In my setting, a war between "rationals" and "irrationals" has led to the establishment of a global society that makes all it's decisions based on logic and science. What is a believable threat to the way of life of a civilization such as this?
Your scientists were so preoccupied with whether they could that they didn’t stop to think if they should. -Ian Malcolm, Jurassic Park (film)
So the threat to this civilization is "Can we vs Should we" and there are all sorts of arguments that are logical to do, but shouldn't be implemented. In the world of Jurassic Park, the first half of the film is devoted to the system put into place to ensure we can control this. But at the start of the initial tour (the only ride that will be available when the park opens to the public) we start to see the wheels come off. The dinosaurs aren't attention whores, there are enough problems to occupy a full time staff at both a theme park and a zoo and they are in the process of downsizing, and safety features were deliberately ignored (the game warden has a wonderful "I told you so" moment when the entire party leave the ride vehicle while the ride is in motion... what theme park ride doesn't tell you to "keep your hands and feet inside the ride at all times" let alone have some kind of barrier to make sure the rider isn't injured should the choose to take that rule under advisement? Well, the rides that have 30 foot tall live animals that eat meat and can swallow humans whole!). And the final nail in the coffin is that the guy charged with implementing all this modern automation has vocally complained about how he's not being paid enough for his trouble and everyone believes he's working towards the best interests of the guy stiffing him.
Yes, these people can bring dinosaurs back to life, but these people should be the last people you want to bring them back.
But that's a big fantastical problem... a more likely society problem: How does this society deal with elderly? Well, operating in pure science and logic, elderly people are drain on the community. They need care, which takes away individuals from care of more healthy and viable populations, they don't contribute all that much to innovation... often stubborn and resistant to any changes of a modern society (the earliest known utterance of "Kids today have no respect" traces back to a tablet found in Mesopotamia, which feared that contemporary youth would doom civilization as we know. Kids Today: Number One threat to Civilization since 4,000 B.C.). Science says they will not likely to amount to much more in their life. Logic says that if resources must go to either an elderly person or a young person, they should go to the youth because they have more potential contribution. And medical spending on senior citizens is always higher than on youth on a one to one basis... and if taken to the full logical conclusion, we must assume that upon retirement, granny moocher should be terminated for the good of society, whether she or her family want to or not.
Now, scientifically and logically speaking, can we kill Granny without her consent? You bet your euthanasia we can. But should we do that? Well, in our modern culture, this is a horrifying idea. We still have trouble with letting terminal patients the right to doctor assisted suicide to allow them mercy for pain, let alone getting anywhere close to "Throw them a nice retirement party before we cast them as the title hero in our remake of 'Old Yeller'."
Even Star Trek's Vulcans, which worked as a society of logic and science still was plagued by emotions and those would factor into their decision making. Spock, the advocate of logic, even said that there are situations that, when logic cannot be relied upon, it was acceptable to act in illogical ways which is a nice way of saying "Be as logical and scientific as you want... just don't be stupid about it."
And since this situation is a human society, a logical course must take into account an illogical emotional creature. As pointed out in Men In Black, "A Human is smart. Humans are stupid." Now, you can try to build a system that can contain human stupidity... but I'd sooner trust the fine folks at InGen to build a dinosaur zoo than I ever would a person who believes she can build an idiot proof system... Human stupidity is a part of life... and as we all know, life finds a way.
The unpredictable human is the threat to the perfectly ordered rational society.
In Star Trek TOS Kirk often beat Spock at chess by making unpredictable moves and going for long shots. This is how humans work! We aim to misbehave!
In recent events, when Trump first declared his candidacy few took him seriously enough to mount an effective opposition. And when attempted to persuade people from their opinions using logic and facts it failed completely. You can't reason someone out of an opinion they did not arrive at using reason and logic, you have to use emotional persuasion. This becomes even more important when science and rationality have been weakened by selling out to the highest bidder (no smoking isn't bad for you we have scientific studies). Lies, damn lies, and statistics; facts lose credibility when anyone can use the same data to show what they want.
