Critical rationalism
Critical rationalism is an epistemological philosophy advanced by Karl Raimund Popper, which is a logical generalization of his approach to science, falsificationism.
Criticism, not support
Critical rationalists hold that scientific theories, and any other claims to knowledge, can and should be rationally criticized, and (if they have empirical content) can and should be subjected to tests which may refute them. Thus claims to knowledge may be contrastively, normatively evaluated. They are either refutable and thus scientific (in a very broad sense), or irrefutable and thus unscientific. Those claims to knowledge that are potentially refutable can then be admitted to the body of science, and then further differentiated according to whether they are (so far) retained or indeed are actually refuted. If retained, yet further differentiation may be made on the basis of how much subjection to criticism they have received, and how severe such criticism has been.
This is not an epistemologically relativist philosophy, then, or a post-modernist or sociological approach to "truth". Critical rationalism has it that knowledge is objective (in the sense of being embodied in various substrates and in the sense of not being reducible to what humans individually "know"), and also that truth is objective (in the sense of being real, and having qualities and consequences not reducible to whatever one prefers "the truth" to be).
However, this contrastive, critical approach to objective knowledge is quite different from more traditional views that also hold knowledge to be objective. (These include the strong rationalism of the Enlightenment, the verificationism of the logical positivists, or approaches to science based on induction, a supposed form of logical inference which critical rationalists reject, in line with David Hume.) For criticism is all that can be done when attempting to differentiate claims to knowledge, according to the critical rationalist. Reason is the organon of criticism, not of support; of refutation, not of proof.
Supposed positive evidence (such as the provision of "good reasons" for a claim, or its making of successful predictions) actually does nothing to bolster, support, or prove a claim, belief, or theory.
In this sense, critical rationalism turns the normal understanding of a traditional rationalist, and a realist, on its head.
Not justificationism
William Warren Bartley compared critical rationalism to the very general philosophical approach to knowledge which he called "justificationism". Most justificationists do not know that they are justificationists. A justificationist confuses truth with certainty, the scientific with the verifiable, the rational with the provable.
According to Bartley, some justificationists are positive about these mistakes. They are naive rationalists, and thinking that their knowledge can indeed be founded, in principle, it may be deemed certain to some degree, and rational.
Other justificationists are negative about these mistakes. They are epistemological relativists, and think (rightly, according to the critical rationalist) that you cannot found knowledge, that there is no source of epistemological absolutism. But they conclude (wrongly, according to the critical rationalist) that there is therefore no rationality, and no objective distinction to be made between the true and the false.
By dissolving justificationism itself, the critical rationalist regards knowledge and rationality, reason and science, as neither foundational nor infallible, but nevertheless does not think we must therefore all be relativists. Knowledge and truth still exist, just not in the way we thought.
The pitfalls of justificationsim and positivism
The rejection of "positivist" approaches to knowledge occurs due to various pitfalls that positivism falls into.
1. The naive empiricism of induction was shown to be illogical by Hume. A thousand observations of some event A coinciding with some event B does not allow one to logically infer that all A's coincide with B's. (According to the critical rationalist, if there is a sense in which humans accrue knowledge positively by experience, it is only by pivoting observations off existing conjectural theories pertinent to the observations, or off underlying cognitive schemas which unconsciously handle perceptions and use them to generate new theories. But these new theories advanced in response to perceived particulars are not logically "induced" from them. These new theories may be wrong. The myth that we induce theories from particulars is persistent because when we do this we are often successful, but this is due to the advanced state of our evolved tendencies. If we were really "inducting" theories from particulars, it would be inductively logical to claim that the sun sets because I get up in the morning, or that all buses must have drivers in them (if you've never seen an empty bus).
2. Karl Popper and David Miller showed in 1983 (Nature 302, April 21, "A Proof of the Impossibility of Inductive Probability") that evidence supposed to partly support a hypothesis can in fact only be neutral to, or counter-supports the hypothesis.
3. Related to the point above, David Miller (in his Critical Rationalism : A Restatement and Defence, Chapter 3 "A Critique of Good Reasons"), attacks the use of "good reasons" in general (including evidence supposed to support the excess content of a hypothesis). He argues that good reasons are neither attainable, nor even desirable. Basically the case, which Miller calls "tediously familiar", is that valid arguments are either circular, or invalid. That is, if one provides a valid deductive argument (an inference from premises to a conclusion) for a given claim, then the content of the claim must already be contained within the premises of the argument (if it is not, then the argument is ampliative and so is invalid). Therefore the claim is already presupposed by the premises, and is no more "supported" than are the assumptions upon which the claim rests.
Popper wrote about critical rationalism in his works, The Open Society and its Enemies Volume 2, and Conjectures and Refutations.