Property of a statement that is written in an empirical language and contradicts some observations, realistic or not, that can be described in that language.
These black swans are a state of affairs that obtains, but even with no black swans to possibly falsify it, "All swans are white" would still be falsifiable—a black swan would still be an observable state of affairs.

In the philosophy of science, a theory is falsifiable (or refutable) if it is contradicted by an observation that is logically possible, i.e., expressible in the language of the theory, and this language has a conventional empirical interpretation.[A] Thus there must exist a state of affairs that obtains or not and can be used as a scientific evidence against the theory, in particular, it must be observable with existing technologies. For example, "All swans are white" is falsifiable, because "Here is a black swan" contradicts it.[B] To make falsifiability more intuitive, one can assume that the state of affairs is allowed by some other law than the one that is falsified. For example, Newton's law of gravitation is also falsifiable—it is falsified by "The brick fell upwards when released",[1][C] which is a state of affairs that can be observed if some hidden force other than gravity acts on the brick.[D] On the other hand, "All men are mortal" is not falsifiable, because, unlike a swan being black, a man being immortal is not an inter-subjective property—there is no shared procedure to systematically conclude to immortality.[2][3]

Falsifiability was introduced by the philosopher of science Karl Popper in his book Logik der Forschung (1934), faithfully translated into English by himself and two other translators[E] in 1959 as The Logic of Scientific Discovery. He proposed it as the cornerstone of a solution to both the problem of induction and the problem of demarcation.

The role of falsifiability in Popper's philosophy is to make a deductive testing and a rational critic of the theory possible. This logical part is used within a methodology that, in Popper's own account, is hardly rigorous, because it involves irrational creative processes and, as pointed out by Duhem and others, definitive experimental falsifications are impossible. In response to the impossibility of a technologically reasonable verification and to avoid the problems of falsification, Popper argued for falsifiability and opposed this to the intuitively similar concept of verifiability. Verifying the claim "All swans are white" would require observing all swans, which is not technologically possible under reasonable assumptions in any theory that has a conventional empirical interpretation. In contrast, the observation of a single black swan is technologically reasonable and sufficient to logically falsify it and steer a critical discussion. Moreover, even if a black swan was in principle impossible due to a fundamental law of biology, it would still be a potential falsifier—i.e., an observation that is logically possible in the empirical language and in contradiction with the claim.[F]

As a key notion in the separation of science from non-science and pseudo-science, falsifiability has featured prominently in many scientific controversies and applications, even being used as legal precedent.

One of the questions in scientific method is: how does one move from observations to scientific laws? This is the problem of induction. Suppose we want to put the hypothesis that all swans are white to the test. We come across a white swan. We cannot validly argue (or induce) from "here is a white swan" to "all swans are white"; doing so would require a logical fallacy such as, for example, affirming the consequent.[G][4]

For Popper, induction is actually never needed in science.[H][I] Instead, in Popper's view, laws are conjectured in a non-logical manner on the basis of expectations and predispositions.[5] This has led David Miller, a student and collaborator of Popper, to write "the mission is to classify truths, not to certify them".[6] In contrast, the logical empiricism movement, which included such philosophers as Moritz Schlick, Rudolf Carnap, Otto Neurath and A.J. Ayer wanted to formalize the idea that, for a law to be scientific, it must be possible to argue on the basis of observations either in favor of its truth or its falsity. There was no consensus among these philosophers about how to achieve that, but the thought expressed by Mach's dictum that "where neither confirmation nor refutation is possible, science is not concerned" was accepted as a basic precept of critical reflection about science.[7][8][9]

Popper said that a demarcation criterion was possible, but we have to use the logical possibility of falsifications, which is falsifiability. He cited his encounter with psychoanalysis in the 1910s. It did not matter what observation was presented, psychoanalysis could explain it. Unfortunately, the reason why it could explain everything is that it did not exclude anything also.[J] For Popper, this was a failure, because it meant that it could not make any prediction. From a logical standpoint, if one finds an observation that does not contradict a law, it does not mean that the law is true. A verification has no value in itself. But, if the law makes risky predictions and these are corroborated, Popper says, there is a reason to prefer this law over another law that makes less risky predictions or no predictions at all.[K][L] In the definition of falsifiability, contradictions with observations are not used for actual falsifications, but for logical "falsifications" that show that the law makes risky predictions, which is completely different.

On the basic philosophical side of this issue, Popper said that some 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: a statement that could not be verified was considered meaningless. In opposition to this view, Popper said that there are meaningful theories that are not scientific, and that, accordingly, a criterion of meaningfulness does not coincide with a criterion of demarcation.[M]

The problem of induction is often called Hume's problem. David Hume studied how human beings obtain new knowledge that goes beyond known laws and observations, including how we can discover new laws. He understood that deductive logic could not explain this learning process and argued in favour of a mental or psychological process of learning that would not require deductive logic. He even argued that this learning process can not be justified by any general rules, deductive or not.[10] Popper accepted Hume's argument and therefore viewed progress in science as the result of quasi-induction, which does the same as induction, but has no inference rules to justify it.[11][12] Philip N. Johnson-Laird, professor of psychology, also accepted Hume's conclusion that induction has no justification. For him induction does not require justification and therefore can exist in the same manner as Popper's quasi-induction does.[13]

When Johnson-Laird says that no justification is needed, he does not refer to a general method of justification that, to avoid a circular reasoning, would not itself require any justification. On the contrary, in agreement with Hume, he refers to the fact that there is no general method of justification for induction and that's ok, because the induction steps do not require justification.[13] Instead, these steps use patterns of induction that may or may not be applicable depending on the background knowledge. Johnson-Laird wrote: "[P]hilosophers have worried about which properties of objects warrant inductive inferences. The answer rests on knowledge: we don’t infer that all the passengers on a plane are male because the first ten off the plane are men. We know that this observation doesn’t rule out the possibility of a woman passenger."[13] The reasoning pattern that was not applied here is enumerative induction.

