The nuclear bomb reality
There is a philosophical irony in the truth about the bomb. To truly understand the threat of nuclear war is transcendental in some ways. The reality of such technology is unfathomable. Not to mention that, but the actual threat is painfully ironic. Now, the constant bickering and wars of world powers have been amplified to the nth degree by weaponry so powerful, it threatens not only one nation or another, but the entire planet. The plot of history has unfolded so that the climax appears to be the destruction of the planet if ever another world war occurs. That's why the book uses the word Doomsday, because that will be an apocalypse of religious proportion.
The infinite risk
Ellsberg teaches something ironic about nuclear war. He teaches that in addition to well-known effects, the absolute annihilation of anything in the blast radius and the threat of death from radiation and cancer, there are even more threats. For instance, he explains that each bomb has a chance of permanently damaging the ozone layer, so that if multiple bombs go off on the planet at once, the planet's ecological balance might swing radically, leading to a post-apocalypse where the sun is not effectively held back by the ozone layer. This could lead to extreme climate change.
Political paranoia
Geopolitical tension is as old as time, since the first time any animal ever killed another for the right for dominion. The hierarchical nature of human beings leads to this ironic paradox: When the threat of conflict rises, paranoia rises, and preparedness begins. Yet, that defensive effort has the appearance of threat to the other party, so that before a bullet is ever fire, an arms race begins. This played out to the world's absolute horror in the Cold War.
Trust and leadership
There is an irony between trust and leadership. This irony is perhaps most famously represented by Hobbes, in his Leviathan where he observes that contrary to common belief, the goal of the government is to secure it's own power by limiting the freedom of its populace to reduce the fear of the imminent threat death. That irony is shown as relevant here, because now the looming threat of death is literally arbitrated by the government itself. One wrong move from the government could mean death for millions of innocent civilians. Remembering the government's sole interest in power will not likely give any reader warm and fuzzy feelings.
The Eisenhower case
As an example of how dramatic irony can easily conceal dubious government decisions, the author mentions a case when President Eisenhower covertly gave allowances to his subordinates saying that, if there were a case of national emergency when it might become necessary for the military to launch nuclear missiles, that they could do so without the president's knowledge or consent. That multiplies the already-horrifying risk of human nature and capricious decision-making, as well as showing how political decisions can sometimes take peculiar, suspicious shape.
