Originally, fallibilism (from

"Fallibilism and Necessity"

''

"Fallibilism"

''

"Karl Popper"

''

From a Logical Point of View

Harvard University Press.Duhem, Pierre Maurice Marie (1954)

The Aim and Structure of Physical Theory

Princeton University Press. As a consequence, statements are held to be underdetermined. Underdetermination explains how evidence available to us may be insufficient to justify our beliefs. The Duhem-Quine thesis should therefore erode our belief in ''logical falsifiability'' as well as in ''methodological falsification''. The thesis can be contrasted with a more recent view posited by philosophy professor Albert Casullo, which holds that statements can be overdetermined. Overdetermination explains how evidence might be considered sufficient for justifying beliefs in absence of auxiliary assumptions. Philosopher Ray S. Percival holds that the Popperian asymmetry is an illusion, because in the action of falsifying an argument, scientists will inevitably verify its

The Popper-Lakatos Controversy in the Light of 'Die Beiden Grundprobleme Der Erkenntnistheorie'

The British Journal for the Philosophy of Science. p. 149–171. Lakatos' critical attitude towards

"Imre Lakatos"

''

Heuristic, Methodology or Logic of Discovery? Lakatos on Patterns of Thinking

MIT Press Direct. p. 314. While critical fallibilism strictly opposes dogmatism, critical rationalism is said to require a limited amount of dogmatism.Lakatos, Imre (1978)

Mathematics, Science and Epistemology

Cambridge University Press. Vol. 2. p. 9–23. Though, even Lakatos himself had been a critical rationalist in the past, when he took it upon himself to argue against the inductivist illusion that

Philosophy of Logics

'. Cambridge University Press. pp. 234; Chapter 12. Critical rationalist Hans Albert argues that it is impossible to prove any truth with certainty, not only in logic, but also in mathematics.

Imre Lakatos and the Guises of Reason.

' Duke University Press. pp. 45, 109, 155, 323. Mathematical fallibilism deviates from traditional views held by philosophers like

The Consistency of the Continuum-Hypothesis

'. Princeton University Press. Vol. 3. The

"Cantor's theorem"

''

"The Independence of the Continuum Hypothesis"

''

"Knowledge, hope, and fallibilism"

''

"Fallibilism"

by Stephen Hetherington in the ''

"Fallibilism"

by Nicholas Rescher in the ''

Medieval Latin
Medieval Latin was the form of Literary Latin used in Roman Catholic Western Europe during the Middle Ages. In this region it served as the primary written language, though local languages were also written to varying degrees. Latin functioned ...

: ''fallibilis'', "liable to err") is the philosophical principle that propositions
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...

can be accepted even though they cannot be conclusively proven or justified,Haack, Susan (1979)"Fallibilism and Necessity"

''

Synthese
''Synthese'' () is a scholarly periodical specializing in papers in epistemology, methodology, and philosophy of science, and related issues. Its subject area is divided into four specialties, with a focus on the first three: (1) "epistemology, me ...

'', Vol. 41, No. 1, pp. 37–63. or that neither knowledge
Knowledge can be defined as awareness of facts or as practical skills, and may also refer to familiarity with objects or situations. Knowledge of facts, also called propositional knowledge, is often defined as true belief that is distin ...

nor belief
A belief is an attitude that something is the case, or that some proposition is true. In epistemology, philosophers use the term "belief" to refer to attitudes about the world which can be either true or false. To believe something is to tak ...

is certain
Certainty (also known as epistemic certainty or objective certainty) is the epistemic property of beliefs which a person has no rational grounds for doubting. One standard way of defining epistemic certainty is that a belief is certain if and ...

.Hetherington, Stephen"Fallibilism"

''

Internet Encyclopedia of Philosophy
The ''Internet Encyclopedia of Philosophy'' (''IEP'') is a scholarly online encyclopedia, dealing with philosophy, philosophical topics, and philosophers. The IEP combines open access publication with peer reviewed publication of original papers ...