Popper was interested in the overall learning process in science, to quasi-induction, which he also called the "path of science".[11] However, Popper did not show much interest in these reasoning patterns, which he globally referred to as psychologism.[14] He did not deny the possibility of some kind of psychological explanation for the learning process, especially when psychology is seen as an extension of biology, but he felt that these biological explanations were not within the scope of epistemology.[N][O] Popper proposed an evolutionary mechanism to explain the success of science,[15] which is much in line with Johnson-Laird's view that "induction is just something that animals, including human beings, do to make life possible",[13] but Popper did not consider it a part of his epistemology.[16] He wrote that his interest was mainly in the logic of science and that epistemology should be concerned with logical aspects only.[P] Instead of asking why science succeeds he considered the pragmatic problem of induction.[17] This problem is not how to justify a theory or what is the global mechanism for the success of science but only what criteria do we use to pick one theory among theories that are already conjectured. His answer to the latter question is that we pick the theory that is the most tested with the available technology.[17] By his own account, because only a negative approach was supported by logic, Popper adopted a negative methodology.[Q] The purpose of his methodology is to prevent "the policy of immunizing our theories against refutation". It also supports some "dogmatic attitude" in defending theories against criticism, because this allows the process to be more complete.[18] This negative view of science was much criticized and not only by Johnson-Laird.

In practice, some steps based on observations can be justified under assumptions, which can be very natural. For example, Bayesian inductive logic[19] is justified by theorems that make explicit assumptions. These theorems are obtained with deductive logic, not inductive logic. They are sometimes presented as steps of induction, because they refer to laws of probability, even though they do not go beyond deductive logic. This is yet a third notion of induction, which overlap with deductive logic in the following sense that it is supported by it. These deductive steps are not really inductive, but the overall process that includes the creation of assumptions is inductive in the usual sense. In a fallibilism perspective, a perspective that is widely accepted by philosophers, including Popper, every learning step only creates or reinforces an assumption—that is all what science does.

Imre Lakatos divided the problems of falsification in two categories. The first category corresponds to decisions that must be agreed upon by scientists before they can falsify a theory. The other category emerges when one tries to use falsifications and corroborations to explain progress in science. Lakatos said that there were two incorrect approaches, which he called dogmatic falsificationism and naive falsificationism. Dogmatic falsificationism ignores both types of problems, whereas naive falsificationism considers the first type only.[20] Lakatos contrasted them with sophisticated falsificationism,[21] his own improvement on Popper's solution. Popper's methodology is not (and has never been) based on one of the two incorrect approaches.[R] On the terminological side of this issue, Popper said that he never referred to his methodology as "falsificationism",[S] tended to avoid this term[T] and proposed instead the term "critical rationalism".[U]

A dogmatic falsificationist ignores that every observation is theory-impregnated. This leads to the critique that it is unclear which theory is falsified. Is it the one that is being studied or the one behind the observation?[V] This is sometimes called the 'Duhem–Quine problem'. An example is Galileo's refutation of the theory that celestial bodies are faultless crystal balls. Many considered that it was the optical theory of the telescope that was false, not the theory of celestial bodies. Another example is the theory that neutrinos are emitted in beta decays. Had they not been observed in the Cowan–Reines neutrino experiment, many would have considered that the strength of the beta-inverse reaction used to detect the neutrinos was not sufficiently high. At the time, Grover Maxwell [es] wrote, the possibility that this strength was sufficiently high was a "pious hope".[22]

A dogmatic falsificationist ignores the role of auxiliary hypotheses, which could explain the contradicting observation. For the falsification to logically occur, a ceteris paribus clause must say that no auxiliary hypothesis is responsible for the contradicting observation. Again, this leads to the critique that it cannot be told if it is the theory or the ceteris paribus clause that is false. Lakatos gives the example of the path of a planet. If the path contradicts Newton's law, we will not know if it is Newton's law that is false or the assumption that no other body influenced the path. Popper was aware that one can always find another auxiliary hypothesis,[W] though he clearly distinguished falsifiable theories such as Newton theory and unfalsifiable theories on this respect.[X]

Lakatos says that Popper's solution to these criticisms requires that one relaxes the assumption that an observation can show a theory to be false:[Y]

If a theory is falsified [in the usual sense], it is proven false; if it is falsified [in the technical sense], it may still be true.

Methodological falsificationism replaces the contradicting observation in a falsification with a "contradicting observation" accepted by convention among scientists, a convention that implies four kinds of decisions that have these respective goals: the selection of all basic statements (statements that correspond to logically possible observations), selection of the accepted basic statements among the basic statements, making statistical laws falsifiable and applying the refutation to the specific theory (instead of the ceteris paribus clause).[Z] The falsifiers and falsifications thus depend on decisions made by scientists in view of the currently accepted technology and its associated theory. So, Popper says that "Science does not rest upon solid bedrock".[AA] He also says (see section ) that it's not an obstacle to the definition of an empirical basis and of falsifiability.

