In
his article, Miller discussed a coincidence between the limits of
one-dimensional absolute judgment and the limits of short-term memory.
In a one-dimensional absolute-judgment task, a person is presented with a
number of stimuli that vary on one dimension (e.g., 10 different tones
varying only in pitch) and responds to each stimulus with a
corresponding response (learned before). Performance is nearly perfect
up to five or six different stimuli but declines as the number of
different stimuli increases. The task can be described as one of
information transmission: The input consists of one out of n possible stimuli, and the output consists of one out of n
responses. The information contained in the input can be determined by
the number of binary decisions that need to be made to arrive at the
selected stimulus, and the same holds for the response. Therefore,
people's maximum performance on a one-dimensional absolute judgment can
be characterized as an information channel capacity with approximately 2
to 3 bits of information, which corresponds to the ability to distinguish between four and eight alternatives.
The second cognitive limitation Miller discusses is memory span.
Memory span refers to the longest list of items (e.g., digits, letters,
words) that a person can repeat back in the correct order on 50% of
trials immediately after the presentation. Miller observed that the
memory span of young adults is approximately seven items. He noticed
that memory span is approximately the same for stimuli with vastly
different amounts of information—for instance, binary digits have 1 bit
each; decimal digits have 3.32 bits each; words have about 10 bits each.
Miller concluded that memory span is not limited in terms of bits but
rather in terms of chunks.
A chunk is the largest meaningful unit in the presented material that
the person recognizes—thus, what counts as a chunk depends on the
knowledge of the person being tested. For instance, a word is a single
chunk for a speaker of the language but is many chunks for someone who
is totally unfamiliar with the language and sees the word as a
collection of phonetic segments.
Miller recognized that the correspondence between the limits of
one-dimensional absolute judgment and of short-term memory span was only
a coincidence, because only the first limit, not the second, can be
characterized in information-theoretic terms (i.e., as a roughly
constant number of bits). Therefore, there is nothing "magical" about
the number seven, and Miller used the expression only rhetorically.
Nevertheless, the idea of a "magical number 7" inspired much theorizing,
rigorous and less rigorous, about the capacity limits of human
cognition. The number seven constitutes a useful heuristic, reminding us
that lists that are much longer than that become significantly harder
to remember and process simultaneously.
The "magical number 7" and working memory capacity
Later research on short-term memory and working memory
revealed that memory span is not a constant even when measured in a
number of chunks. The number of chunks a human can recall immediately
after presentation depends on the category of chunks used (e.g., span is
around seven for digits, around six for letters, and around five for
words), and even on features of the chunks
within a category. Chunking is used by the brain's short-term memory as
a method for keeping groups of information accessible for easy recall.
It functions and works best as labels that one is already familiar
with—the incorporation of new information into a label that is already
well rehearsed into one's long-term memory. These chunks must store the
information in such a way that they can be disassembled into the
necessary data.
The storage capacity is dependent on the information being
stored. For instance, span is lower for long words than it is for short
words. In general, memory span for verbal contents (digits, letters,
words, etc.) strongly depends on the time it takes to speak the contents
aloud. Some researchers have therefore proposed that the limited
capacity of short-term memory for verbal material is not a "magic
number" but rather a "magic spell," i.e. a period of time. Baddeley used this finding to postulate that one component of his model of working memory, the phonological loop, is capable of holding around 2 seconds of sound. However, the limit of short-term memory cannot easily be characterized
as a constant "magic spell" either, because memory span also depends on
other factors besides speaking duration. For instance, span depends on
the lexical status of the contents (i.e., whether the contents are words
known to the person or not). Several other factors also affect a person's measured span, and
therefore it is difficult to pin down the capacity of short-term or
working memory to a number of chunks. Nonetheless, Cowan has proposed
that working memory has a capacity of about four chunks in young adults
(and less in children and older adults).
Tarnow finds that in a classic experiment typically argued as
supporting a 4 item buffer by Murdock, there is in fact no evidence for
such and thus the "magical number", at least in the Murdock experiment,
is 1. Other prominent theories of short-term memory capacity argue against measuring capacity in terms of a fixed number of elements.
Other cognitive numeric limits
Cowan also noted a number of other limits of cognition that point to a "magical number four", and different from Miller, he argued that this correspondence is no
coincidence. One other process that seems to be limited at about four
elements is subitizing,
the rapid enumeration of small numbers of objects. When a number of
objects are flashed briefly, their number can be determined very
quickly, at a glance, when the number does not exceed the subitizing
limit, which is about four objects. Larger numbers of objects must be
counted, which is a slower process.