''. The term was coined in the late nineteenth century by the American philosopher Charles Sanders Peirce
Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism".
Educated as a chemist and employed as a scientist for ...

, as a response to foundationalism. Theorists, following Austrian-British philosopher Karl Popper
Sir Karl Raimund Popper (28 July 1902 – 17 September 1994) was an Austrian-British philosopher, academic
An academy (Attic Greek: Ἀκαδήμεια; Koine Greek Ἀκαδημία) is an institution of secondary education, secondary or ...

, may also refer to fallibilism as the notion that knowledge might turn out to be false. Furthermore, fallibilism is said to imply corrigibilism, the principle that propositions are open to revision. Fallibilism is often juxtaposed with infallibilism
Infallibilism is the epistemological view that propositional knowledge is incompatible with the possibility of being wrong.
Definition
In philosophy, infallibilism (sometimes called "epistemic infallibilism") is the view that knowing the truth o ...

.
Infinite regress and infinite progress

According to philosopher Scott F. Aikin, fallibilism cannot properly function in the absence ofinfinite regress
An infinite regress is an infinite series of entities governed by a recursive principle that determines how each entity in the series depends on or is produced by its predecessor. In the epistemic regress, for example, a belief is justified bec ...

. The term, usually attributed to Pyrrhonist philosopher Agrippa Agrippa may refer to:
People Antiquity
* Agrippa (mythology), semi-mythological king of Alba Longa
* Agrippa (astronomer), Greek astronomer from the late 1st century
* Agrippa the Skeptic, Skeptic philosopher at the end of the 1st century
* ...

, is argued to be the inevitable outcome of all human inquiry, since every proposition requires justification. Infinite regress, also represented within the regress argument
In epistemology, the regress argument is the argument that any proposition requires a justification. However, any justification itself requires support. This means that any proposition whatsoever can be endlessly (infinitely) questioned, result ...

, is closely related to the problem of the criterion and is a constituent of the Münchhausen trilemma
In epistemology, the Münchhausen trilemma, also commonly known as the Agrippan trilemma, is a thought experiment intended to demonstrate the theoretical impossibility of proving any truth, even in the fields of logic and mathematics, without a ...

. Illustrious examples regarding infinite regress are the cosmological argument, turtles all the way down, and the simulation hypothesis. Many philosophers struggle with the metaphysical implications that come along with infinite regress. For this reason, philosophers have gotten creative in their quest to circumvent it.
Somewhere along the seventeenth century, English philosopher Thomas Hobbes
Thomas Hobbes ( ; 5/15 April 1588 – 4/14 December 1679) was an English philosopher, considered to be one of the founders of modern political philosophy. Hobbes is best known for his 1651 book '' Leviathan'', in which he expounds an influen ...

set forth the concept of "infinite progress". With this term, Hobbes had captured the human proclivity to strive for perfection
Perfection is a state, variously, of completeness, flawlessness, or supreme excellence.
The term is used to designate a range of diverse, if often kindred, concepts. These have historically been addressed in a number of discrete disciplines, ...

. Philosophers like Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mat ...

, Christian Wolff, and Immanuel Kant
Immanuel Kant (, , ; 22 April 1724 – 12 February 1804) was a German philosopher and one of the central Enlightenment thinkers. Born in Königsberg, Kant's comprehensive and systematic works in epistemology, metaphysics, ethics, and a ...

, would elaborate further on the concept. Kant even went on to speculate that immortal
Immortality is the ability to live forever, or eternal life.
Immortal or Immortality may also refer to:
Film
* ''The Immortals'' (1995 film), an American crime film
* ''Immortality'', an alternate title for the 1998 British film '' The Wisdom of ...

species should hypothetically be able to develop their capacities to perfection. This sentiment is still alive today. Infinite progress has been associated with concepts like science
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earliest archeological evidence f ...

