Karl Popper was a philosopher in Vienna during the reign of Logical Positivism, but he was not himself a Positivist. Popper is best known for his contributions to two issues, the problem of induction and the demarkation problem. In both cases his views were critical of the Logical Positivist's views.
The problem of induction is first raised by Hume. Hume argues that inductive arguments fail to provide rational support for their conclusions. His reason for taking induction to be irrational is that every inductive argument assumes that unobserved events will follow the pattern of observed events and this assumption cannot be supported either deductively or inductively. No purely deductive support can be given for this principle of induction because it is not a mere truth of logic. And any inductive argument offered in support of the inductive principle that unobserved cases will be like observed cases will be circular because it will also employ the very principle of induction it tries to support as a premise.
Popper accepted Hume's conclusion that inductive inference is not rationally justifiable. He takes the problem of induction to have no adequate solution. But he rejects the further conclusion that science therefore yields no knowledge of the nature of the world. With Hume, Popper holds that no number of cases offered as confirmation of a scientific hypothesis yields knowledge of the truth of that hypothesis. But just one observation that disagrees with a hypothesis can refute that hypothesis. So while empirical inquiry cannot provide knowledge of the truth of hypotheses through induction, it can provide knowledge of the falsity of hypotheses through deduction.
In place of induction, Popper offers the method of conjecture and refutation. Scientific hypotheses are offered as bold conjectures (guesses) about the nature of the world. In testing these conjectures through empirical experiment, we cannot give positive inductive reasons for thinking that they are true. But we can give reasons for thinking they are false. So see how this works, lets look at the pattern of reasoning employed in testing a scientific hypothesis using induction on the one hand, and Poppers 'deductivist' method of conjecture and refutation on the other. First, in designing an experiment, we determine what we should expect to observe if the hypothesis is true. Using induction, if our observation agrees with our expectation, we take the hypothesis to be inductively confirmed. The pattern of reasoning looks like this:
If H then O
This pattern of reasoning is not deductively valid (generate a counterexample) and as an inductive argument it faces the problem of induction. So this pattern of reasoning fails to provide us with rational grounds for accepting H as true. But suppose that when we carry out our experiment, we observe 'not O'. In this case our pattern of reasoning looks like this
If H then O
not O .
Therefore, not H
This pattern of reasoning is deductively valid (to see this try to suppose that the premises are true and the conclusion is false. If the conclusion were false, then 'H' would be true. And, given this and the truth of the first premise, 'O' would follow. But 'O' contradicts ‘not O” which is asserted by the second premise. So it is not possible for the premises to be true and the conclusion false. In other words, the pattern of reasoning here is deductively valid.)
The later is the pattern of reasoning used in the method of conjecture and refutation. It is a deductively valid pattern that makes no use of inductive confirmation. It should now be clear how Popper's method of conjecture and refutation works and how empirical inquiry making use of this method can provide us with knowledge of the world while avoiding the problem of induction.
According to Popper, there is no rational methodology or logic for evaluating how scientists come up with hypotheses. They are just conjectures. And no amount of evidence is capable of inductively confirming hypotheses in the sense of giving us positive reason for thinking our hypotheses are true. Evidence in agreement with a hypothesis never provides it with inductive confirmation. If all the evidence is in agreement with a hypothesis, we can say that it is “corroborated”. But to say that a hypothesis is corroborated is just to say that it has survived our best attempts at refutation. But contrary evidence can decisively refute hypotheses.
Popper's other major contribution was to provide an alternative to the Positivist’s verificationist theory of meaning in answering the demarkation problem. Again, the demarkation problem is just the problem of distinguishing legitimate scientific inquiry from things like religion, pseudo science (like explaining the difference between living and dead things by positing an elan vitale, or vital force), 19th century German metaphysics (Hegel's absolute or Heidigger's nothing which “noths”), literature etc. Since claims of ethics fail to pass the positivist criteria for meaningfulness given in the verificationist theory of meaning, the Positivist’s solution to the demarkation problem has the downside of denying that we can assert
as true that it is wrong to torture innocent babies just for fun.
Popper's method of conjecture and refutation suggests another criterion for distinguishing science from non-science. That is, that we can take a hypothesis, a proposed explanation, to be investigated scientifically if and only if it is falsifiable. For a hypothesis to be falsifiable does not mean that that it will be proven false or that it can be shown to be false (either of these confusions would lead to the absurd view that a claim is only scientific if it is false). Rather, to say that a claim is falsifiable is just to say that we can state some possible observable conditions under which we would judge the claim to be false.
We can state possible observable conditions under which we would judge that it is false that all crows are black. Specifically, we would judge this claim false if we observed a white crow. So the claim that all crows are black can be investigated using the methods of science. But we can not state observable conditions under which we would deny that the nothing noths. Hence, this is not a scientific claim. Likewise for the claim that opium causes people to sleep because it has the dormative virtue. This is a pseudo-scientific explanation because we can not state any possible observable conditions under which we would judge that it is false.