox News. A division between news and opinion is standard — and healthy — at many media outlets, but facts and alternative facts cannot coexist.

I asked a Fox News spokeswoman whether the network supports Smith's Uranium One fact-check and, if so, why it permitted Hannity to provide contradictory information hours later.

After this article was published, I received the following response: “We support both of our talent. Shep is an outstanding journalist, and Sean is an outstanding opinion commentator.”

In a similar situation, in May, Hannity said he would stop promoting a conspiracy theory related to the 2016 shooting death of Democratic National Committee staffer Seth Rich, after Fox News retracted an online report that lent credence to the theory.

In March, Fox News suspended legal analyst Andrew Napolitano for making an unsupported claim that British intelligence officials spied on Donald Trump during the 2016 campaign at the request of President Barack Obama.

In that case, as in the current one involving Clinton and Uranium One, it was Smith who set the record straight on his afternoon show.

“Fox News knows of no evidence of any kind that the now-president of the United States was surveilled at any time, in any way. Full stop,” he told viewers.

This recent history would seem to bode well for Smith. When Hannity and Napolitano made assertions that were contrary to the reporting of others at Fox News, it was the pundits who had to back down or face a penalty.

But as The Washington Post's Fred Barbash noted, some Fox News viewers don't appreciate Smith's debunking. They say Smith belongs on another channel. (MSNBC would probably take him, by the way. That network's president, Phil Griffin, told the Hollywood Reporter in 2012 that if he could poach an on-air personality from a rival, he would steal Smith. “I just like his way,” Griffin said of Smith. “I like everything about him.")

Hannity is a Fox News institution and, since moving into the 9 p.m. time slot in September, has regularly outrated MSNBC's Rachel Maddow, something “The Five” often failed to do in the same lineup spot.

In the big picture, it is no guarantee that Smith's brand of reporting will win out over Hannity's kind of punditry.

“Were Fox forced to choose, it would choose Hannity over Shep,” said Nicole Hemmer, a contributing opinion editor at U.S. News & World Report, and author of “Messengers from the Right,” a book about conservative media. “The network has clearly placed its chips on Trump — witness the reshuffling of the lineup to highlight Trumpier voices, including Tucker Carlson — and you’d be hard-pressed to find a more pro-Trump voice than Hannity.”

“But there are some mitigating factors,” Hemmer continued. “Rupert Murdoch highly prizes and highly pays Shep Smith because he brings credibility to the network. ... Hannity gets away with a lot but also toes the line when he’s asked to. The reality is that neither man is likely to be shown the door. Each plays a vital role at the network, and if there’s one thing Fox is good at doing, it’s overlooking contradictions in service of its broader agenda.”

This post has been updated with a statement from Fox News.