In a tradition that can be traced back to John Stuart Mill (1806–1873) and Francis Bacon (1561–1626), many have taken the scientific method to be inductive. An inductive inference is ampliative (i.e., the content of the conclusion goes beyond the content of the premises) and non-demonstrative (i.e., all true premises do not guarantee a true conclusion; at best they render the conclusion more probable). For example, suppose that one has observed a large number of mammals and every kind of mammal that one has observed has teeth; from this evidence one might make the inductive generalization that all mammals have teeth. It is possible, however, that the next mammal one observes (say, an anteater) might turn out not to have teeth. The fallibility of inductive inferences is often referred to as Hume’s problem of induction, after the philosopher David Hume (1711–1776).

Carl Hempel (1905–1997) argues that the scientific method begins not with observations but with hypotheses. According to this hypothetico-deductive method one deduces certain observational predictions from the hypothesis and then rigorously tests them through further observation and experimentation. If the predictions are borne out, then the hypothesis is confirmed. Thus Hempel’s method is still broadly inductive. Although the conclusion of an inductive argument is not certain, one would like to determine quantitatively how probable the conclusion is, given its premises (the evidence). The logical positivist Rudolf Carnap (1891–1970) sought to develop such a logic of confirmation. Other models of confirmation, such as Bayesian and bootstrapping models, are reviewed in John Earman’s Testing Scientific Theories (1983).

Karl Popper (1902–1994) insists that the scientific method is deductive, not inductive. Observation always requires a prior point of view or problem. Like Hempel, Popper believes science begins with a bold hypothesis or conjecture. The way in which the scientist comes to the hypothesis (context of discovery) is irrelevant (e.g., it could come to the scientist in a dream); all that matters is the way in which it is tested (context of justification).

Unlike Hempel, Popper does not think that hypotheses can be confirmed. If the observational prediction is borne out, deductively the scientist is unable to conclude anything (to conclude that the hypothesis is confirmed is to commit the deductive fallacy of affirming the consequent). If, however, the predictions are falsified, then, by the valid deductive inference modus tollens (if p then q, not q, therefore not p) one can conclude that the hypothesis is falsified. Hence, Popper’s method is known as falsificationism. According to Popper, the scientist should not seek to confirm theories but rather, refute them. A theory that has survived repeated attempts of falsification—especially in those cases where it has made risky predictions—has been corroborated, though not confirmed. On this view, a theory is demarcated as scientific if there are observational conditions under which one would be willing to reject the theory as falsified.

As a matter of historical fact, however, scientists typically do not abandon their theories in the face of falsifying evidence. Furthermore, in many cases it turns out to be sound scientific judgment to continue developing and modifying a theory in the face of recalcitrant evidence. In response to these sorts of difficulties, Popper’s student, Imre Lakatos (1922–1974), developed a sophisticated falsificationism known as the “methodology of scientific research programs.” For Lakatos, instead of evaluating an individual theory or modification of a theory as scientific or ad hoc, one should evaluate a whole series of theories developed over time. This series, called a research program, consists of a hard core, which defines the research program and is taken to be irrefutable, and a protective belt, which consists of auxiliary hypotheses and background assumptions to be modified in the face of falsifying data, thereby protecting the hard core. According to Lakatos, a research program is demarcated as scientific if it is progressive—that is, it continues to make new predictions that become corroborated. Once a research program ceases to make new corroborated predictions it becomes degenerative and its hard core should be abandoned.

Paul Feyerabend (1924–1994) was a close friend of Lakatos and also a student of Popper’s. In his book Against Method (1978) he denies that there is such thing as the scientific method. He writes, “the idea of a fixed method, or a fixed theory of rationality, rests on too naïve a view … there is only one principle that can be defended under all circumstances. … It is the principle: anything goes” (pp. 27–28). Feyerabend’s view is known as epistemological anarchism.