But the notion of fairness requires that before it comes to blows, the attacker should try and steelman his opponent's position. So in the following, I try to briefly summarize what I think Libertarianism is, and to put special emphasis on the ideas I like in it.
Libertarians' fundamental belief is that individual freedom should be regarded as the highest aspiration in society. Everyone should be free to do anything she likes as long as it doesn't harm others. "My freedom to swing my arm ends where the other fellow’s nose begins" - as the famous quote says. The government's sole purpose is to provide the conditions under which individual freedom can flourish. It should guarantee national security and ensure the rule of the law domestically, and with some exceptions (some libertarians are quite ok with seatbelts), that's all it should interfere with people's life. Taxation for any other purposes, like building an orphanage or funding basic research, is theft or plain extortion. Charity, like business, should be strictly voluntary.
This belief logically leads to the faith in free-market capitalism. Economy, just like any other public endeavor in society, should be organized bottom-up. Here I find the ideas and arguments of free-market advocates very persuasive. The theory behind why voluntary exchange between individuals leads to higher prosperity for the whole society than central planning is elegant, parsimonious, and powerful. It is also proven.
Let's demonstrate it with an example. If the French government tried to calculate how much a Parisian restaurant should ask for oysters, it would need to unpick a very intricated net of interactions. It would need to know how much oyster people generally wish to eat, how much oyster fishermen in Marseille can produce in a day - and their fellow fishermen elsewhere. How much compensation they deserve for the work. How much transporting the goods would cost (the best cost/benefit ratio among the alternative routes). How much is the demand in Paris for oysters and what is the fair price customers would be willing to pay for them. How much the cook would ask for preparing them.
With voluntary exchange, these calculations (and a myriad more with varying relevance, like what's producers of oyster-fishing equipment would want for their goods) happen between individuals, who know best how much value they put on their time and work.
The price of an oyster dish condenses a huge amount of information in a single number no one alone could come up with. And if the government forced the price to the half by decree, the ripple effects would wide and far-fetching and would affect more people than it could be foreseen. Most of the time, not for the better.
Accepting the supremacy of voluntary exchange, the impossible burden of knowing every detail and consequence of a financial action vanishes. And the free exchange-based system indeed rules supreme over any other economic alternative. Never before the modern times had humanity it as good as now in terms of GDP, nutrition, life-expectancy, or any other meaningful materialistic metric one can name. Capitalist countries are free and prosperous, while there is not a single self-proclaimed socialist state that didn't manage to make its people poorer and its prisons more crowded.
Libertarian thinkers, however, are rarely satisfied by stating their case not solely on economic terms. They claim that their way of doing things is not only economically more efficient than the alternatives, but it's also morally justified. Here they are not so different from socialists or even liberal democrats. Every camp believes that their political/economic ideas would yield the highest prosperity, and, conveniently, they are also morally superior.
So as far as I can see, Libertarians tend to state their case in four different ways.
1. Reaping the rewards of your work is just and it requires no further explanation.
2. (going from the other direction) although strict adherence to the first principle might be a justification for being selfish and greedy, but it's still immoral to forcefully disappropriate someone else's possession.
3. Value is in the eyes of the beholder. No one has the right to impose her priorities on her fellow citizens.
4. Finally, life is terribly complicated. Solutions based on strictly voluntary bases might not always be the best, but it's practically impossible to build a framework of rules that would on average perform better. In the same vein, it's better to accept that the value of something is simply what people are willing to give for it than attempting to come up with some arbitrary and complicated way to measure it. It's not only futile, but it also leads to corruption and injustice.
So that is at least my interpretation of Libertarianism. My rebuttal below runs broadly along the following lines. For some problems, states sometimes have better solutions at their disposal than what free markets could provide. Ensuring that economic choices are really uncoerced sometimes requires more and not less state intervention. Libertarianism weakens social cohesion, which is in cases desirable and in others not. Finally, values are not entirely subjective, and they shouldn't be treated as such.
Let's start with a generic observation. I think every single libertarian argument above stands well on its own. Some are very convincing, but all of them are at least defendable. My problem is with the almost religious fervor Libertarians cling to them. Principles in life always have a sphere of validity. Take them too far, and they will either collide with the extreme end of another dear principle or will just lose power. Don't kill is a good guideline in life until you see a serial killer with a blood-stained axe in his hand coming at your kids and you have only one bullet left in the chamber. Being compassionate and always telling the truth are also mutually exclusive in certain situations.