I think it is pushing believability that such a utopian society could ever actually form. But if it did, it would quickly fall when all of the human people failed to act rationally and logically in unpredictable ways. Most human decisions (including political ones) are not made rationally they are made emotionally and we use logic to justify and rationalize the decision we have already made. Social scientist (I'm looking at you economists) often lament that people aren't acting rationally according to their current theories, rather than studying how people actually act and changing their theories accordingly.
A rational society that you propose would most likely simply be a despotism of the scientific elites, which can choose to ignore the input of those deemed irrational. This would quickly lead to an underground movement of the oppressed who would make all sorts or un-rational decisions like choosing to fight and die against perceived injustices.
To those above:
Rational/Logical government does not mean what people think it does.
The kind of totalitarian, bulldozer, Borg-like version of 'Logic' is not, by definition logical. Because a rational system would aim to maximize things like total economic productivity, and average human wellbeing (including psychological well-being), whilst also putting in place ways of improving these things.
This is hard, but not impossible in theory. There would be a lot of empiricism involved - perhaps 'Try Policy A in one area, Policy B in another, see what works'. Testing by experiment is very much a 'rational/scientific' thing, as is changing your mind when these experiments give data that doesn't agree with your pet theory.
The best example I can think of in fiction would be The Culture
So.. our rational society can deal with physical challenges - indeed, by acting rationally it can avert self-destructive things like environmental destruction. The threat could be something like a fanatical religion ; or a break away part of this society that is dedicated to war; any group that will irrationally place their group above themselves (to a much greater extent than is normal for humans). This would give them a short term advantage over a more rational society.
How does logic apply to government?
Logic applies when you have a stated aim and you need the simplest path to achieve it. The straight line from A to B is logical. It's illogical to have emotional attachment to the things that are destroyed along that path.
For example, it's economically logical to introduce euthanasia for people over the age of 85 or earlier if they become a burden on society rather than a benefit. Is that really a logical path you want to follow? One could say it's not emotionally logical, it's certainly not logical if one wants to remain in power.
Is remaining in power one of your specifically required outcomes? If so that will lead down a path of repression of freedoms, if not then some outrageous character appealing to the basest emotions of the public could take power and undo all your hard work.
Technocracies don't tend to last long, people are too emotional.
What you're asking for is basically a technocracy, hopefully with a hint of meritocracy. Logically run by the best and brightest. Government however is about compromise, it's about negotiation, it's about allowing for arbitrary special interest groups, supporting uneconomic sections of society because they improve society, not the economy. Logic and science don't know this, only emotion does.
A few ideas that jumped into my head would be:
Actual magic, if some members of the society started displaying irrefutable and provable demonstrations of "scientifically impossible" phenomena that they themselves brought into existence then this could cause a threat to the way of life.
A system of leadership designed to make the most logical decisions for their society and in so doing they debate the logic of each decision to the point that nothing gets done and starvation and disease are running rampant. this would all be a result of their inability to make a decision that could potentially be illogical.
Stagnation in their science, the road to scientific discovery in our society is paved with people doing things that are not logical. These people end up discovering something previously thought impossible or illogical, and those discoveries lead to innovation. no illogical "doers" no innovation.
The biggest threat to such a culture is existential dread.
There's no way to actually define "scientific" and "logical" using only scientific and logical terminology. The logical limits were explored by Tarski in the earth 20th century. Science's limits have been explored many times. Thus there will always be at least one point in their society which is not perfectly nailed down with logic and science.
What you are proposing is Technocracy (or Scientocracy). The main problem is that it is not a natural form of government. People need to delegate their power to trusted scientists and experts. And this delegation, like any other delegation, has its flaws.
- Representatives may have inadequate expertise;
- People may disagree with the representatives;
- Representatives may get into a stalemate between themselves;
- Representatives may try to usurp power.
Take look also at how to design a realistically looking technocracy