According to Lakatos, naive falsificationism is the claim that methodological falsifications can by themselves explain how scientific knowledge progresses. Very often a theory is still useful and used even after it is found in contradiction with some observations. Also, when scientists deal with two or more competing theories which are both corroborated, considering only falsifications, it is not clear why one theory is chosen above the other, even when one is corroborated more often than the other. In fact, a stronger version of the Quine-Duhem thesis says that it's not always possible to rationally pick one theory over the other using falsifications.[23] Considering only falsifications, it is not clear why often a corroborating experiment is seen as a sign of progress. Popper's critical rationalism uses both falsifications and corroborations to explain progress in science.[AB] How corroborations and falsifications can explain progress in science was a subject of disagreement between many philosophers, especially between Lakatos and Popper.[AC]

Popper distinguished between the creative and informal process from which theories and accepted basic statements emerge and the logical and formal process where theories are falsified or corroborated.[AD][AE][AF] The main issue is whether the decision to select a theory among competing theories in the light of falsifications and corroborations could be justified using some kind of formal logic.[AG] It is a delicate question, because this logic would be inductive: it justifies a universal law in view of instances. Also, falsifications, because they are based on methodological decisions, are useless in a strict justification perspective. The answer of Lakatos and many others to that question is that it should.[AH][AI] In contradistinction, for Popper, the creative and informal part is guided by methodological rules, which naturally say to favour theories that are corroborated over those that are falsified,[AJ] but this methodology can hardly be made rigorous.[AK]

Popper's way to analyze progress in science was through the concept of verisimilitude, a way to define how close a theory is to the truth, which he did not consider very significant, except (as an attempt) to describe a concept already clear in practice. Later, it was shown that the specific definition proposed by Popper cannot distinguish between two theories that are false, which is the case for all theories in the history of science.[AL] Today, there is still on going research on the general concept of verisimilitude.[24]

Hume explained induction with a theory of the mind[25] that was in part inspired by Newton's theory of gravitation.[AM] Popper rejected Hume's explanation of induction and proposed his own mechanism: science progresses by trial and error within an evolutionary epistemology. Hume believed that his psychological induction process follows laws of nature, but, for him, this does not imply the existence of a method of justification based on logical rules. In fact, he argued that any induction mechanism, including the mechanism described by his theory, could not be justified logically.[26] Similarly, Popper adopted an evolutionary epistemology, which implies that some laws explain progress in science, but yet insists that the process of trial and error is hardly rigorous and that there is always an element of irrationality in the creative process of science. The absence of a method of justification is a built-in aspect of Popper's trial and error explanation.

As rational as they can be, these explanations that refer to laws, but cannot be turned into methods of justification (and thus do not contradict Hume's argument or its premises), were not sufficient for some philosophers. In particular, Russell once expressed the view that if Hume's problem cannot be solved, “there is no intellectual difference between sanity and insanity”[26] and actually proposed a method of justification.[27][28] He rejected Hume's premise that there is a need to justify any principle that is itself used to justify induction.[AN] It might seem that this premise is hard to reject, but to avoid circular reasoning we do reject it in the case of deductive logic. It makes sense to also reject this premise in the case of principles to justify induction. Lakatos' proposal of sophisticated falsificationism was very natural in that context.

Therefore, Lakatos urged Popper to find an inductive principle behind the trial and error learning process[AO] and sophisticated falsificationism was his own approach to address this challenge.[AP][AQ] Kuhn, Feyerabend, Musgrave and others mentioned and Lakatos himself acknowledged that, as a method of justification, this attempt failed, because there was no normative methodology to justify—Lakatos' methodology was anarchy in disguise.[AR][AS][AT][AU][AV]

Popper's philosophy is sometimes said to fail to recognize the Quine-Duhem thesis, which would make it a form of dogmatic falsificationism. For example, Watkins wrote "apparently forgetting that he had once said 'Duhem is right [...] ', Popper set out to devise potential falsifiers just for Newton's fundamental assumptions".[29] But, Popper's philosophy is not always qualified of falsificationism in the pejorative manner associated with dogmatic or naive falsificationism.[30] The problems of falsification are acknowledged by the falsificationists. For example, Chalmer points out that falsificationists freely admit that observation is theory impregnated.[31] Thornton, referring to Popper's methodology, says that the predictions inferred from conjectures are not directly compared with the facts simply because all observation-statements are theory-laden.[32] For the critical rationalists, the problems of falsification are not an issue, because they do not try to make experimental falsifications logical or to logically justify them, nor to use them to logically explain progress in science. Instead, their faith rests on critical discussions around these experimental falsifications.[5] Lakatos made a distinction between a "falsification" (with quotation marks) in Popper's philosophy and a falsification (without quotation marks) that can be used in a systematic methodology where rejections are justified.[33] He knew that Popper's philosophy is not and has never been about this kind of justifications, but he felt that it should have been.[AO] Sometimes, Popper and other falsificationists say that when a theory is falsified it is rejected,[34][35] which appears as dogmatic falsificationism, but the general context is always critical rationalism in which all decisions are open to critical discussions and can be revised.[36]

Popper distinguished between the logic of science and its applied methodology.[AD] The logical part consists of theories, statements and their purely logical relationship. The methodological part consists, in Popper's view, of informal rules, which are used to guess theories, accept observation statements as factual, etc. When this distinction is applied to the term "falsifiability", it corresponds to a distinction between two completely different meanings of the term. The same is true for the term "falsifiable". Popper said that he only uses "falsifiability" or "falsifiable" in reference to the logical side and that, when he refers to the methodological side, he speaks instead of "falsification" and its problems.[Y]

Popper said that methodological problems require proposing methodological rules. For example, one such rule is that, if one refuses to go along with falsifications, then one has retired oneself from the game of science.[37] The logical side does not have such methodological problems, in particular with regard to the falsifiability of a theory, because basic statements are not required to be possible. Methodological rules are only needed in the context of actual falsifications.