The film 1988 Rain Man portrayed an autistic savant, who was able to rapidly determine the number of toothpicks
from an entire box spilled on the floor, apparently subitizing a much
larger number than four objects. A similar feat was informally observed
by neuropsychologistOliver Sacks and reported in his book 1985 The Man Who Mistook His Wife for a Hat. Therefore, one might suppose that this limit is an arbitrary limit imposed by our cognition rather than necessarily being a physical limit. Autism expert Daniel Tammet has suggested, however, that the children Sacks observed may have pre-counted the matches in the box. There is also evidence that even four chunks is a high estimate: Gobet and Clarkson at Brunel University London conducted an experiment and found that over half of the memory recall conditions yielded only about two chunks. Research also shows that the size, rather than the number, of chunks
that are stored in short-term memory is what allows for enhanced memory
in individuals.
Here are two black swans,
but even with no black swans, "All swans are white" would still be
shown falsifiable by "Here is a black swan"—it would still be a valid
observation statement in the empirical language, even if empirically
false.
Falsifiability is a standard of evaluation of scientific theories and hypotheses. A hypothesis is falsifiable if it belongs to a language or logical structure capable of describing an empirical observation that contradicts it. It was introduced by the philosopher of scienceKarl Popper in his book The Logic of Scientific Discovery (1934). Popper emphasized that the contradiction is to be found in the logical
structure alone, without having to worry about methodological
considerations external to this structure. He proposed falsifiability as
the cornerstone solution to both the problem of induction and the problem of demarcation.
Popper also emphasized the related asymmetry created by the relation of a universal law with basic observation statements and contrasted falsifiability with the intuitively similar concept of verifiability that was then current in the philosophical discipline of logical positivism.
He argued that the only way to verify a claim such as "All swans are
white" would be if one could empirically observe all swans, which is not
possible. On the other hand, the observation of a single black swan is
enough to refute this claim.
This asymmetry can only be seen rigorously when methodological
falsification issues are put aside. Otherwise, a stated observation of
one or even more black swans constitute at best a problematic refutation
of the claim. Accordingly, to be rigorous, falsifiability is a
logical criterion within an empirical language that is accepted by
convention and allows these methodological considerations to be avoided.
Only then the asymmetry and falsifiability are rigorous. Popper argued
that it should not be conflated with falsificationism, which is a
methodological approach where scientists actively try to find evidence
to disprove theories. Falsifiability is distinct from Lakatos' falsificationism. Its purpose is to make theory predictive, testable and useful in practice.
By contrast, the Duhem–Quine thesis says that definitive experimental falsifications are impossible and that no scientific hypothesis is by itself capable of making predictions, because an empirical
test of the hypothesis requires background assumptions, which
acceptations are methodological decisions in Lakatos' falsificationism.
Popper's response was that falsifiability is a logical criterion.
Experimental research has the Duhem problem and other problems, such as
the problem of induction, but, according to Popper, logical induction is a fallacy and statistical tests, which are possible only when a theory is falsifiable, are useful within a critical discussion.
Popper's distinction between logic and methodology has not
allowed falsifiability to escape some criticisms aimed at methodology.
For example, Popper's rejection of Marxism as unscientific because of
its resistance to negative evidence is a methodological position, but
the problems with this position are nevertheless presented as a
limitation of falsifiability. Others, despite the unsuccessful proposals of Russell, the Vienna Circle, Lakatos, and others to establish a rigorous way of justifying scientific theories or research programs and thus demarcating them from non-science and pseudoscience,
criticize falsifiability for not following a similar proposal and for
supporting instead only a methodology based on critical discussion.
As a key notion in the separation of science from non-science and
pseudoscience, falsifiability has featured prominently in many
controversies and applications, used as legal precedent.
One concern about the scientific method is how to move from observations to scientific laws.
This is the problem of induction. Considering the hypothesis that all
swans are white, given an observation of a white swan, there is no
logical path from "here is a white swan" to "all swans are white"; doing
so would involve a logical fallacy such as, for example, affirming the consequent.
Popper's idea to solve this problem was that while it is
impossible to verify that every swan is white, finding a single black
swan shows that not every swan is white. Such falsification uses the valid inference modus tollens: if, from a law , can be logically deduced, but is observed, is false. Thus, given "all swans are white", "the specific swan here is white", but if what is observed is "the observed swan is not white", then "all swans are white" is false. More precisely, the deducible statement can be broken into an initial condition and a prediction as in in which "the thing here is a swan" and "the thing here is a white swan". If what is observed is C being true while P is false (formally, ), the law is false by modus tollens.
Popper claimed that induction is not needed in science, that is,
he rejected that we learn by the repetition of observations and
considered that logical induction was a fallacy. Instead, laws are conjectured in a non-logical manner on the basis of
expectations and predispositions and trials continue as long as there
are problems. This led Popper's student and collaborator David Miller to write "the mission is to classify truths, not to certify them". In contrast, the logical empiricism movement, led by philosophers such 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 in favor of its truth
or falsity. No consensus emerged 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
scientific precept.
Popper said that a demarcation criterion for the laws of science
was possible, but that what matters is the logical possibility of
falsification of these laws, which is falsifiability. He cited his
encounter with psychoanalysis in the 1910s, especially with Alfred Adler.
