So far, Peirce has identified three inferior methods that we all use to fixate our beliefs: authority, tenacity, a priori. He then argues that there is one method that is more likely to settle our opinions: the method of science.
First of all, Peirce is an epistemological realist, which means that he believes that there is something out there called reality whose nature remains constant relative to our beliefs, and that truth and falsity relate to that external reality. In other words, at least some of our individual and/or collective beliefs are true and others are false, whether we like it or not. Now, Peirce realizes that many philosophers in his time rejected realism and that it's impossible to "prove" that the real world exists. His argument is that the process of inquiry, however, can provide us with some guidance. The fact of the matter is that the overwheming majority of individuals and collectives believe that there is a real world (of some kind), and therefore, we do not doubt that it exists. A few philosophers doubt it, but so what? The pragmatic (common-sense) truth of the matter is that we all act based on our beliefs, unless that belief is in fact doubted. Doubt cannot be turned on and off. It's something that naturally arises, whether we like it or not! Therefore, Peirce argues that we are entitled to believe in a real world, at least until we actually doubt it.
So what is the "Method of Science?" Well, it's nothing more than the "process of elimination," or "trial and error." If something "works" we keep it! If it "doesn't work" we don't keep it. In short, Peirce is proposing an evolutionary epistemology, whereby Truth and Falsity are sorted out by the process of inquiry over time. Methodologically, Peirce argues that human knowledge advances based on evolution, especially variation and selection. Over time, our individual and collective bodies of belief evolve by weeding out the unfit. Later philosophers called this process "creative destruction." Hence, nature "creatively destroyed" dinosaurs, buggy whips, the geocentric map of the universe. Within the realm of belief, the process of inquiry requires that we willingly expose our beliefs to the falsification process, which implies avoiding the methods of tenacity, authority, and a priori. We can't know for certain what's true, but we can know what's false.
What's important here is that Peirce observes that we do (in fact) employ those inferior methods, however, we must deliberately fight that natural impulse. As Thomas Kuhn later observed, scientific theories (beliefs) are often willfully protected from the forces of creative destruction by self-interested scientists, scientific organizations, and governments that have invested their time, effort, and resources in maintaining the status quo. In short, there is a sociology of knowledge that often works against scientific human inquiry. Peirce also argued that scientific knowledge is highly fallible (his epistemological doctrine of fallibilism) and that we ought to guard against the rising tide of scientific positivism.
How many of our cultural beliefs are overdue for creative destruction but remain intact because they have been propped up by sociopolitical power structures? Libertarians argue that socialism is long overdue for creative destruction. I would add, that I seriously doubt that the Cincinnati Bengals will make it to the playoffs this year.