So observations have two purposes in Popper's view. On the methodological side, observations can be used to show that a law is false, which Popper calls falsification. On the logical side, observations, which are purely logical constructions, do not show a law to be false, but contradict a law to show its falsifiability. Unlike falsifications and free from the problems of falsification, these contradictions establish the value of the law, which may eventually be corroborated. He wrote that an entire literature exists because this distinction was not understood.[AW]

In Popper's view of science, statements of observation can be analyzed within a logical structure independently of any factual observations.[AX][AY] The set of all purely logical observations that are considered constitutes the empirical basis. Popper calls them the basic statements or test statements. They are the statements that can be used to show the falsifiability of a theory. Popper says that basic statements do not have to be possible in practice. It is sufficient that they are accepted by convention as belonging to the empirical language, a language that allows intersubjective verifiability: "they must be testable by intersubjective observation (the material requirement)".[38][AZ] See the examples in section § Examples of demarcation and applications.

In more than twelve pages of The Logic of Scientific Discovery,[39] Popper discusses informally which statements among those that are considered in the logical structure are basic statements. A logical structure uses universal classes to define laws. For example, in the law "all swans are white" the concept of swans is a universal class. It corresponds to a set of properties that every swan must have. It is not restricted to the swans that exist, existed or will exist. Informally, a basic statement is simply a statement that concerns only a finite number of specific instances in universal classes. In particular, an existential statement such as "there exists a black swan" is not a basic statement, because it is not specific about the instance. On the other hand, "this swan here is black" is a basic statement. Popper says that it is a singular existential statement or simply a singular statement. So, basic statements are singular (existential) statements.

Thornton says that basic statements are statements that correspond to particular "observation-reports". He then gives Popper's definition of falsifiability:

"A theory is scientific if and only if it divides the class of basic statements into the following two non-empty sub-classes: (a) the class of all those basic statements with which it is inconsistent, or which it prohibits—this is the class of its potential falsifiers (i.e., those statements which, if true, falsify the whole theory), and (b) the class of those basic statements with which it is consistent, or which it permits (i.e., those statements which, if true, corroborate it, or bear it out)."

As in the case of actual falsifiers, decisions must be taken by scientists to accept a logical structure and its associated empirical basis, but these are usually part of a background knowledge that scientists have in common and, often, no discussion is even necessary.[BA] The first decision described by Lakatos[40] is implicit in this agreement, but the other decisions are not needed. This agreement, if one can speak of agreement when there is not even a discussion, exists only in principle. This is where the distinction between the logical and methodological sides of science becomes important. When an actual falsifier is proposed, the technology used is considered in detail and, as described in section § Dogmatic falsificationism, an actual agreement is needed. This may require using a deeper empirical basis,[BB] hidden within the current empirical basis, to make sure that the properties or values used in the falsifier were obtained correctly (Andersson 2016 gives some examples).

Popper says that despite the fact that the empirical basis can be shaky, more comparable to a swamp than to solid ground,[BB] the definition that is given above is simply the formalization of a natural requirement on scientific theories, without which the whole logical process of science[AX] would not be possible.

In his analysis of the scientific nature of universal laws, Popper arrived at the conclusion that laws must "allow us to deduce, roughly speaking, more empirical singular statements than we can deduce from the initial conditions alone."[41] A singular statement that has one part only can not contradict a universal law. A falsifier of a law has always two parts: the initial condition and the singular statement that contradicts the prediction.

However, there is no need to require that falsifiers have two parts in the definition itself. This removes the requirement that a falsifiable statement must make prediction. In this way, the definition is more general and allows the basic statements themselves to be falsifiable.[41] Criteria that require that a law must be predictive, just as is required by falsifiability (when applied to laws), Popper wrote, "have been put forward as criteria of the meaningfulness of sentences (rather than as criteria of demarcation applicable to theoretical systems) again and again after the publication of my book, even by critics who pooh-poohed my criterion of falsifiability."[42]

In response to Lakatos who suggested that Newton's theory was as hard to show falsifiable as Freud's psychoanalytic theory, Popper gave the example of an apple that moves from the ground up to a branch and then starts to dance from one branch to another.[C] It is clearly impossible, yet a basic statement that is a valid potential falsifier for Newton's theory, because the position of the apple at different times can be measured.