It did not matter what observation was presented, psychoanalysis could
explain it. The reason it could explain everything is that it did not
exclude anything. Popper claimed that this was a failure, because it meant that the
criterion could not lead to a prediction. From a logical standpoint,
observations that do not contradict a law does not mean that the law is
true. A verification has no value in itself. But, if a hypothesis makes
risky predictions and these are corroborated, Popper stated, that was a
reason to prefer this hypothesis over others that makes less risky
predictions or no predictions at all. In the definition of falsifiability,
contradictions with observations are not used to support
falsifications, but for logical "falsifications" that show that the law
makes risky predictions.
Popper said that some philosophers of the Vienna Circle had conflated two 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. Popper instead said that meaningful
non-scientific theories exist, and that, accordingly, a criterion of
meaningfulness does not coincide with a criterion of demarcation.
Hume's problem
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 to 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 argued that this learning process cannot
be justified by any general rules, deductive or not. Popper accepted Hume's argument and therefore viewed progress in
science as the result of quasi-induction, which is induction without
inference rules and which he also called the "path of science".
Philip N. Johnson-Laird
agreed with Hume that no general method of justification for induction
is possible but that induction does not require justification. Instead, these steps use patterns of induction, which are not expected to have a general justification: they may or may not be applicable depending on context.
[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.
Johnson-Laird's view was that "induction is just something that animals, including human beings, do to make life possible".
Popper accepted the possibility of a psychological explanation
for the learning process, especially when psychology is seen as an
extension of biology, but claimed that biological explanations were not
within the scope of epistemology. In line with Johnson-Laird's view, Popper proposed an evolutionary mechanism to explain science's success, but he did not consider it part of his epistemology. He referred to this as psychologism. He wrote that his interest was mainly in the logic of science and that epistemology should be concerned with logical aspects only. Instead of asking why science succeeds, he considered induction pragmatically. He asked what methodology should be used to accept one among multiple
hypotheses. He proposed that it be the one that was the most tested:
"the one, which in the light of our critical discussion, appears to be the best so far". By his own account, because only a negative approach was supported by logic, Popper adopted a negative methodology to prevent the "policy of immunizing our theories against refutation".
It also supported a "dogmatic attitude" in defending theories against
criticism, because this would allow the process to be more complete. This view was much criticized.
A different notion of induction
In practice, some steps based on observations can be justified under assumptions. For example, Bayesian inductive logic is justified by theorems that make explicit assumptions. These theorems
are obtained with deductive logic. They are sometimes presented as
supporting steps of induction, because they refer to laws of
probability, even though they do not extend beyond deductive logic. This
is a different notion of induction, which overlaps with deductive logic
in the sense of being supported by it. Hume's argument does not reject
the possibility of a general procedure that relies on hypotheses to
explain the progress of science, but it says the problem of how to
choose the initial hypotheses and prove their validity creates an
infinite regress.
Logic of science versus applied methodology
Popper distinguished the logic of science from its applied methodology. For example, the falsifiability of Newton's law of gravitation, as
defined by Popper, depends purely on the logical relation it has with a
statement such as "The brick fell upwards when released". A brick that falls upwards would not alone falsify Newton's law of
gravitation. The capacity to verify the absence of conditions such as a
hidden string attached to the brick is also needed for this state of affairs to eventually falsify Newton's law of gravitation. However, these
applied methodological considerations are irrelevant in falsifiability,
because it is a logical criterion. The empirical requirement on the
potential falsifier, also called the material requirement, is only that it is communicable inter-subjectively.
The potential falsifier is not required to actually show the law to be
false. The purely logical contradiction, together with the material
requirement, are sufficient. The logical part consists of theories,
statements, and their logical relationship together with this material
requirement, which is needed for a connection with the methodological
part.
The methodological part consists, in Popper's view, of informal
rules, which are used to formulate hypotheses, accept observations as
factual, etc. These include statistical tests: Popper is aware that
observation statements are accepted with the help of statistical methods
and that these involve methodological decisions. When this distinction is applied to the term "falsifiability", it
corresponds to a distinction between two 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.
Popper said that methodological problems require methodology
rules. One such rule is that, refusing to go along with falsifications
is equivalent to retiring from science. The logical side has no 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. On the methodological side,
observations can show that a law is false. On the logical side,
observations, which are purely logical constructions, 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.
Popper wrote that an entire literature exists because this
distinction between the logical and the methodological was not observed. This survives in later literature. For example, in their 2019 article
"Evidence based medicine as science", Vere and Gibson wrote
"[falsifiability has] been considered problematic because theories are
not simply tested through falsification but in conjunction with
auxiliary assumptions and background knowledge."
According to Thornton, Popper's distinction between logic and
methodology did not allow falsifiability, even when presented as a
logical criterion, to escape criticism aimed at methodology. For example, the dismissal of Marxism as unscientific because it was
not abandoned despite the evidence was a methodological position adopted
by Popper, and problems with this position have been presented by
Lakatos and others as a limitation of his criterion of demarcation.
