An anomalous phenomenon is an observed event which deviates from what is expected (an anomaly) according to existing rules or scientific theory. Sometimes the anomalous phenomenon is expected, but the reason for the deviation is unclear (See section on anomalies in science). The scientific study of the field is called anomalistics, though it has been (incorrectly) associated with pseudoscience.
Anomalous phenomena as falsification of a theory
In work beginning in the 1930s, noted philosopher Karl Popper gave falsifiability a renewed emphasis as a criterion of empirical statements in science.  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 which is a swan, and x is white.
The second type of statement of interest to scientists categorizes all instances of something, for example '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 the second type. Perhaps the most 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 somehow avoided observation. Yet some philosophers of science claim that science is based on such an inductive method.
Popper held that science could not be grounded on such an invalid inference. He proposed falsification as a solution to the problem of induction.  Popper noticed that although a singular existential statement such as 'there is a white swan' cannot be used to affirm a universal statement, it can be used to show that one is false: the singular existential observation of a black swan serves to show that the universal statement 'all swans are white' is false - in logic this is called modus tollens. 'There is a black swan' implies 'there is a non-white swan' which in turn implies 'there is something which is a swan and which is not white', hence 'all swans are white' is false, because that is the same as 'there is nothing which is a swan and which is not white'.
One notices a white swan. From this one can conclude: At least one swan is white. From this, one may wish to infer that: All swans are white. It may be impossible 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 not be true; it would be falsified by the counterexample of the single black swan.
The falsification of statements occurs through modus tollens, via some observation. Suppose some universal statement U implies an observation O:
U → O
An observation conflicting with O, however, is made:
So by modus tollens,
One anomalous phenomenon
A statement is only complete insofar as it accurately describes something free from anomalies. As in the example where the truth of the statement 'all swans are white' is falsified by the counterexample of the single black swan, any theory is shown to be falsified by a verified singular anomaly. For example, the statement 'dinosaurs are extinct' would be falsified by the discovery of just one remaining dinosaur. The burden of cryptozoologists, then, would be to find a single example of a cryptid to disprove the statement 'cryptids do not exist'. Of course, as cryptozoologist Loren Coleman notes, every time a cryptid is "discovered" (e.g. giant panda, mountain gorilla, okapi, coelacanth, megamouth shark, saola) then that species becomes part of zoology, and not cryptozoology.
Reversely, falsification is why various fields that pursue anomalous phenomena are often seen as not being worthwhile in mainstream science, or, by extreme skeptics, as pseudoscientific. If the aim of science is to move observations to laws, or to weed out singular existential statements in favor of universal statements through testing against falsifying propositions, anomalies suffer from a missing component of the scientific method. A falsifying proposition of anomalies as deviations from the norm would be the norm. Testing for the norm is seen as redundant.
Although the logic of 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 anticipated criticism from Duhem and Carnap.  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, 'the bird watchers are incompetent'. Falsification does not enable scientists to present a definitive falsification of universal statements.
These 'compensatory adjustments' have been noticed in criticisms of James Randi's One Million Dollar Paranormal Challenge. The challenge offers the proposition that if one can show a single verifiable anomalous phenomenon (termed paranormal in the challenge), then one will be awarded the prize of one million dollars. To date no one has been able to do so. Some of Randi's detractors claim that the challenge is insincere, and that Randi will ensure he never has to pay out. The October 1981 issue of Fate magazine, quoted him as saying "I always have an out". Although some critics interpret this to mean he will never let his organization lose such a challenge, the "out" could easily be a variation on Popper's idea that it is always possible to change the universal statement or the existential statement so that falsification does not occur. A statement about the debatable existence of psychic phenomena, for example, can be changed from 'psychic phenomena do not exist' to 'psychic phenomena do not exist in the laboratory under these conditions', and so on. It should be noted that Randi claims that the phrase "I always have an out" refers to the fact that he does not allow test subjects to cheat.
Anomalies in science
The occurrence of anomalies in science are often expected, but why the event deviates from what should occur is not always clear. For example, in the Pioneer anomaly, observations have pointed to a deviation in the expected trajectories of various spin-stabilised unmanned spacecraft visiting the outer solar system, notably number 10 and 11 of the Pioneer program. After several observations of the effect, one can reasonably expect the deviation to occur. However, when all known forces acting on the spacecraft are taken into consideration, a very small but unexplained force remains. It causes a constant sunward acceleration of (8.74 ± 1.33) × 10−10 m/s² for both spacecraft. The reason for this anomaly remains unknown beyond speculation.