Another example of a basic statement is "The inert mass of this object is ten times larger than its gravitational mass." This is a basic statement because the inert mass and the gravitational mass can both be measured separately, even though it never happens that they are different. It is, as described by Popper, a valid falsifier for Einstein's equivalence principle.[BD]

An example of a basic statement in the theory of evolution is "In this industrial area, the relative fitness of the white-bodied peppered moth is high." Here "fitness" means "reproductive success over the next generation".[BE][BF] This is an example of a basic statement, because it is possible to separately determine the kind of environment, industrial vs natural, and the relative fitness of the white-bodied form (relative to the black-bodied form) in an area, even though it never happens that the white-bodied form has a high relative fitness in an industrial area. "In industrial areas, the black form of the peppered moth has higher relative fitness (due to a better camouflage)" is a famous example of a falsifiable statement that illustrates the effect of natural selection.[46]

A famous example of a basic statement from J. B. S. Haldane is "[These are] fossil rabbits in the Precambrian era." This is a basic statement because it is possible to find a fossil rabbit and to determine that the date of a fossil is in the Precambrian era, even though it never happens that the date of a rabbit fossil is in the Precambrian era. Despite opinions to the contrary,[47] some times wrongly attributed to Popper,[BG] this shows the scientific character of paleontology or the history of the evolution of life on Earth, because it contradicts the hypothesis in paleontology that all mammals existed in a much more recent era. Richard Dawkins adds that any other modern animal, such as a hippo, would suffice.[48][49][50]

Even if it is accepted that angels exist, the sentence "All angels have large wings" is not falsifiable, because though it is possible to observe the absence of large wings, no technology (independent of the presence of wings) exists to identify these angels.

A simple example of a non-basic statement is "this angel does not have large wings". It is not a basic statement, because though the absence of large wings can be observed, no technology (independent of the presence of wings[BH]) exists to identify angels. Even if it is accepted that angels exist, the sentence "All angels have large wings" is not falsifiable.

Another example from Popper of a non-basic statement is "This human action is altruistic." It is not a basic statement, because no accepted technology allows us to determine whether or not an action is motivated by self-interest. Because no basic statement falsifies it, the statement that "All human actions are egotistic, motivated by self-interest" is thus not falsifiable.[BI]

Some adherents of young-Earth creationism make an argument (called the Omphalos hypothesis after the Greek word for navel) that the world was created with the appearance of age; e.g., the sudden appearance of a mature chicken capable of laying eggs. This ad hoc hypothesis introduced into young-Earth creationism makes it unfalsifiable because it says that the time of creation (of a species) measured by the accepted technology is illusory and no accepted technology is proposed to measure the claimed "actual" time of creation. Popper says that it's fine to modify a theory by the introduction of an auxiliary hypothesis, but the new theory must at the least remain falsifiable, which is not the case here. One can also present the Omphalos hypothesis as an auxiliary hypothesis that is introduced into the accepted theory. In this view, the new theory remains falsifiable, but its falsifiability does not increase, because no additional observations are predicted. In both views, the ad hoc hypothesis, seen by itself, is not falsifiable because there is no way to measure the claimed "actual" time of creation that is proposed by this hypothesis. This is discussed in details by Dienes in the case of a variation on the Omphalos hypothesis, which, in addition, specifies that God made the creation in this way to test our faith.[51]

Grover Maxwell [es] discussed statements such as "All men are mortal".[2] This is not falsifiable, because it does not matter how old a man is, maybe he will die next year.[3] Maxwell said that this statement is nevertheless useful, because it is often corroborated. He coined the term "corroboration without demarcation". Popper's view is that it is indeed useful, but only because it is indirectly corroborated by the corroboration of the falsifiable law "All men die before the age of 150." For Popper, if no such a falsifiable law exists, then the metaphysical law is not useful, because it's not indirectly corroborated.[BJ] This kind of non-falsifiable statements in science was noticed by Carnap as early as 1937.[52]

Maxwell also used the example "All solids have a melting point." This is not falsifiable, because maybe the melting point will be reached at a higher temperature.[2][3] The law is falsifiable and more useful if we specify an upper bound on melting points or a way to calculate this upper bound.

Another example from Maxwell is "All beta decays are accompanied with a neutrino emission from the same nucleus." This is also not falsifiable, because maybe the neutrino can be detected in a different manner. The law is falsifiable and much more useful from a scientific point of view, if the method to detect the neutrino is specified.[22] Maxwell said that most scientific laws are metaphysical statements of this kind,[53] which, Popper said, need to be made more precise before they can be indirectly corroborated.[BJ] In other words, specific technologies must be provided to make the statements inter-subjectively-verifiable, i.e., so that scientists know what the falsification or its failure actually means.

In his critique of the falsifiability criterion, Maxwell considered the requirement for decisions in the falsification of, both, the emission of neutrinos (see § Dogmatic falsificationism) and the existence of the melting point.[22] For example, he pointed out that had no neutrino been detected, it could have been because some conservation law is false. Popper did not argue against the problems of falsification per se. He always acknowledged these problems. Popper's response was at the logical level. For example, he pointed out that, if a specific way is given to trap the neutrino, then, at the level of the language, the statement is falsifiable, because "no neutrino was detected after using this specific way" formally contradicts it (and it is inter-subjectively-verifiable—people can repeat the experiment).