, religion
Religion is usually defined as a social- cultural system of designated behaviors and practices, morals, beliefs, worldviews, texts, sanctified places, prophecies, ethics, or organizations, that generally relates humanity to supernatural, t ...

, technology
Technology is the application of knowledge to reach practical goals in a specifiable and reproducible way. The word ''technology'' may also mean the product of such an endeavor. The use of technology is widely prevalent in medicine, science, ...

, economic growth
Economic growth can be defined as the increase or improvement in the inflation-adjusted market value of the goods and services produced by an economy in a financial year. Statisticians conventionally measure such growth as the percent rate of ...

, consumerism
Consumerism is a social and economic order that encourages the acquisition of goods and services in ever-increasing amounts. With the Industrial Revolution, but particularly in the 20th century, mass production led to overproduction—the s ...

, and economic materialism
Materialism can be described as either a personal attitude which attaches importance to acquiring and consuming material goods or as a logistical analysis of how physical resources are shaped into consumable products.
The use of the term materi ...

. All these concepts thrive on the belief that they can carry on endlessly. Infinite progress has become the panacea to turn the vicious circles of infinite regress into virtuous circles. However, vicious circles have not yet been eliminated from the world; hyperinflation, the poverty trap
In economics, a cycle of poverty or poverty trap is caused by self-reinforcing mechanisms that cause poverty, once it exists, to persist unless there is outside intervention. It can persist across generations, and when applied to developing count ...

, and debt accumulation for instance still occur.
Already in 350 B.C.E, Greek philosopher Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ph ...

made a distinction between potential and actual infinities. Based on his discourse, it can be said that actual infinities do not exist, because they are paradoxical. Aristotle deemed it impossible for humans to keep on adding members to finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
:\
is a finite set with five elements. T ...

s indefinitely. It eventually led him to refute some of Zeno's paradoxes
Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c. 490–430 BC) to support Parmenides' doctrine that contrary to the evidence of one's senses, the belief in plura ...

. Other relevant examples of potential infinities include Galileo's paradox and the paradox of Hilbert's hotel. The notion that infinite regress and infinite progress only manifest themselves potentially pertains to fallibilism. According to philosophy professor Elizabeth F. Cooke, fallibilism embraces uncertainty, and infinite regress and infinite progress are not unfortunate limitations on human cognition
Cognition refers to "the mental action or process of acquiring knowledge and understanding through thought, experience, and the senses". It encompasses all aspects of intellectual functions and processes such as: perception, attention, thoug ...

, but rather necessary antecedents for knowledge acquisition. They allow us to live functional and meaningful lives.
Critical rationalism

In the mid-twentieth century, several important philosophers began to critique the foundations oflogical positivism
Logical positivism, later called logical empiricism, and both of which together are also known as neopositivism, is a movement in Western philosophy whose central thesis was the verification principle (also known as the verifiability criterion o ...

. In his work ''The Logic of Scientific Discovery
''The Logic of Scientific Discovery'' is a 1959 book about the philosophy of science by the philosopher Karl Popper. Popper rewrote his book in English from the 1934 (imprint '1935') German original, titled ''Logik der Forschung. Zur Erkenntnisth ...

'' (1934), Karl Popper, the founder of critical rationalism, tried to solve the problem of induction
First formulated by David Hume, the problem of induction questions our reasons for believing that the future will resemble the past, or more broadly it questions predictions about unobserved things based on previous observations. This inferen ...

by arguing for falsifiability
Falsifiability is a standard of evaluation of scientific theories and hypotheses that was introduced by the philosopher of science Karl Popper in his book ''The Logic of Scientific Discovery'' (1934). He proposed it as the cornerstone of a sol ...

as a means to devalue the verifiability criterion. He adamantly proclaimed that scientific truths are not inductively inferred from experience
Experience refers to conscious events in general, more specifically to perceptions, or to the practical knowledge and familiarity that is produced by these conscious processes. Understood as a conscious event in the widest sense, experience invol ...

