Not all statements that are falsifiable in principle are falsifiable in practice. For example, "it will be raining here in one million years" is theoretically falsifiable, but not practically. On the other hand, a statement like "there exist parallel universes which cannot interact with our universe" is not falsifiable even in principle; there is no way to test whether such a universe does or does not exist.
Popper noticed that two types of statements are of particular value to scientists.
The first are statements of observations, such as "this is a white swan". Logicians call these statements singular existential statements, since they assert the existence of some particular thing. They can be parsed in the form: There is an x that is a swan, and x is white.
The second are statements that categorize all instances of something, such as "all swans are white". Logicians call these statements universal. They are usually parsed in the form: For all x, if x is a swan, then x is white. Scientific laws are commonly supposed to be of this type. One difficult question in the methodology of science is: How does one move from observations to laws? How can one validly infer a universal statement from any number of existential statements?
Inductivist methodology supposed that one can somehow move from a series of singular existential statements to a universal statement. That is, that one can move from 'this is a white swan', 'that is a white swan', and so on, to a universal statement such as 'all swans are white'. This method is clearly deductively invalid, since it is always possible that there may be a non-white swan that has eluded observation.
One notices a white swan. From this one can conclude:
From this, one may wish to conjecture:
It is impractical to observe all the swans in the world to verify that they are all white.
Even so, the statement all swans are white is testable by being falsifiable. For, if in testing many swans, the researcher finds a single black swan, then the statement all swans are white would be falsified by the counterexample of the single black swan.
Observation O, however, is made:
So by modus tollens,
Although the logic of naïve falsification is valid, it is rather limited. Nearly any statement can be made to fit the data, so long as one makes the requisite 'compensatory adjustments'. Popper drew attention to these limitations in The Logic of Scientific Discovery in response to criticism from Pierre Duhem. W. V. Quine expounded this argument in detail, calling it confirmation holism. In order to logically falsify a universal, one must find a true falsifying singular statement. But Popper pointed out that it is always possible to change the universal statement or the existential statement so that falsification does not occur. On hearing that a black swan has been observed in Australia, one might introduce the ad hoc hypothesis, 'all swans are white except those found in Australia'; or one might adopt another, more cynical view about some observers, 'Australian bird watchers are incompetent'.
Thus, naïve falsification ought to, but does not, supply any way of handling competing hypotheses for many subject controversies (for instance conspiracy theories and urban legends). People arguing that there is no support for such an observation may argue that there is nothing to see, that all is normal, or that the differences or appearances are too small so as to be statistically insignificant. On the other side are those who concede that an observation has occurred and that a universal statement has been falsified as a consequence. Therefore, naïve falsification does not enable scientists, who rely on objective criteria, to present a definitive falsification of universal statements.
Naïve falsification considers scientific statements individually. Scientific theories are formed from groups of these sorts of statements, and it is these groups that must be accepted or rejected by scientists. Scientific theories can always be defended by the addition of ad hoc hypotheses. As Popper put it, a decision is required on the part of the scientist to accept or reject the statements that go to make up a theory or that might falsify it. At some point, the weight of the ad hoc hypotheses and disregarded falsifying observations will become so great that it becomes unreasonable to support the base theory any longer, and a decision will be made to reject it.
In place of naïve falsification, Popper envisioned science as evolving by the successive rejection of falsified theories, rather than falsified statements. Falsified theories are to be replaced by theories that can account for the phenomena that falsified the prior theory, that is, with greater explanatory power. For example, Aristotelian mechanics explained observations of everyday situations, but were falsified by Galileo’s experiments, and were themselves replaced by Newtonian mechanics, which accounted for the phenomena noted by Galileo (and others). Newtonian mechanics' reach included the observed motion of the planets and the mechanics of gases. Or at least most of them; the size of the precession of the orbit of Mercury was not predicted by Newtonian mechanics, but was by Einstein's general relativity. The Youngian wave theory of light (i.e., waves carried by the luminiferous aether) replaced Newton's (and many of the Classical Greeks') particles of light but in its turn was falsified by the Michelson-Morley experiment, whose results were eventually understood as incompatible with an ether and which was superseded by Maxwell's electrodynamics and Einstein's special relativity, which did account for the new phenomena. Furthermore, Newtonian mechanics applied to the atomic scale was replaced with Quantum Mechanics, when the old theory couldn't provide an answer to the Ultraviolet catastrophe, the Gibbs paradox, or how electron orbits could exist without the particles radiating away their energy and spiraling towards the centre. Thus the new theory had to posit the existence of unintuitive concepts such as Energy levels, Quanta and Heisenberg's Uncertainty Principle.
At each stage, experimental observation made a theory untenable (i.e., falsified it) and a new theory was found that had greater explanatory power (i.e., could account for the previously unexplained phenomena), and as a result, provided greater opportunity for its own falsification.
Popper claimed that, if a theory is falsifiable, then it is scientific; if it is not falsifiable, then it is not open to falsification.