Practical value of a logical criterion
The
fact that science must be based on empirical evidence does not mean
that a logical criterion fails to address the real concerns of science.
As explained by Thornton, there is no methodological falsifications in the process of science
except in those tests that compare the theory with factual observations,
but in these tests too the procedure is mostly logical and involves
observations that are only logical constructions. Popper distinguishes
four different lines along which the testing of a theory could be
carried out. First there is the logical comparison of deduced statements
by which the internal consistency of the system is tested. Second,
there is the determination whether it has the character of an empirical
or scientific theory, or whether it is, for example, tautological, as
required by the falsifiability criterion. Third, there is the
comparison with other theories to determine whether the theory would
constitute a scientific advance if not empirically rejected. Finally and
most importantly, there is the testing of the theory by way of
empirical applications of the conclusions which can be derived from it,
which would not be possible without falsifiability. Here, too, the
testing procedure turns out to require deductive logic, essentially
because it is needed to properly conceive, implement, and interpret
observations. The logical aspect of the criterion makes it practical, because it goes along the usual activities in science.
Basic statements
In Popper's view, observation statements can be analyzed within a logical structure independent of factual observations. The set of all purely logical observations that are considered
constitutes the empirical basis. Popper calls them the basic statements
or test statements. They can be used to show the falsifiability of a
theory. Popper says that basic statements do not have to be possible. It
is sufficient that they are accepted by convention as belonging to the
empirical language, a language that "must be testable by intersubjective
observation (the material requirement)".
When there is a technological advance, a technology that was
previously only hypothetical, for example, the use of a rocket to
examine the hidden surface of the moon, might become available with all
the details known. This has led Herbert Keuth [de] to write: "a hypotheses that was first untestable may become testable later on." Formally, though, one might counter argue that it's not the same
theory, because the basic statements are interpreted slightly
differently with the actual details.
In The Logic of Scientific Discovery, 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.
Definition of falsifiability
Thornton
says that basic statements 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, discussion is not necessary. The first decision described by Lakatos is implicit in this agreement, but the other decisions are not needed.
The agreement exists only in principle. This is where the
logical/methodological distinction becomes important. When an actual
falsifier is proposed, the technology used is considered in detail and
an actual agreement is needed. This may require using a deeper empirical
basis, to make sure that the properties or values used in the falsifier were obtained correctly.
Popper says that despite the fact that the empirical basis can be shaky, the above definition is simply the formalization of a natural requirement on scientific theories, without which the whole logical process of science would not be possible.
Initial condition and prediction in falsifiers of laws
In his analysis of universal laws, Popper conclusion was that laws must "allow us to deduce, roughly speaking, more empirical singular statements than we can deduce from the initial conditions alone." A singular statement that has only one part cannot contradict a
universal law. To contradict an universal law, a falsifier must have
two parts: the initial condition and the singular statement that
contradicts the prediction as in in which "the thing here is a swan" and "the thing here is a white swan".
However, falsifiers need not have two parts in the definition
itself. In this way, the definition is more general and allows basic
statements to be falsifiable. For example, "the thing here is a black swan" is falsifiable, because
it is contradicted by the falsifier "the thing here is a white swan".
Popper wrote that criteria that require that a law must be predictive,
just as is required by falsifiability when applied to laws, "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."
Necessity to strengthen laws to make them falsifiable
Grover Maxwell [es] discussed statements such as "All men are mortal." This is not falsifiable, because it does not matter how old a man is, he might die next year. 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, because he
considers that metaphysical statements can be useful, but also because
it is indirectly corroborated by its agreement with the falsifiable law
"All men die before the age of 150." For Popper, if no such falsifiable
law exists, then the metaphysical law is less useful, because it is not
indirectly corroborated by a stronger law, a law that prohibits more. This kind of non-falsifiable statements in science was noticed by Carnap as early as 1937.
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. 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. Maxwell said that most scientific laws are metaphysical statements of this kind, which, Popper said, need to be made more precise before they can be indirectly corroborated. 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 and the existence of the melting point. 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. His 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).
Falsifiability in model theory
Herbert A. Simon studied the semantic aspects of falsifiability. There it is proposed that two formal requirements govern a formally
defined and stringent falsifiability that a theory must satisfy to
qualify as scientific: that they be finitely and irrevocably testable. These studies were done in the perspective that a logic is a relation
between formal sentences in languages and a collection of mathematical
structures, each of which is considered a model within model theory. The relation, usually denoted , says the formal sentence is true when interpreted in the structure —it provides the semantic of the languages. According to Rynasiewicz,
in this semantic perspective, Popperian falsifiability means that in
some observation structure (in the collection) there exists a set of
observations which refutes the theory.
A stronger notion of falsifiability was considered, which
requires that all structures in the collection that cannot be expanded
to a structure that satisfies
contain such a contradicting set of observations. This stronger
definition makes sense, because it says that we want a falsifier
whenever the theory is actually false. It also implies other interesting
properties, but it is not the usual falsifiability. For example, « all
swans are white and there exists a white raven » is falsifiable in the
usual sense, but not strongly falsifiable, because in the case where all
swans are white and all ravens are black we cannot find a falsifier.