Another example, from the pepper moth example, is "In all areas, the white vs black trait of the pepper moth affects its fitness." This is also not falsifiable, because maybe the right environmental factor was not yet considered. When it is specified, namely, fitness in polluted industrial areas vs non polluted areas, then the law is falsifiable and it says which environmental factor should be considered to actually see an effect.[BL]

In the 5th and 6th editions of On the origin of species, following a suggestion of Alfred Russel Wallace, Darwin used "Survival of the fittest", an expression first coined by Herbert Spencer, as a synonym for "Natural Selection".[BM] Popper and others said that, if one uses the most widely accepted definition of "fitness" in modern biology (see subsection § Evolution), namely reproductive success itself, the expression "survival of the fittest" is a tautology.[BN][BO][BP]

In practice, as illustrated by the peppered moth example of section § Evolution, the questions asked are of the kind how specific traits affect the survival rate or fitness of a species when confronted by an environmental factor such as industrial pollution. Great Darwinist Ronald Fisher worked out mathematical theorems to help answer this kind of questions. But, for Popper and others, there is no (falsifiable) law of Natural Selection in this, because it only applies to some rare traits.[BQ][BR] Instead, for Popper, the work of Fisher and others on Natural Selection is part of an important metaphysical research program.

Popper said that not all unfalsifiable statements are useless in science. Mathematical statements are good examples. Like all formal sciences, mathematics is not concerned with the validity of theories based on observations in the empirical world, but rather, mathematics is occupied with the theoretical, abstract study of such topics as quantity, structure, space and change. Methods of the mathematical sciences are, however, applied in constructing and testing scientific models dealing with observable reality. Albert Einstein wrote, "One reason why mathematics enjoys special esteem, above all other sciences, is that its laws are absolutely certain and indisputable, while those of other sciences are to some extent debatable and in constant danger of being overthrown by newly discovered facts."[54]

Popper made a clear distinction between the original theory of Marx and what came to be known as Marxism later on.[55] For Popper, the original theory of Marx contained genuine scientific laws. Though they could not make preordained predictions, these laws constrained how changes can occur in society. One of them was that changes in society cannot "be achieved by the use of legal or political means".[BS] For Popper, this was testable, and in fact falsified. "Yet instead of accepting the refutations", Popper wrote, "the followers of Marx re-interpreted both the theory and the evidence in order to make them agree. ... They thus gave a 'conventionalist twist' to the theory; and by this stratagem they destroyed its much advertised claim to scientific status."[BT][BU] Popper's attacks was not directed toward Marxism, or Marx's theories, which were falsifiable, but toward Marxists who ignored the falsifications which had happened.[56] Popper more fundamentally criticized 'historicism' in the sense of any preordained prediction of history, given what he saw as our right, ability and responsibility to control our own destiny.[56]

Falsifiability has been used in the McLean v. Arkansas case (in 1982),[57] the Daubert case (in 1993)[58] and other cases. A survey of 303 federal judges conducted in 1998[BV] found that "[P]roblems with the nonfalsifiable nature of an expert's underlying theory and difficulties with an unknown or too-large error rate were cited in less than 2% of cases."[59]

In the ruling of the McLean v. Arkansas case, Judge William Overton used falsifiability as one of the criteria to determine that "creation science" was not scientific and should not be taught in Arkansas public schools as such (it can be taught as religion). In his testimony, philosopher Michael Ruse defined the characteristics which constitute science as (see Pennock 2000, p. 5 and Ruse 2010):

In his conclusion related to this criterion Judge Overton stated that

While anybody is free to approach a scientific inquiry in any fashion they choose, they cannot properly describe the methodology as scientific, if they start with the conclusion and refuse to change it regardless of the evidence developed during the course of the investigation.

William Overton, McLean v. Arkansas 1982, at the end of section IV. (C)

In several cases of the United States Supreme Court, the court described scientific methodology using the five Daubert factors, which include falsifiability.[BW] The Daubert result cited Popper and other philosophers of science:

Ordinarily, a key question to be answered in determining whether a theory or technique is scientific knowledge that will assist the trier of fact will be whether it can be (and has been) tested. Green 645. See also C. Hempel, Philosophy of Natural Science 49 (1966) (); K. Popper, Conjectures and Refutations: The Growth of Scientific Knowledge 37 (5th ed. 1989) () (emphasis deleted).

Scientific methodology today is based on generating hypotheses and testing them to see if they can be falsified; indeed, this methodology is what distinguishes science from other fields of human inquiry.[T]he statements constituting a scientific explanation must be capable of empirical test[T]he criterion of the scientific status of a theory is its falsifiability, or refutability, or testability

David H. Kaye[BX] said that references to the Daubert majority opinion confused falsifiability and falsification and that "inquiring into the existence of meaningful attempts at falsification is an appropriate and crucial consideration in admissibility determinations."[BY]

Considering the specific detection procedure that was used in the neutrino experiment, without mentioning its probabilistic aspect, Popper wrote "it provided a test of the much more significant falsifiable theory that such emitted neutrinos could be trapped in a certain way". In this manner, in his discussion of the neutrino experiment, Popper did not raise at all the probabilistic aspect of the experiment.[60] Together with Maxwell, who raised the problems of falsification in the experiment,[22] he was aware that some convention must be adopted to fix what it means to detect or not a neutrino in this probabilistic context. This is the third kind of decisions mentioned by Lakatos.[61] For Popper and most philosophers, observations are theory impregnated. In this example, the theory that impregnates observations (and justifies that we conventionally accept the potential falsifier "no neutrino was detected") is statistical. In statistical language, the potential falsifier that can be statistically accepted (not rejected to say it more correctly) is typically the null hypothesis, as understood even in popular accounts on falsifiability.[62][63][64]