Known anomalies in science
- In astronomy (and also in astrodynamics), the anomaly is the angular distance of a planet from its periapsis:
- * The true anomaly is observed from the primary (I.e., from the orbit's focus);
- * The eccentric anomaly is observed from the centre of the ellipse, using a projection of the planet's position onto the circumscribing circle; and
- The mean anomaly is the fraction of an orbital period that has elapsed, expressed as an angle.
- In an anomaly time series the times are not uniformly spaced.
- In physics, an anomaly is a classical symmetry that is not a symmetry of the quantum theory. See anomaly (physics) and conformal anomaly, chiral anomaly, gravitational anomaly, gauge anomaly, mixed anomaly, global anomaly, parity anomaly.
- The Pioneer anomaly is the observed deviation of the trajectories of various spin-stabilised unmanned spacecraft, notably number 10 and 11 of the Pioneer program.
Main article: Forteana
Charles Fort (1874 – 1932) is perhaps the best known researcher of unexplained anomalous phenomena. Fort is said to have compiled as many as 40,000 notes on unexplained phenomena, though there were no doubt many more than this. These notes came from what he called "the orthodox conventionality of Science", which were odd events originally printed in respected mainstream scientific journals or newspapers such as Scientific American, The Times, Nature and Science. From these researches Fort wrote seven books, though only four survive. These are: The Book of the Damned (1919), New Lands (1923), Lo! (1931) and Wild Talents (1932); one book was written between New Lands and Lo! but it was abandoned and absorbed into Lo!.
Collectively these phenomena are now referred to as 'Fortean' phenomena, or Forteana. Reported events include teleportation (a term Fort is generally credited with coining); poltergeist events, falls of frogs, fishes, inorganic materials of an amazing range; crop circles; unaccountable noises and explosions; spontaneous fires; levitation; ball lightning (a term explicitly used by Fort); unidentified flying objects; mysterious appearances and disappearances; giant wheels of light in the oceans; entombed animals; and animals found outside their normal ranges (see phantom cat). He offered many reports of OOPArts, abbreviation for "out of place" artifacts: strange items found in unlikely locations. He also is perhaps the first person to explain strange human appearances and disappearances by the hypothesis of alien abduction, and was an early proponent of the extraterrestrial hypothesis.
There are many phenomena in Fort's works which have now been partially or entirely "recuperated" by mainstream science — ball lightning, for example, was largely rejected as impossible by the scientific consensus of Fort's day, but is now generally recognized as a genuine phenomenon. However, many of Fort's ideas remain on the very borderlines of "mainstream science". This is unsurprising, as Fort resolutely refused to abandon the territory beyond "acceptable" science.
Main article: Paranormal phenomena
Purported phenomena with explanations considered to be outside the scope of conventional science can be classified as paranormal phenomena. Because these anomalies are difficult to explain in terms of science, their existence is often challenged by skeptics. Even when there is a widespread belief that the anomaly is real, explanations for the anomaly are subjected to frequent debate and controversy.
Within the umbrella term of paranormal, there are subclasses of phenomena as well. For example, phenomena studied in the limited field of parapsychology would be a subclass, which in turn can be divided into three classes of its own:
- Mental phenomena: often described as extrasensory perception, this class includes unusual mental states or abilities, such as telepathy, clairvoyance and precognition.
- Physical phenomena: this class includes unusual physical occurrences, such as psychokinesis, poltergeists, stigmata, paranormal vanishing, reality shifts or materializations.
- Survival phenomena: this class deals with the survival of consciousness after physical death and includes ghosts, hauntings, out-of-body experiences (OBEs), reincarnation and near-death experiences (NDEs).
Phenomena considered outside the scope of parapsychology, but possibly within the scope of paranormal includes:
To be classified as paranormal, a phenomenon must lack a scientific explanation. When an anomaly receives a valid scientific explanation, it becomes "perinormal." Perinormal is a term that has been suggested to describe previously unknown forces which at first appeared to be paranormal and were later verified scientifically. For example, while the idea of stones falling from the sky was once considered anomalous, meteorites are now acknowledged and well understood.