The fact that all ravens are black contradicts the theory, but it is not
a singular statement, not a possible observation.
Examples
Newton's theory
Lakatos suggested that Isaac Newton's law of universal gravitation was as difficult to show to be falsifiable as Sigmund Freud's theory of psychoanalysis.
To refute this, Popper gave the example of an apple that moves from the
ground up to a branch and then starts to dance from branch to branch.
According to Popper's definition, this is a basic statement and a
potential falsifier for Newton's theory, because the position of the
apple at different times can be measured. This appears controversial,
because Newtonian physics does not deny that forces stronger than
Earth's gravity can act on the apple. However, the definition of
falsifiability only asserts the well-established fact that Newton's law
is predictive, that is, it prohibits certain states of affairs. To this
end, it restricts itself to what the law formally says about the basic
statements, just as when theoretical physicists compute the directory of
a rocket and ignores all methodological considerations that are not a
part of the initial conditions that they decided to consider, including
these extra forces, because these have been known since at least Duhem
to imply the impossibility of predictions.
Equivalence principle
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 the equivalence principle, a concept associated with Albert Einstein.
In a discussion of the theory of evolution, Popper mentioned industrial melanism as an example of a falsifiable law. A corresponding basic statement
that acts as a potential falsifier 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". It is 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.
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. This shows the scientific character of paleontology,
because it contradicts the hypothesis in paleontology that all mammals
existed in a much more recent era, despite opinions to the contrary.Richard Dawkins adds that any other modern animal, such as a hippo, would suffice.
Unfalsifiable statements
Even
if it is accepted that angels exist, "All angels have large wings" is
not falsifiable, because no technology exists to identify and observe
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) 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.
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 is
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.
Moreover, if the ad hoc hypothesis says that the world was created as
we observe it today without stating further laws, by definition it
cannot be contradicted by observations and thus is not falsifiable. This
is discussed 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.
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.[78] Popper and others said that the most widely accepted definition of "fitness" in modern biology, namely reproductive success, the expression "survival of the fittest" is a tautology.
Darwinist Ronald Fisher
worked out mathematical theorems to help answer questions regarding
natural selection. But, for Popper and others, no (falsifiable) law of
Natural Selection has been offered, because these tools apply only to
certain rare traits. Instead, for Popper, the work of Fisher and others on Natural Selection
is part of an important and successful metaphysical research program.
Popper said that some unfalsifiable statements may be useful to 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 the study of abstract topics such as quantity, structure, space and change. Mathematical methods are, however, applied in constructing and testing 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."
Popper distinguished the original theory of Marx and what came to be known as Marxism. He claimed that the original theory contained genuine scientific laws.
Though they could not make preordained predictions, these laws
constrained how changes occur in society. One of them was that changes
cannot "be achieved by the use of legal or political means". In Popper's view, this was both testable and subsequently 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."Popper's attacks were not directed toward Marxism, or Marx's theories,
which were falsifiable, but toward Marxists who he considered to have
ignored the falsifications which had happened. Popper more fundamentally criticized 'historicism' in the sense of any
preordained prediction of history, given what he saw as human's right,
ability and responsibility to control its destiny.
Courts of law
Falsifiability was used in the McLean v. Arkansas case (in 1982), the Daubert case (in 1993) and other cases. A survey of 303 federal judges conducted in 1998 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."
McLean v. Arkansas case
In the McLean v. Arkansas case, Judge William Overton used falsifiability as a criterion to determine that "creation science" was not scientific and should not be taught in Arkansaspublic schools as such (it can be taught as religion). In his testimony, philosopher Michael Ruse defined the characteristics which constitute science as:
It is guided by natural law;
It has to be explanatory by reference to natural law;
It is testable against the empirical world;
Its conclusions are tentative, i.e., are not necessarily the final word; and
It is falsifiable.
In his conclusion related to this criterion Judge Overton stated:
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.
In several cases, the United States Supreme Court described scientific methodology using the five Daubert factors, which include falsifiability. The Daubert case 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. 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. Green 645. See also Carl Hempel, Philosophy of Natural Science 49 (1966) ([T]he statements constituting a scientific explanation must be capable of empirical test); Karl Popper, Conjectures and Refutations: The Growth of Scientific Knowledge 37 (5th ed. 1989) ([T]he criterion of the scientific status of a theory is its falsifiability, or refutability, or testability) (emphasis deleted).
— Harry Blackmun
David H. Kaye 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."
Statistical theories and falsifiability
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".
Popper was not concerned with the probabilistic aspect of the
experiment.[55] Together with Maxwell, who raised the problems of falsification, he was aware that some convention must be adopted to fix what it means
to detect or not detect a neutrino. This is Lakatos' third kind of
decision. For Popper and most philosophers, some theory underpins observations.