Libertarians sometimes seem to think that the world is a just bunch of nails and they have found the perfect hammer - even if it's admittedly an unfair comparison. Free-markets are such a powerful and versatile invention that they reject physical parallels. But they are not a panacea. An apter analogy could be found in engineering. Software engineers know that decentralization is a powerful technique for building simple, robust, and efficient systems. But they also know that there are no silver bullets in engineering. Sometimes centralized control is the best option. If no single approach can solve every problem in engineering, it's unlikely that one such exists in a world much more messy and complex, that is, economics.
In turning from the abstract to the concrete, I can think of at least three areas where free markets don't perform very well.
The first is that seemingly voluntary life-choices are not really voluntary. The libertarian argument for the legalization of prostitution is that as it happens between consenting adults, it's no one else's business. But neat ideology is a thin veneer on the grim reality: the large majority of prostitutes are drug-addicts. Women are often forced into selling their bodies and are abused by their pimps. Arguments for and against polygamy follow the same lines. Fundamentalist Mormon women can claim that they are happy in a polygamous relationship, but in those communities, underage girls forced to marry men 50 years their senior. It's not their free choice, and they are brainwashed into it from childhood and punished severely if they try to dissent.
The second area is the problems for which individual actions just don't provide adequate solutions. Sometimes the sum of the parts is less than the whole. Recycling your waste will make you feel better about yourself and will do nothing to solve the problem of climate change. To stop lethal viruses, a large majority of the population needs to be vaccinated. If the government refuses or unable to enforce that, the fate of the whole society stays in the hands of a small group of delusional anti-vaxxers. Another example is banning cigarettes from public places. On their own, few smokers would have followed guidelines, but even the majority of smokers are now happy to be able to spend a night in a bar without stinking the next day. The same can be said about enforcing helmets in professional cycling or ice-hockey. A hockey player would not have worn a helmet when others don't because it constraints her vision and she would have had to trade efficiency for safety. But being mandatory, it poses no disadvantage to anyone and provides higher safety for everyone.
The third area is where places where the financial incentives of individual actions are missing. Funding basic research in, let's say mathematics, is not a rational business decision anyone would make if those researches are expected to bring results in a hundred years.
The element of libertarian ideology I take issue with the most is the definition of value. Libertarians say, if you produce movies for which people are willing to pay to see, you have exactly the same moral right to enjoy your wealth as if you had amassed it by producing medicaments for disabled children. The majority has no right to impose their system of values on individuals. After all, who has the right to claim to be the judge of other people's preferences, and second, value systems change over time. What once was regarded as the deepest depravity, like homosexuality or atheism, is today considered perfectly normal.
And, in general, they have a point. Who is there to say how much a painting should cost? What is worth more, a computer or a motorbike? Caring for the elderly or nursing children? But even though I agree with the idea in general, I don't believe that values are purely subjective. And neither really do those who say so. Let's take the dilemma raised by Breaking Bad. If people are willing to pay much more for producers of crystal-meth than for chemistry teachers, does it mean the former deserve more? It's futile to try to align moral and material worth. But it doesn't mean they are truly independent and every attempt to reconcile them deserves scorn.
Finally, libertarianism weakens cohesion in society. This is sometimes a good thing, the world would have preferred a bit less unity in the Third Reich. But oftentimes it's not. Public and mandatory schooling, universal health care, national television, and the like do forge national unity, and the only thing that makes that hard to see is that we take them for granted. I once heard the famous American libertarian, John Stossel arguing for a privatized postal service. Stossel cited the massive losses the national postal service suffers every year, and argued, very sensibly, that private companies would do the job much better. I'm sure it's true. But they surely wouldn't be universal. Scarcely populated, remote places with old and possibly lonely people would unlikely be covered by them. I accept that there are some ideological hermits who would find infuriating that they can't get away from the overbearing state, but their well-being concerns me less than those who would rightly feel just left out in the cold by the nation. I'm not saying that receiving letters from the state would fill the gap of loneliness in people's hearts. But I think the idea that there is a community that regards you as a member comforts more people than how many it incenses. Also, what would a libertarian utopia do if its less enlightened neighbor decided to invade it? The "every man for himself" approach would bring a quick end to the glorious experiment.
In the end, although I don't want to live in a country ruled by libertarians, I want to hear their voice in opposition. Libertarianism is the best antidote against theocracy, an ossified class-system, or socialist enthusiasm in government. They have a lot to say both in the area of morality and economics. But unlike moderate liberals or conservatives, they think what they have to say is all that's out there.