The Popperian criterion excludes from the domain of science not unfalsifiable statements but only whole theories that contain no falsifiable statements; thus it leaves us with the Duhemian problem of what constitutes a 'whole theory' as well as the problem of what makes a statement 'meaningful'. Popper's own falsificationism, thus, is not only an alternative to verificationism, it is also an acknowledgement of the conceptual distinction that previous theories had ignored.
In the philosophy of science, verificationism (also known as the verifiability theory of meaning) holds that a statement must be in principle empirically verifiable in order to be both meaningful and scientific. This was an essential feature of the logical positivism of the so-called Vienna Circle that included such philosophers as Moritz Schlick, Rudolf Carnap, Otto Neurath, the Berlin philosopher Hans Reichenbach, and the logical empiricism of A.J. Ayer.
Popper noticed that the philosophers of the Vienna Circle had mixed two different problems, that of meaning and that of demarcation, and had proposed in verificationism a single solution to both. In opposition to this view, Popper emphasized that there are meaningful theories that are not scientific, and that, accordingly, a criterion of meaningfulness does not coincide with a criterion of demarcation. Verifiability came to be replaced by falsifiability as the criterion of demarcation.
Falsificationism strictly opposes the view that non-falsifiable statements are meaningless or otherwise inherently bad.
Some falsificationists saw Kuhn’s work as a vindication, since it provided historical evidence that science progressed by rejecting inadequate theories, and that it is the decision, on the part of the scientist, to accept or reject a theory that is the crucial element of falsificationism. Foremost amongst these was Imre Lakatos.
Lakatos attempted to explain Kuhn’s work by arguing that science progresses by the falsification of research programs rather than the more specific universal statements of naïve falsification. In Lakatos' approach, a scientist works within a research program that corresponds roughly with Kuhn's 'paradigm'. Whereas Popper rejected the use of ad hoc hypotheses as unscientific, Lakatos accepted their place in the development of new theories.
Some philosophers of science, such as Paul Feyerabend, take Kuhn's work as showing that social factors, rather than adherence to a purely rational method, decide which scientific theories gain general acceptance. Many other philosophers of science dispute such a view, such as Alan Sokal and Kuhn himself.
In their book Fashionable Nonsense (published in the UK as Intellectual Impostures) the physicists Alan Sokal and Jean Bricmont criticized falsifiability on the grounds that it does not accurately describe the way science really works. They argue that theories are used because of their successes, not because of the failures of other theories. Their discussion of Popper, falsifiability and the philosophy of science comes in a chapter entitled "Intermezzo," which contains an attempt to make clear their own views of what constitutes truth, in contrast with the extreme epistemological relativism of postmodernism.
Sokal and Bricmont write, "When a theory successfully withstands an attempt at falsification, a scientist will, quite naturally, consider the theory to be partially confirmed and will accord it a greater likelihood or a higher subjective probability. ... But Popper will have none of this: throughout his life he was a stubborn opponent of any idea of 'confirmation' of a theory, or even of its 'probability'. ... [but] the history of science teaches us that scientific theories come to be accepted above all because of their successes." (Sokal and Bricmont 1997, 62f)
They further argue that falsifiability cannot distinguish between astrology and astronomy, as both make technical predictions that are sometimes incorrect.
David Miller, a contemporary philosopher of critical rationalism, has defended Popper against these claims.
Non-falsifiable theories can usually be reduced to a simple uncircumscribed existential statement, such as there exists a green swan. It is entirely possible to verify whether or not this statement is true, simply by producing the green swan. But since this statement does not specify when or where the green swan exists; it is simply not possible to show that the swan does not exist, and so it is impossible to falsify the statement. That such theories are unfalsifiable says nothing about either their validity or truth. But it does assist us in determining to what extent such statements might be evaluated. If evidence cannot be presented to support a case, and yet the case cannot be shown to be indeed false, not much credence can be given to such a statement. However, you can also look at this case from another perspective. Let's say that the statement is "all swans are not green". An attempt to verify this positively would require a search for non-green swans, which you are sure to find. However, having rounded up and examined every known swan, there is always the possibility that there is at least one more swan but we will never know for sure until we find it and if we do, there may be yet, one more swan, and it may be green. On the other hand, we may say that "all swans are not green" but instead of attempting to positively verify this statement we attempt to falsify it by looking for a green swan. In that case, we need only find one swan (a green one), in the absence of which we can accept the original statement as a working hypothesis until such a swan is discovered.
The most common argument is made against rational expectations theories, which work under the assumption that people act to maximize their utility. However, under this viewpoint, it is impossible to disprove the fundamental theory that people are utility-maximizers. The political scientist Graham T. Allison, in his book Essence of Decision, attempted to both quash this theory and substitute other possible models of behavior.
Another construct that has been accused of being irrefutable is the principle of comparative advantage.