The theory that justifies that we conventionally accept the potential
falsifier "no neutrino was detected" is statistical. In statistical
language, the potential falsifier that can be not rejected statistically
is typically the null hypothesis, as understood even in popular accounts on falsifiability.
Statisticians use various techniques to draw conclusions about hypotheses on the basis of available evidence. Fisher, Neyman, and Pearson proposed approaches that require no prior probabilities. In contrast, Bayesian inference emphasizes the importance of prior probabilities. Any approach that provides a way to accept or reject a potential
falsifier can be used, including Bayes' theorem and estimates of prior
probabilities that are made using critical discussions and reasonable
assumptions taken from background knowledge. No general rule considers a
hypothesis with small Bayesian revised probability to be falsified,
because the individual outcomes described will have small probabilities
under available evidence without qualifying as genuine anomalies (Mayo and Popper). Nevertheless, Mayo added, "they can indirectly falsify hypotheses by adding a methodological falsification rule". In general, Bayesian statistics can play a role in the context of inductive logic, which is said to be inductive because implications are generalized to conditional probabilities. According to Popper and others 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". Inductive logic is not precluded, especially not when it is a
deductively valid application of Bayes' theorem that is used to evaluate
the probability 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.
Because statisticians often associate statistical inference with
induction, Popper's philosophy is often said to have a hidden form of
induction. Mayo wrote "The falsifying hypotheses ... necessitate an
evidence-transcending (inductive) statistical inference. This is hugely
problematic for Popper". Yet, also according to Mayo, Popper [as a non-inductivist] acknowledged
the useful role of statistical inference in the falsification problems:
she mentioned that when Popper wrote her "I regret not studying
statistics", her thought was "not as much as I do".
Lakatos's falsificationism
Imre Lakatos
divided the problems of falsification into two categories. The first
corresponds to decisions that must be agreed upon by scientists before
they can falsify a theory. The other is the use of falsifications and
corroborations to explain progress in science. Lakatos described four kind of falsificationisms:
Dogmatic falsificationism – ignores both types of problems.
Methodological falsificationism – addresses the first type of problem by accepting that decisions must be taken by scientists.
Naive methodological falsificationism or naive falsificationism – does not do anything to address the second type of problems.
Sophisticated falsificationism – attempts to address both problems.
Lakatos used dogmatic and naive falsificationism to describe how
Popper changed over time and viewed sophisticated falsificationism as
his refinement, but also said that Popper sometimes presents as a
sophisticated falsificationist. Popper responded that Lakatos was misrepresenting his intellectual history.
Dogmatic falsificationism
A
dogmatic falsificationist rejects that every observation is
theory-impregnated, which means that it goes beyond direct experience.
For example, the statement "Here is a glass of water" goes beyond
experience, because the concepts of glass and water "denote physical
bodies which exhibit a certain law-like behaviour" (Popper). This leads to the critique that it is unclear which theory is
falsified: the one under study or the one behind the observation. 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
claimed that the optical theory of the telescope was false, not the
reigning 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 that the possibility that this strength was sufficiently high was a "pious hope".
A dogmatic falsificationist ignores the role of auxiliary
hypotheses. The assumptions or auxiliary hypotheses of a particular test
are all the hypotheses that must be correct in order for the test to
perform as expected. The predicted observation that is contradicted depends on the theory
and these auxiliary hypotheses. Therefore, whether it is the theory or
an auxiliary hypothesis that is falsified by the observation is
undetermined. Lakatos gives the example of the path of a planet. If the
path contradicts Newton's law, it is not clear what should be rejected:
Newton's law or the hypothesis that no other body influenced the path. A
dogmatic falsificationist would ignore that and consider that Newton's
law is falsified.
Lakatos says that Popper's solution to these criticisms of the
dogmatic falsificationist requires relaxing the assumption that an
observation can show a theory to be false:
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.
— Imre Lakatos
Popper's
solution, in his own words, is to distinguish between states of affairs
as logical falsifications and actual methodological falsifications.
Popper never relaxed the notion that logical falsifications show that a
theory is falsifiable. He always accepted the existence of
methodological problems. He always maintained that actual falsifications
are not possible. He wrote that one should nor read Lakatos to understand his intellectual history.
Methodological falsificationism
Methodological
falsificationism replaces the usual notion of contradicting observation
in a falsification with a new notion of 'contradicting observation'
(now in quotes), a convention that requires four kinds of decisions with
accompanying goals:
selecting all basic statements (statements that correspond to logically possible observations),
selecting the accepted basic statements among those basic statements,
making statistical laws falsifiable and
applying the refutation to the specific theory (instead of an auxiliary hypothesis).
The experimental falsifiers and falsifications thus depend on
decisions in view of accepted technology and associated theory. A fifth
decision is mentioned by Lakatos to allow even more theories to be
falsified.
