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Modernism in philosophy refers to the period from Descartes to logical positivism, with a general focus on skeptical, materialist, empiricist, or positivist doctrines of truth. That which attacked the supposed "orthodoxy" or "dogmatism" in pre-modern philosophy (scholasticism, Aristotle, etc). Its characteristic set of claims would be, that real knowledge is scientific knowledge, that metaphysical knowledge is impossible but scientific knowledge can be clearly distinguished from it and is possible. A narrower position often conflated with all of modernism, is the typical positivist claim that everything can be reduced to sense impressions as data and logic handling various abstractions from it, all consistently axiomatized and made one connected system. Sometimes the typical 19th century faith in determinist reduction (analysis) is also meant (meaning, things necessarily just do whatever their component bits do, added up).

Postmodernism in that context then means, where people arrive after they come to doubt the knowledge claims of modernism, as well. Typically because they see no rigorous distinction between metaphyiscal and scientific knowledge, or they reduce both to opinion, or deny knowledge properly so called applies to either. Skepticism alone does not have to arrive here, but subjectivist idealisms often do, including skeptical idealisms. Relativism arrives here, historicism likewise. The basic diagnosis is the reason's attempts to reliably and exhaustively grasp a real external world fail. For logical reasons, or because of limits on knowledge or certainty, or because of subjective distortions, or because there isn't an unambiguous external reality to grasp in the first place, or because real relations don't have the necessary stability or determinism to be grasped unless we artificially impose it on nature, etc. Arguments of that sort.

In the continental philosophy tradition, modernism would end around Hegel and Marx (and later neo-Kantians) and post modernism would arrive with Nietzsche and Heidegger, and would include present day French versions of their thought like Foucault and Derrida. In the English philosophy tradition, modernism would include Russel and Whitehead and the early Wittgenstein, post modernism would come in with the later Wittgenstein, Quine, and Kuhn. In mathematics, Hilbert is the paradigmatic modernist and Godel passes for "post". In reality, Godel was a mathematical platonist who went back to pre-modern philosophy for his philosophic opinions, but he certainly rejected Hilbert's "constructivist" program to ground mathematics in logic. He thought it is essentially bigger. (In pure formal computation terms, he was right - basic logic is not Turing complete and Peano arithmetic is).

Boiled down to one sentence: Postmoderns suffer from cognitive dissonance (the mental confusion that results from actually holding polar opposite attitudes and beliefs simultaneously).

Quintessential example:

They don't believe that there is such a thing as "absolute truth", EXCEPT the absolute truth that there us no such thing as absolute truth.

Ding! Ding!

Such confused mentalities can be found on forums like Free Republic actually claiming to want to uphold and defend the Constitution --- which, of course, is a meaningless document unless it is actively guarding the absolute moral (self-evident) truth that man has inalienable (because they're God-given) rights.

Any who are foolish enough to think it is possible to use reason and logic in a debate with them are also suffering from cognitive dissonance.