- James E. Alcock, Behavioral and Brain Sciences (1998), 21: 303-303 Cambridge University Press
- Hacohen, Malachi Haim (2002). Karl Popper - The Formative Years, 1902-1945: Politics and Philosophy in Interwar Vienna. Cambridge, UK: Cambridge University Press, 206-208. ISBN 0-521-89055-1.
- Popper, Karl Raimund (2002). The Logic of Scientific Discovery. Cambridge, UK: Routledge, 3-25. ISBN 0-415-27843-0.
- Esfeld, Michael (2001). Holism in Philosophy of Mind and Philosophy of Physics. Boston: Kluwer Academic Publishers. ISBN 0-7923-7003-1.
- Rawlins, Dennis (October 1981). ""sTARBABY"". FATE Magazine (No. 34). Reprinted in "sTARBABY". Centre Universitaire de Recherche en Astrologie (The International Astrology Research Center). Retrieved on 2006-09-09.
- Mike Hutchinson (2000-01-22). "Geller: the "after bending" effect". sci.skeptic. (Google Groups). Retrieved on 2006-11-28.
- Classical civilization included unique signs and prodigies of nature in works of paradoxography such as The Phaenomena (240 BC) by Aratus of Soli.
- Inoue Enryo, a Japanese educator and philosopher, authored the six-volume The Study of Yōkai (妖怪学). As a result, he was best known as Dr. Ghost (お化け博士) or Dr. Yōkai (妖怪博士).
- Charles Fort, in his four works on anomalies (The Book of the Damned, New Lands, Lo!, and Wild Talents) lambasted and ridiculed the scientists of his day for their shortsightedness. Some of the anomalies listed in his work have been explained and incorporated into modern science, while others continue to be unexplained.
- International Fortean Organization, successor to the original Fortean Society, sponsors the annual FortFest, the world's first, most prestigious and longest-running conference on anomalous phenomena.
- William R. Corliss' Science Frontiers has covered reports in the scientific literature regarding anomalies for years. He, through his Sourcebook Project, has published a large body of reports collected in many of the scientific disciplines.
- Leonard George, a psychologist who specializes in anomalous phenomena, compiled an authoritative encyclopedia of unusual experiences, activities, and beliefs in his 1995 book Alternative Realities.
- Fortean Times, a British monthly magazine, continues in the spirit of Fort's work by publishing reports of anomalous phenomena and longer investigative articles.
- The Anomalist, edited by Patrick Huyghe and Dennis Stacy, is another magazine/journal devoted to the study of anomalies (which may be called anomalistics).
- Strange Magazine is another magazine devoted to the study of anomalies in the spirit of Fort's work.
- Fate magazine, with the slogan "True Reports of the Strange and Unknown" has been published continuously since 1948, and is the longest-running publication of its kind.
- The Society for Psychical Research - Founded in London in 1882
- Fort, Charles (1975). The Complete Books of Charles Fort (paperback), Dover Publications, 1126 pages. ISBN 0-486-23094-5.
- Corliss, William R. (1983). Handbook of Unusual Natural Phenomena: Eyewitness Accounts of Nature's Greatest Mysteries (paperback), Anchor Press, Doubleday, 423 pages. ISBN 0-385-14754-6.
- Rickard, Bob; Michell, John (2000). The Rough Guide to Unexplained Phenomena (paperback), Rough Guides, 400 pages. ISBN 1-85828-589-5.
- Dash, Mike (1997). Borderlands: The Ultimate Exploration of the Unknown (paperback), Dell, 544 pages. ISBN 0-440-23656-8.
- Remote Viewing Secrets, Joseph McMoneagle, Hampton Roads, 2000, paperback, 296 pages, ISBN 1-57174-159-3
- The Conscious Universe, Dean Radin PhD, Harper, 1997, hardback, 362 pages, ISBN 0-06-251502-0
- Channeling: Investigations on Receiving Information from Paranormal Sources. John Klimo, St. Martins Press, 1987. ISBN 0-87477-431-4
- Mr. Wilson's Cabinet Of Wonder: Pronged Ants, Horned Humans, Mice on Toast, and Other Marvels of Jurassic Technology, Lawrence Weschler, 1996, trade paperback, 192 pages, ISBN 0-679-76489-5 (see web site link above)
- Patrick Grim, ed. Philosophy of Science and the Occult, 1982, SUNY, ISBN 0-87395-573-0