and conclusively verified by experiment
An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs when ...

ation, but rather deduced from statements and justified by means of deliberation
Deliberation is a process of thoughtfully weighing options, usually prior to voting. Deliberation emphasizes the use of logic and reason as opposed to power-struggle, creativity, or dialogue. Group decisions are generally made after deliberation ...

and intersubjective consensus within a particular scientific community. Popper also tried to resolve the problem of demarcation by asserting that all knowledge is fallible, except for knowledge that was acquired by means of falsification. Hence, Popperian falsifications are temporarily infallible, until they have been retracted by an adequate research community. Although critical rationalists dismiss the fact that all claims are fallible, they do belief that all claims are provisional. Counterintuitively, these provisional statements can become conclusive once logical
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...

contradictions
In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle ...

have been turned into methodological refutations.Thornton, Stephen (2021)"Karl Popper"

''

Stanford Encyclopedia of Philosophy
The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. Eac ...

''. The claim that all assertions are provisional and thus open to revision in light of new evidence
Evidence for a proposition is what supports this proposition. It is usually understood as an indication that the supported proposition is true. What role evidence plays and how it is conceived varies from field to field.
In epistemology, evide ...

is widely taken for granted in the natural sciences
Natural science is one of the branches of science concerned with the description, understanding and prediction of natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer review and repeatabil ...

.
Popper insisted that verification and falsification are logically asymmetrical. However, according to the Duhem-Quine thesis, statements can neither be conclusively verified nor falsified in isolation from auxiliary assumptions (also called a ''bundle of hypotheses'').Quine, W. V. O. (1953)From a Logical Point of View

Harvard University Press.Duhem, Pierre Maurice Marie (1954)

The Aim and Structure of Physical Theory

Princeton University Press. As a consequence, statements are held to be underdetermined. Underdetermination explains how evidence available to us may be insufficient to justify our beliefs. The Duhem-Quine thesis should therefore erode our belief in ''logical falsifiability'' as well as in ''methodological falsification''. The thesis can be contrasted with a more recent view posited by philosophy professor Albert Casullo, which holds that statements can be overdetermined. Overdetermination explains how evidence might be considered sufficient for justifying beliefs in absence of auxiliary assumptions. Philosopher Ray S. Percival holds that the Popperian asymmetry is an illusion, because in the action of falsifying an argument, scientists will inevitably verify its

negation
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and fals ...

. Thus, verification and falsification are perfectly symmetrical. It seems, in the philosophy of logic, that neither syllogisms
A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true ...

nor polysyllogisms will save underdetermination and overdetermination from the perils of infinite regress.
Furthermore, Popper defended his critical rationalism as a normative
Normative generally means relating to an evaluative standard. Normativity is the phenomenon in human societies of designating some actions or outcomes as good, desirable, or permissible, and others as bad, undesirable, or impermissible. A norm in ...

and methodological theory, that explains how objective, and thus mind-independent, knowledge ought to work. Hungarian philosopher Imre Lakatos
Imre Lakatos (, ; hu, Lakatos Imre ; 9 November 1922 – 2 February 1974) was a Hungarian philosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its "methodology of proofs and refutations" in its pr ...

built upon the theory by rephrasing the problem of demarcation as the ''problem of normative appraisal''. Lakatos' and Popper's aims were alike, that is finding rules that could justify falsifications. However, Lakatos pointed out that critical rationalism only shows how theories can be falsified, but it omits how our belief in critical rationalism can itself be justified. The belief would require an inductively verified principle. When Lakatos urged Popper to admit that the falsification principle cannot be justified without embracing induction, Popper did not succumb.Zahar, E. G. (1983)The Popper-Lakatos Controversy in the Light of 'Die Beiden Grundprobleme Der Erkenntnistheorie'

The British Journal for the Philosophy of Science. p. 149–171. Lakatos' critical attitude towards

rationalism
In philosophy, rationalism is the epistemological view that "regards reason as the chief source and test of knowledge" or "any view appealing to reason as a source of knowledge or justification".Lacey, A.R. (1996), ''A Dictionary of Philosophy' ...

has become emblematic for his so called ''critical fallibilism''.Musgrave, Alan; Pigden, Charles (2021)"Imre Lakatos"

''

Stanford Encyclopedia of Philosophy
The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. Eac ...