Naive falsificationism
According
to Lakatos, naive falsificationism is the claim that methodological
falsifications can by themselves explain scientific progress. 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 is not always
possible to rationally pick one theory over another using
falsifications. Considering only falsifications, it is not clear why a corroborating
experiment should be seen as progress. Lakatos described Popper as being
in part a naive falsificationist and in part a sophisticated
falsificationist." In his own words, Popper's critical rationalism used both
falsifications and the value of a theory in practice to explain
progress.
Popper distinguishes between the creative, informal process from
which accepted basic statements and accepted theories emerge, and the
logical, formal process that compares all theories to all basic
statements and defines falsifiability without saying how to
methodologically accept basic statements and theories. The main issue addressed by Lakatos is whether the methodological side
could be made more formal, in particular, whether the decision to select
one among competing theories in the light of falsifications and
corroborations could be justified using a formal logic. Such logic would be inductive: it justifies a universal law in view of
instances. Lakatos and many others claimed that the decision should be
so justified. In contradistinction, for Popper, the creative and informal part is
guided by methodological rules, which naturally favour theories that are
corroborated and have shown their merit over those that are falsified, but this methodology cannot be made rigorous.
Popper's way to analyze progress was through verisimilitude,
a way to define how close a theory is to the truth, which he did not
consider significant, except as an attempt to describe a concept already
accepted in practice. Later, it was shown that the specific definition
proposed by Popper cannot distinguish two false theories, as is the case
for all theories in the history of science.
Sophisticated falsificationism
Hume explained induction with a theory of the mind that was in part inspired by Newton's theory of gravitation. Popper rejected Hume's explanation 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 that 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 one his theory described could not be
justified logically. Similarly, Popper adopted an evolutionary epistemology that implies
that some laws explain progress, but insists that the process of trial
and error is not rigorous and that an element of irrationality is
unavoidable.
Though they be rational, these explanations cannot be turned into
methods of justification. This was insufficient for philosophers such
as Bertrand Russell,
who once expressed the view that if Hume's problem cannot be solved,
"there is no intellectual difference between sanity and insanity" and discussed what is needed for induction to be possible. He argued that for induction to be possible, a law to make inferences
from matters of fact is needed and such law, unlike the principles of
deductive logic, must be synthetic. He said "the only alternative to
this hypothesis is complete scepticism". Lakatos approved Russell's justificationist view. His proposal of sophisticated falsificationism was natural in that context.
Therefore, Lakatos urged Popper to find an inductive principle behind the trial and error learning process and sophisticated falsificationism was his own approach to this challenge. Kuhn, Feyerabend, Musgrave and others mentioned and Lakatos himself
acknowledged that this attempt failed, because no normative methodology
existed—Lakatos' methodology was anarchy in disguise.
Popperian falsificationism
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". But, Popper's philosophy is not always qualified of falsificationism in
the pejorative manner associated with dogmatic or naive
falsificationism. The problems of falsification are acknowledged by the
falsificationists. For example, Chalmers pointed out that
falsificationists freely admit that observation relies on theory. Thornton, referring to Popper, says that the predictions inferred from
conjectures are not directly compared with the facts simply because all
observation statements are theory-laden. 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. Instead, they rely on critical discussions around
experimental falsifications. 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. He knew that Popper's philosophy had never been about this kind of justification, but claimed that it should have been. Sometimes, Popper and other falsificationists said that when a theory is falsified it is rejected (dogmatic falsificationism), but they said that in the general context
of critical rationalism, in which all decisions are open to critical
discussions and can be revised. Popper complained that his discussions of rejection are often taken out of context.
As discussed in the section § Naive falsificationism,
Lakatos and Popper agreed that universal scientific laws cannot be
derived logically, except from broader laws that encompass them.
However, unlike Popper, Lakatos believed that induction was the only
alternative to deduction. He encouraged Popper to explicitly adopt an
inductive approach and sought such an inductive method. However, Lakatos' method never provided precise inductive rules. In response to critiques from Thomas Kuhn, Paul Feyerabend, and Alan Musgrave, Lakatos admitted that his methodology relied on scientists' judgment. Feyerabend, in Against Method, argued that Lakatos' methodology was essentially epistemological anarchism in disguise, a view echoed by Musgrave. Later, Feyerabend noted that Lakatos proposed rules, but these rules
did not specify when they must be applied and Feyerabend maintained his
position.
Popper also proposed a methodology with rules, but these were
non-inductive, as they did not independently confirm the validity of
scientific laws. Instead, they relied on scientists' creativity or
judgment to determine which theories to study, identify significant
problems, and propose testable hypotheses. Citing Einstein, Popper argued that this creative process eliminated
the need for an inductive methodology or a logical pathway to discover
scientific laws.
Lakatos' methodology built significantly on Popper's by incorporating
a historical perspective. Lakatos supported his methodology with
examples from the history of science. He defined what he called research
programmes, that can be either pursued or abandoned. Research
programmes are classified as progressive or degenerative; the latter are
eventually abandoned. Lakatos claimed that this classification is
largely supported by historical evidence. In contrast, Popper did not
design his methodology to rigorously analyze the history of science.