Different ways are used by statisticians to draw conclusions about hypotheses on the basis of available evidence. Fisher, Neyman and Pearson proposed approaches that require no prior probabilities on the hypotheses that are being studied. In contrast, Bayesian inference emphasizes the importance of prior probabilities.[65] But, as far as falsification as a yes/no procedure in Popper's methodology is concerned, any approach that provides a way to accept or not a potential falsifier can be used, including approaches that use Bayes's theorem and estimations of prior probabilities that are made using critical discussions and reasonable assumptions taken from the background knowledge.[BZ] There is no general rule that considers has falsified an hypothesis with small Bayesian revised probability, because as pointed out by Mayo and argued before by Popper, the individual outcomes described in detail will easily have very small probabilities under available evidence without being genuine anomalies.[66] Nevertheless, Mayo adds, "they can indirectly falsify hypotheses by adding a methodological falsification rule".[66] In general, Bayesian statistic can play a role in critical rationalism in the context of inductive logic,[67] which is said to be inductive because implications are generalized to conditional probabilities.[68] According to Popper and other philosophers such as Colin Howson, Hume's argument precludes inductive logic, but only when the logic makes no use "of additional assumptions: in particular, about what is to be assigned positive prior probability".[69] Inductive logic itself is not precluded, especially not when it is a deductively valid application of Bayes' theorem that is used to evaluate the probabilities of the hypotheses using the observed data and what is assumed about the priors. Gelman and Shalizi mentioned that Bayes' statisticians do not have to disagree with the non-inductivists.[70]

Because statisticians often associate statistical inference with induction, Popper's philosophy is often said to have an hidden form of induction. For example, Mayo wrote "The falsifying hypotheses [...] necessitate an evidence-transcending (inductive) statistical inference. This is hugely problematic for Popper".[71] Yet, also according to Mayo, Popper [as a non-inductivist] acknowledged the useful role of statistical inference in the falsification problems: she mentioned that Popper wrote her (in the context of falsification based on evidence) "I regret not studying statistics" and that her thought was then "not as much as I do".[72]

For Popper, the problems of falsification belong to the inductive perspective, which he also calls the bucket view of science,[73] and the correct application of falsifiability, i.e., his methodology, is as free from the problems of falsification as falsifiability itself, because it relies on a different perspective, the searchlight view of science.[74] Both views incorporate some unknown aspect in its explanation of progress in science. In the inductive or bucket view of science, this unknown aspect takes the form of uncertainty or lack of universality in the inductive logic. In the searchlight view of science, what is unknown is the influence of biological expectations and predispositions on the conjectures. Popper describes these biological expectations and predispositions as knowledge that has not taken (and perhaps cannot fully take) an objective form and in as such cannot participate in any logic that scientists can use.

Bucket view: Observations enter into the bucket and turn into valid statements. Next, (not shown) inference rules generate valid laws.

In the bucket view of science, observations are the basis to justify laws or theories. In this view, observation statements accumulate in a bucket through observations and various procedures are used to make sure that they are valid, so that they can fulfill their purpose.[75] These observation statements are used as evidence to justify new laws through inference rules. This justificational picture was criticized by Hume on the basis of reasonable premises: non-deductive rules are in need of justification, circular arguments are not valid, etc. If we accept Hume's premises, even probabilistic attempts to explain the growth of knowledge in terms of the bucket view of science, Popper stated, are doomed to fail.[76]

Popper argued that an even greater weakness of this view is the level of certainty that it expects from the observation statements as falsifiers (and even as verifiers) in a logical inductive process.[77] The potential falsifiers, which play perfectly their role to show falsifiability or to steer creativity and critical discussions in the case of falsifications, are much less useful in this justificational perspective, because of all the problems of falsifications.

Popper's solution to this problem is simply to reject the bucket view of science. His main argument is basically that he accepts Hume's argument, which shows that the bucket view fails to explain the growth of objective knowledge. Popper said that the processes in the bucket are better seen as physical processes and the laws that govern these processes are biological.[CA] To help people get rid of the limitations associated with the bucket view, Popper brought out the main issue with this view: it ignores the organismic aspect of knowledge.[CB]

Searchlight view: Expectations and predispositions turn into conjectures that act like a searchlight and lead to observations (not shown).

Popper proposed to replace the bucket view of science with what he called the searchlight view of science. In that view, Popper wrote, there is no reason why any methodology should work. It is easy, Popper said, to imagine universes where no methodology can work or even only exist.[CC] If one wants to believe that the methodology will work, it must be postulated as an axiom. In Popper's case, the axiom is that the methodology of conjectures and refutations is going to work.[CD] The conjectures are the searchlight, because they lead to observational results. But this axiom will not help any objective rule in the justification of scientific knowledge.[CE] There is no point in attempting any justification in the searchlight view. For a popperian, the absence of these objective rules is expected. It is not a failure. In this line of thought, Einstein wrote that there is no logical path to science.[CF][CG]

Popper's scientific methodology that accompanies falsifiability contains rules such as "He who decides one day that scientific statements do not call for any further test, and that they can be regarded as finally verified, retires from the game."[37] In general, the rules of Popper's methodology influence which theories will be chosen or rejected, but these rules do that only through decisions taken by the scientists.[CH] As described in , every rule to determine or choose theories must rely on the good judgement of the scientists.