''.Kiss, Ogla (2006)Heuristic, Methodology or Logic of Discovery? Lakatos on Patterns of Thinking

MIT Press Direct. p. 314. While critical fallibilism strictly opposes dogmatism, critical rationalism is said to require a limited amount of dogmatism.Lakatos, Imre (1978)

Mathematics, Science and Epistemology

Cambridge University Press. Vol. 2. p. 9–23. Though, even Lakatos himself had been a critical rationalist in the past, when he took it upon himself to argue against the inductivist illusion that

axioms
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...

can be justified by the truth of their consequences. In summary, despite Lakatos and Popper picking one stance over the other, both have oscillated between holding a critical attitude towards rationalism as well as fallibilism.
Fallibilism has also been employed by philosopher Willard V. O. Quine to attack, among other things, the distinction between analytic and synthetic statements. British philosopher Susan Haack, following Quine, has argued that the nature of fallibilism is often misunderstood, because people tend to confuse fallible ''propositions'' with fallible ''agents''. She claims that logic is revisable, which means that analyticity does not exist and necessity (or a priority) does not extend to logical truths. She hereby opposes the conviction that propositions in logic are infallible, while agents can be fallible.Haack, Susan (1978). Philosophy of Logics

'. Cambridge University Press. pp. 234; Chapter 12. Critical rationalist Hans Albert argues that it is impossible to prove any truth with certainty, not only in logic, but also in mathematics.

Mathematical fallibilism

In '' Proofs and Refutations: The Logic of Mathematical Discovery'' (1976), philosopherImre Lakatos
Imre Lakatos (, ; hu, Lakatos Imre ; 9 November 1922 – 2 February 1974) was a Hungarian philosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its "methodology of proofs and refutations" in its pr ...

implemented mathematical proofs
A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proo ...

into what he called Popperian "critical fallibilism". Lakatos's mathematical fallibilism is the general view that all mathematical theorems are falsifiable.Kadvany, John (2001). Imre Lakatos and the Guises of Reason.

' Duke University Press. pp. 45, 109, 155, 323. Mathematical fallibilism deviates from traditional views held by philosophers like

Hegel
Georg Wilhelm Friedrich Hegel (; ; 27 August 1770 – 14 November 1831) was a German philosopher. He is one of the most important figures in German idealism and one of the founding figures of modern Western philosophy. His influence extends a ...

, Peirce, and Popper. Although Peirce introduced fallibilism, he seems to preclude the possibility of us being mistaken in our mathematical beliefs. Mathematical fallibilism appears to uphold that even though a mathematical conjecture cannot be proven true, we may consider some to be good approximations or estimations of the truth. This so called verisimilitude
In philosophy, verisimilitude (or truthlikeness) is the notion that some propositions are closer to being true than other propositions. The problem of verisimilitude is the problem of articulating what it takes for one false theory to be closer ...

may provide us with consistency
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent i ...

amidst an inherent incompleteness in mathematics. Mathematical fallibilism differs from quasi-empiricism, to the extent that the latter does not incorporate inductivism, a feature considered to be of vital importance to the foundations of set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...

.
In the philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people' ...

, the central tenet of fallibilism is '' undecidability'' (which bears resemblance to the notion of ''isostheneia''; the antithesis of appearance and judgement). Two distinct types of the word "undecidable" are currently being applied. The first one relates to the continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...