However, he occasionally called on historical examples. For instance, he
noted that highly successful scientific theories were often disproven.
He also introduced, before Lakatos, the notion of metaphysical research
programs. He never intended to use them in a rigorous inductive methodology, but
in a text that inspired Lakatos he discussed how they change over
time. Later, he compared them to Kuhn's paradigms. Lakatos removed the "metaphysical" part and sought rigorous inductive
rules. Later, he abandoned this research and adopted rules that require
judgment. According to Eli Zahar, Lakatos "admitted that the
difference between his position and Popper's was so small as to become
purely verbal".
State of affairs as falsifier versus actual falsification
In
1974, Lakatos challenged Popper to demonstrate that his theory of
falsifiability was itself falsifiable, asking, "Under what conditions
would you abandon your demarcation criterion?" Popper responded, "I shall abandon my theory if Professor Lakatos
succeeds in showing that Newton's theory is no more falsifiable by
observable states of affairs than Freud's." In Popper's terminology, a "falsifier" is not an actual observation,
but instead a hypothetical situation, a state of affairs, a logical
concept that as such is not open to evasive methodological
explanations. For example, Newton's law of gravitation states that a
brick falls downward. A hypothetical observation that the brick falls
upward is a falsifier (demonstrating falsifiability) even if something
else such as strings attached to the brick could explain the
observation.
In 1982, David Stove
criticized Popper, claiming that Lakatos' challenge was successful.
Stove contended that any observation appearing to contradict Newtonian
physics could be explained by other laws or mechanisms, suggesting no
truly "non-Newtonian" behavior was possible. Stove claimed that Popper's
counterexamples, such as missiles following a "non-Newtonian
trajectory" or objects not falling due to obvious counteracting forces,
were either flawed (e.g., begging the question) or consistent with Newtonian physics. Popper reiterated that his falsifiability criterion was a purely
logical concept, distinct from the practical ability to disprove a
theory. He wrote, "An entire literature rests on the failure to observe
this distinction."
Kuhn examined periods of normal science and the transitions (revolutions), that separate them. Popper was interested only in revolutions. He claimed that the purpose of science, mathematics, and metaphysics–all forms of knowledge—is to address and solve problems. Kuhn noted that during periods of normal science, scientists use
established theories to routinely solve problems, questioning their
validity only when the theory fails. This perspective aligns with
Popper's view of problem solving, but places more emphasis on solving
routine problems that do not challenge the underlying theory. Kuhn
claimed that Popper focused too heavily on formal or logical
falsifications and did not adequately explain the social and informal
aspects of scientific progress.
Popper often used astrology as an example of a pseudoscience. He says that it is not falsifiable because both the theory itself and its predictions are too imprecise. Kuhn remarked that many predictions made by astrologers were quite precise and were often falsified.
Epistemological anarchism vs the scientific method
Feyeraband entirely rejected prescriptive methodology, including Lakatos' argument for ad hoc hypotheses,
arguing that science could not have progressed without making use of
every available method. He rejected reliance on a scientific method,
along with any special authority for science that might derive from such
a method. He said that the only possible universally valid methodological rule was epistemological anarchism (anything goes) was the only candidate. Ultimately, he claimed that science's special status derives from the value of the results rather than its method.
At
the core of Popper's falsification principle, Karl Popper's principle
of falsification centers on figuring out what specific observation would
prove a hypothesis wrong, and then trying to find it. The research
process, therefore, is ideally structured to test a hypothesis by
seeking the specific result that would falsify it. If a prediction is
observed, the hypothesis is corroborated (supported); repeated failures
to observe the predicted result may lead to its rejection in a critical
discussion.
Compromised methodology
However, numerous authors have highlighted that due to the reproducibility crisis, the rejection is compromised by poor practices through the research process, including vague or ill-defined hypotheses, inadequate or under-reported data processing, and problematic data analysis. The issue is that the methodology maybe negatively influenced by the researcher degrees of freedom in which the procedures are carried out.
Open Science Solutions
The Open Science
movement has introduced valuable tools and practices aimed at
mitigating these biases and restoring the integrity required for
meaningful falsification. By increasing transparency and rigor, Open
Science measures help to reduce researcher degrees of freedom and
improve rejection of wrong hypotheses. Key tools and practices include:
Pre-registration: Researchers commit to their hypotheses, methods, and analysis plans before data collection. This eliminates the possibility of post hoc (after the fact) changes and prevents the selective reporting of results, known as cherry picking.
Open Data
and Open Methods: The sharing of data and materials facilitate
independent attempts to replicate findings, which directly tests the
robustness and potential falsity of previous results and help contribute
to reduce biases for assessing biases.
Reporting Standards:
Checklists and standardized guidelines improve the quality and
completeness of research reports, ensuring all procedural steps and data
analyses are transparently detailed, allowing others to fully assess the study's adherence to falsification standards
