Lyotard's theory of postmodernism basically says that, "all the world's cultures, rituals, races, databanks, myths, and musical motifs, are intermixing like a smorgasbord in an earthquake" (Postmodernism for Beginners). He also says that it's the postmodernists job to represent the individual groups and cultures. Essentially postmodern, to my understanding, is the antithesis to finding a central "truth" in society, and letting the individual mind run.
I see this theory throughout Cat's Cradle. The biggest representation of this theory I see is the entire 'religion' of Bokononism. Before the novel even begins, Vonnegut starts by quoting the Books of Bokonon which says "live by the foma (harmless untruths) that make you brave and kind and healthy and happy". Throughout the rest of the novel, Vonnegut makes it a point to stress how Bokononism believes in lying to oneself in order to live a happy life. The point I think he is trying to make, by contrasting Bokonon with all the scientists looking for "absolute truth" which only kills hundreds of thousands of people (atom bomb), you can lie to yourself and live a perfectly normal, happy life. I believe he is making the postmodern point that individuals may live how they want to, and they do not have to be entirely structured the way the rest of society is, and still be happy without knowing the "absolute truth" about the universe (which is NOT protein).
These theories are not only present in literature, but other forms as well. Architecture is one main way the postmodern theory is expressed in today's society. Frank Gehry is a very notable postmodern architect. He does not like to conform to the standard, efficient, square way of designing buildings, but instead takes on a very individualistic, and artistic approach. This is one of the most concrete comparison's I can make with postmodernism away from literature.