Many creationists have claimed that evolution is unfalsifiable. Numerous examples of potential ways to falsify common descent have been proposed. Richard Dawkins said that "If there were a single hippo or rabbit in the Precambrian, that would completely blow evolution out of the water. None have ever been found. Similarly, J.B.S. Haldane, when asked what hypothetical evidence could disprove evolution, replied "fossil rabbits in the Precambrian era". In contrast, many religious beliefs are not falsifiable, because no testable prediction has been made about the supernatural.
Similarly, the evolution of the great apes and humans from a common ancestor predicts a (geologically) recent common ancestor of apes and humans. This assertion could have been disproven with the invention of DNA analysis. Molecular biology identifies DNA as the mechanism for inherited traits. Therefore if common descent is true, human DNA should be more similar to great apes than other mammals. If this is not the case, then common descent is falsified. DNA analysis has shown that humans and the great apes share a large percentage of their DNA, and hence human evolution has passed a falsifiable test.
Popper himself drew a distinction between common descent and the process of natural selection. While he agreed common descent was falsifiable (he used the even more drastic example of the remains of a car in cambrian sediments), Popper said that natural selection "is not a testable scientific theory but a metaphysical research programme". However, Popper later said "I have changed my mind about the testability and logical status of the theory of natural selection, and I am glad to have the opportunity to make a recantation." He went on to formulate natural selection in a falsifiable way and offered a more nuanced view of its status. He still felt that "Darwin's own most important contribution to the theory of evolution, his theory of natural selection, is difficult to test." However, "[t]here are some tests, even some experimental tests; and in some cases, such as the famous phenomenon known as 'industrial melanism', we can observe natural selection happening under our very eyes, as it were. Nevertheless, really severe tests of the theory of natural selection are hard to come by, much more so than tests of otherwise comparable theories in physics or chemistry."
Logic and the other normative sciences, although they ask, not what is but what ought to be, nevertheless are positive sciences since it is by asserting positive, categorical truth that they are able show that what they call good really is so; and the right reason, right effort, and right being of which they treat derive that character from positive categorical fact. (Peirce, EP 2, 144).On the other hand, Peirce distinguishes mathematics proper from all positive sciences, and reckons it more fundamental than any of them, saying that any positive science "must, if it is to be properly grounded, be made to depend upon the Conditional or Hypothetical Science of Pure Mathematics, whose only aim is to discover not how things actually are, but how they might be supposed to be, if not in our universe, then in some other" (Peirce, EP 2, 144).
In this way of looking at things, logic is a science that seeks after knowledge of how we ought to conduct our reasoning if we want to achieve the goals of reasoning. As such, the logical knowledge that we have at any given time can easily fall short of perfection. Thus rules of logical procedure, as normative claims about the fitness of this or that form of inference, are falsifiable according to whether their actual consequences are successful or not.
Pure mathematics, on the contrary, contains no propositions that are not contingent on prior assumptions. Its apparent certainty is but a relative certainty, relative to the certainty of its axioms. One can say that its theorems are tautologies, so long as one remembers the original meaning of tautology, which is a repetition of something previously asserted. Mathematical theorems merely say more acutely what the axioms more obtusely already say.
Applied mathematics, in particular, mathematics as applied in empirical science, is still another thing. The application of mathematical abstractions to a domain of experiential phenomena involves a critical comparison of many different mathematical models, not all of them consistent with each other, and it normally leads to a judgment that some of the hypothetical models are better analogues or more likely icons than others of the empirical domain in question. This is, of course, an extremely fallible business, and each judgment call is subject to revision as more empirical data comes in.
How well a mathematical formula applies to the physical world is a physical question, and thus testable, within certain limits. For example, the proposition that all objects follow a parabolic path when thrown into the air is falsifiable; indeed, it is false. To see this, one has but to think of a feather. A slightly better proposition is that all objects follow a parabolic path when thrown in a vacuum and acted upon by gravity, which is itself falsified in regard to paths whose lengths are not negligible in proportion to a given planet's radius.
What is the conclusion then? Are mathematical theorems falsifiable or not? The most that can be said of them is that they are true of what they are true of, but what they are true of may not be the object of a given experience.
The above discussion addressed the nature of mathematical theorems in and of themselves, and then took up their application to empirical phenomena. But the actual practice of mathematics involves yet another level of consideration, and it may yet involve activities that are very similar to empirical science. Many working mathematicians, from Peirce in his day to Stephen Wolfram in ours, have remarked on the active, observational, and even experimental character of mathematical work. Imre Lakatos brings the concept of falsifiability to bear on the discipline of mathematics in his Proofs and Refutations. The question of whether mathematical practice is a quasi-experimental science depends in part on whether proofs are fundamentally different from experiments. Lakatos argues that often axioms, definitions, and proofs evolve through criticism and counterexample in a manner not unlike the way that a scientific theory evolves in response to experiments.
Anti-solipsism--the position that an external world does exist--is similarly non-falsifiable because regardless of what evidence is produced, it is always possible that there exists an external world outside one's experiences that does not interact with them.