I have often heard the "argument" for the use of the Atomic Bombs in WWII was that 'Operation Downfall' would have cost far more lives. Is this a commonly held belief among historians or is it up for contention?
Upon doing my own research, it's hard to find credible sources as many use emotional arguments to justify/condemn the use of the atomic bombs.
Further questions:
Kind regards all.
Late to the story (was in Japan!) but some quick responses:
There is considerable evidence that the declaration of war by the USSR was the "event" that caused Japan to negotiate for surrender when they did. The first atomic bomb may have played a role in their feeling as well. Untangling the two influences is not easy, but it is clear that the Soviet invasion definitely did have an effect on the Japanese high command. Hasegawa's Racing the Enemy elaborates this thesis and gives much of the evidence.
The atomic bombs were largely dropped because they were ready to be used and because the US thought they might influence the Japan to surrender. That they would also show the USSR what "cards" the US was playing with in the postwar was certainly recognized as well. Historians tend to view the use of the atomic bombs as "overdetermined": there were many reasons to use them seen at the time, very few seen not to use them. (That does not mean they were necessary, just that they were likely to be used once ready.)
At the top level of policy, e.g. between Sec. War Stimson and President Truman, there was no discussion of whether the atomic bombs would be used. There was only some discussion on the timing and (I think importantly) about the targets (specifically whether Kyoto would be a target). That's basically it. At a slightly lower level, e.g. that of the Interim Committee (in which Stimson was the highest-level government representative, but it also included military and scientific participants) there was some discussion about whether there should be a warning, whether the bomb should be "demonstrated" in a way that would have no loss of life, and so on. Interestingly the idea of having a warning was endorsed but not acted upon. The idea of a demonstration was ruled out (by the four scientists consulted).
The first atomic bomb attack (on Hiroshima) was meticulously planned for and considered and executed according to very high level strategy. The order to use it, though, had an open-ended clause that said that the military could use further atomic bombs as they became available. The second bomb was made available a bit sooner than expected, and it was used based on the judgment of the military forces at Tinian, without any input from the Secretary of War, much less the President (neither Stimson nor Truman were even told it was going to be used before it was; it is not clear Truman actually even knew another was ready to be used so soon after the first). There is no evidence that the Nagasaki attack (which was actually originally scheduled for Kokura) had a significant impact on the discussions of the Japanese; many scholars believe that even if you think Hiroshima was unavoidable, or necessary for surrender, that Nagasaki was clearly not. (See, e.g., Barton Bernstein, "Understanding the atomic bomb and Japanese surrender: Missed opportunities, little-known near disasters, and modern memory," Diplomatic History 19, no. 2, Spring 1995.)
Last brief thought: the "use two atomic bombs or invade" argument was specifically constructed in order to justify the actions taken during World War II, and it was only constructed well after World War II ended. It was made by the people who dropped the bomb to deflect the criticism they were getting (esp. criticism from top leaders of the US military, who said the bombs weren't necessary). One should take care not to fall into its rhetorical trap. If the only options available are "use two atomic bombs on cities within three days of each other" or "have a full, bloody invasion of Kyushu and Honshu" then yes, the bomb is always the better choice.
But if your other options include: use an atomic bomb on a purely military target (e.g. the Harbor of Truk); use an atomic bomb above Tokyo Bay (which would minimize casualties but make for an awesome display); wait for the USSR to declare war on Japan; use an atomic bomb on a city but wait more than 3 days for a reply; and so on, then it becomes a more complex set of options and possible outcomes.
There was no "big decision" to use the bomb by Truman in 1945. He was actually mostly peripheral to the entire process. There were smaller decisions made by others, but for the most part everyone involved in the decision-making side of the project assumed that the bomb would be used. That it was a weapon most amenable to using on cities (and not, say, the Harbor of Truk) was due to technical decisions made early in its development. Truman made literally two significant decisions the entire time: a) he agreed with Stimson that Kyoto would be removed from the list of targets, and b) he ordered, on August 10th, that no further atomic bombs would be dropped without his express permission (they were about to send a third nuclear core to Tinian, for use on August 17th).
More can be said here, but check out the FAQ section here for a starting point.