Camus is an absurdist who many classify as an existentialist, a label he himself rejected. To Camus, who had a potent form of anti-intellectualism by appealing to and writing in the style of the emotionally poignant, absurdism as a philosophy tends to dismiss the greater enterprise of philosophy as, well, absurd rationalization. He had a falling out with Sartre who embodies a more traditional notion of philosophy with his views on existentialism.
Absurd reasoning is engaging in reason from the vantage point of an absurdist. It's straight forward intellectually. But you have to infer through The Myth of Sisyphus and other works of fiction like The Stranger and The Plague what Camus was driving at. As a French partisan who saw the horrors of the Third Reich up close, and as journalist with an eye for what goes on in the world, as opposed to what merely goes on between the cover of philosophical books, the first thing that absurd reasoning entails is the rejection of any sort of comprehensive meta-narrative or meaning of life. There isn't one. The myths of the various religions are not just opiates of the masses, but they are a way of dodging life itself by living in a fantasy. It is absurd to live in a fantasy, and yet most people aspire to live in fantasies of their own construction.
Life, as Hobbes noted, has the capacity for being brutish and short, and the WWII certainly revealed all of the horrors humans inflict on each other. But Camus did not become a nihilist (PhilSE). It doesn't make sense to do so. He did not reject value. He simply asked if life without meaning is all there is, should you kill yourself literally? No being his answer, he then asks about philosophical suicide (Medium.com). Should one simply just stop asking questions and stick one's head in the ground? The answer again is no. Here then, an absurdist notes that much of what masquerades as intellectual and philosophical discourse is just institutional rationalization. Economists with their theories. Philosophers with theirs. Scientists have their theories. Politicians have political theories. And most of it is just nonsense. To be in a world with no meaning assigned, no self-assigning of meaning possible, and to be surrounded by people in the Plato's cave, as it were, is absurd. Such facts are the starting point of absurd reasoning. It is the way to cope with the Absurd Condition.
Once one has witnessed Absurdity, one then must confront it and rebel against it. Live and live well with emotional vigor. The only task that matters is this living in rebellion against the Absurd Condition. That doesn't mean claiming no values exist. Camus had values and lived in accordance with them stridently. Fascism was bad. Family was good. Truth was important. He was no nihilist, and urged others to rebel against the silliness of nihilism and to live out authentic lives as the human condition allows. But never fall for the delusion of meta-narrative. That would be philosophical suicide.
This task is not easy. He chose Sisyphus to convey the idea that the struggle never ends. The rebellion is continuous. The quest to be authentic was never ending. Religion is constantly trying to acquire fresh converts. Governments spill propaganda on to its people to encourage obedience. Corporations are trying to separate you from your wealth and autonomy with advertisement and marketing. Philosophers are trying to persuade you to think in convoluted ways. The rebellion against these delusional vested interests is perpetual.
If one applies one's heart and mind to the process in accordance with the call to Rebellion, one does manage to eek out some form of freedom commensurate with one's efforts. One is free from religious doctrine, government propaganda, and corporate interests. One may not be part of the Great Chain of Being, but one has the satisfaction of knowing that such a chain fetters minds. And in a world and society that values freedom, it is absurd so many gravitate to the delusions cultivated and practiced by the institutions of society. This is the process of absurd reasoning.