The usefulness of falsifiability is that falsifiable conjectures say more, because they prohibit more and, in the case of their falsification, they lead to useful problems, which steer the creative process of science. For Popper, who knew most of section § Examples of demarcation and applications, this is exactly what we should expect from a scientific theory.

In , it is seen that Lakatos reached the same conclusion in the following sense that he said that his methodology did not offer any "firm heuristic advice about what to do".[AR] Before Popper's time, in 1906, being aware of the problems of falsification, Pierre Duhem reached the same conclusion.[CI] Popper reemphasized non-justificationism, which was a good match for his added falsifiability criterion and associated critical methodology.

As described in section § Naive falsificationism, Lakatos and Popper agreed that universal laws cannot be logically deduced (except from laws that say even more). But unlike Popper, Lakatos felt that if the explanation for new laws can not be deductive, it must be inductive. He urged Popper explicitly to adopt some inductive principle[AO] and sets himself the task to find an inductive methodology.[CJ] However, the methodology that he found did not offer any exact inductive rules. In a response to Kuhn, Feyerabend and Musgrave, Lakatos acknowledged that the methodology depends on the good judgment of the scientists.[AR] Feyerabend wrote in "Against Method" that Lakatos' methodology of scientific research programmes is epistemological anarchism in disguise[AS] and Musgrave made a similar comment.[AT] In more recent work, Feyerabend says that Lakatos uses rules, but whether or not to follow any of these rules is left to the judgment of the scientists.[AU] This is also discussed elsewhere.[AV]

Popper also offered a methodology with rules, but these rules are also not inductive rules, because they are not by themselves used to accept laws or establish their validity. They do that through the creativity or "good judgment" of the scientists only. For Popper, the required non deductive component of science never had to be an inductive methodology. He always viewed this component as a creative process beyond the explanatory reach of any rational methodology, but yet used to decide which theories should be studied and applied, find good problems and guess useful conjectures.[CK] Quoting Einstein to support his view, Popper said that this renders obsolete the need for an inductive methodology or logical path to the laws.[CL][CF][CG] For Popper, no inductive methodology was ever proposed to satisfactorily explain science.

Section says that both Lakatos's and Popper's methodology are not inductive. Yet Lakatos's methodology extended importantly Popper's methodology: it added a historiographical component to it. This allowed Lakatos to find corroborations for his methodology in the history of science. The basic units in his methodology, which can be abandoned or pursued, are research programmes. Research programmes can be degenerative or progressive and only degenerative research programmes must be abandoned at some point. For Lakatos, this is mostly corroborated by facts in history.

In contradistinction, Popper did not propose his methodology as a tool to reconstruct the history of science. Yet, some times, he did refer to history to corroborate his methodology. For example, he remarked that theories that were considered great successes were also the most likely to be falsified. Zahar's view was that, with regard to corroborations found in the history of science, there was only a difference of emphasis between Popper and Lakatos.

As an anecdotal example, in one of his articles Lakatos challenged Popper to show that his theory was falsifiable: he asked "Under what conditions would you give up your demarcation criterion?".[78] Popper replied "I shall give up my theory if Professor Lakatos succeeds in showing that Newton's theory is no more falsifiable by 'observable states of affairs' than is Freud's."[79]

Thomas Kuhn analyzed what he calls periods of normal science as well as revolutions from one period of normal science to another,[80] whereas Popper's view is that only revolutions are relevant.[CM][CN] For Popper, the role of science, mathematics and metaphysics, actually the role of any knowledge, is to solve puzzles.[CO] In the same line of thought, Kuhn observes that in periods of normal science the scientific theories, which represent some paradigm, are used to routinely solve puzzles and the validity of the paradigm is hardly in question. It's only when important new puzzles emerge that cannot be solved by accepted theories that a revolution might occur. This can be seen as a viewpoint on the distinction made by Popper between the informal and formal process in science (see section § Naive falsificationism). In the big picture presented by Kuhn, the routinely solved puzzles are corroborations. Falsifications or otherwise unexplained observations are unsolved puzzles. All of these are used in the informal process that generates a new kind of theory. Kuhn says that Popper emphasizes formal or logical falsifications and fails to explain how the social and informal process works.

Popper often uses astrology as an example of a pseudo-science. He says that it is not falsifiable because both the theory itself and its predictions are too imprecise.[CP] Kuhn, as an historian of science, remarked that many predictions made by astrologers in the past were quite precise and they were very often falsified. He also said that astrologers themselves acknowledged these falsifications.[CQ]

Paul Feyerabend rejected any prescriptive methodology at all. He rejected Lakatos' argument for ad hoc hypothesis, arguing that science would not have progressed without making use of any and all available methods to support new theories. He rejected any reliance on a scientific method, along with any special authority for science that might derive from such a method.[81] He said that if one is keen to have a universally valid methodological rule, epistemological anarchism or anything goes would be the only candidate.[82] For Feyerabend, any special status that science might have, derives from the social and physical value of the results of science rather than its method.[83]

In their book Fashionable Nonsense (from 1997, published in the UK as Intellectual Impostures) the physicists Alan Sokal and Jean Bricmont criticised falsifiability.[84] They include this critique in the "Intermezzo" chapter, where they expose their own views on truth in contrast to the extreme epistemological relativism of postmodernism. Even though Popper is clearly not a relativist, Sokal and Bricmont discuss falsifiability because they see postmodernist epistemological relativism as a reaction to Popper's description of falsifiability, and more generally, to his theory of science.[85]