; the hypothesis that a statement can neither be proved nor be refuted in a specified deductive system.Gödel, Kurt (1940). The Consistency of the Continuum-Hypothesis

'. Princeton University Press. Vol. 3. The

continuum
Continuum may refer to:
* Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes
Mathematics
* Continuum (set theory), the real line or the corresponding cardinal numbe ...

hypothesis was proposed by mathematician Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...

in 1873. This type of undecidability is used in the context of the ''independence of the continuum hypothesis'', namely because this statement is said to be independent from the axioms in Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...

combined with the axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection o ...

(also called ZFC). Both the hypothesis and its negation are consistent with these axioms. Many noteworthy discoveries have preceded the establishment of the continuum hypothesis.
In 1877, Cantor introduced the diagonal argument A diagonal argument, in mathematics, is a technique employed in the proofs of the following theorems:
*Cantor's diagonal argument (the earliest)
* Cantor's theorem
* Russell's paradox
* Diagonal lemma
** Gödel's first incompleteness theorem
** Tar ...

to prove that the cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

of two finite sets is equal, by putting them into a one-to-one correspondence. Diagonalization reappeared in Cantors theorem, in 1891, to show that the power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is pos ...

of any countable set
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...

must have strictly higher cardinality.Hosch, William L"Cantor's theorem"

''

Encyclopædia Britannica
The (Latin for "British Encyclopædia") is a general knowledge English-language encyclopaedia. It is published by Encyclopædia Britannica, Inc.; the company has existed since the 18th century, although it has changed ownership various t ...

''. The existence of the power set was postulated in the axiom of power set; a vital part of Zermelo–Fraenkel set theory. Moreover, in 1899, Cantor's paradox was discovered. It postulates that ''there is no set of all cardinalities''. Two years later, polymath
A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose knowledge spans a substantial number of subjects, known to draw on complex bodies of knowledge to solve specific pro ...

Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ar ...

would invalidate the existence of the universal set
In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory inc ...

by pointing towards Russell's paradox, which implies that ''no set can contain itself as an element (or member)''. The universal set can be confuted by utilizing either the axiom schema of separation or the axiom of regularity
In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set ''A'' contains an element that is disjoint from ''A''. In first-order logic, the ...

. In contrast to the universal set, a power set does not contain itself. It was only after 1940 that mathematician Kurt Gödel showed, by applying inter alia the diagonal lemma, that the continuum hypothesis cannot be refuted, and after 1963, that fellow mathematician Paul Cohen revealed, through the method of forcing, that the continuum hypothesis cannot be proved either.Cohen, Paul (1963)"The Independence of the Continuum Hypothesis"

''

Proceedings of the National Academy of Sciences of the United States of America
''Proceedings of the National Academy of Sciences of the United States of America'' (often abbreviated ''PNAS'' or ''PNAS USA'') is a peer-reviewed multidisciplinary scientific journal. It is the official journal of the National Academy of Scie ...

''. Vol. 50, No. 6. pp. 1143–1148. In spite of the undecidability, both Gödel and Cohen suspected the continuum hypothesis to be false. This sense of suspicion, in conjunction with a firm belief in the consistency of ZFC, is in line with mathematical fallibilism. Mathematical fallibilists suppose that new axioms, for example the axiom of projective determinacy, might improve ZFC, but that these axioms will not allow for dependence of the continuum hypothesis.
The second type of undecidability is used in relation to computability theory (or recursion theory) and applies not solely to statements but specifically to decision problems; mathematical questions of decidability. An undecidable problem
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. The halting problem is an ...

is a type of computational problem in which there are countably infinite
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...

sets of questions, each requiring an effective method
In logic, mathematics and computer science, especially metalogic and computability theory, an effective method Hunter, Geoffrey, ''Metalogic: An Introduction to the Metatheory of Standard First-Order Logic'', University of California Press, 1971 ...

to determine whether an output is either "yes or no" (or whether a statement is either "true or false"), but where there cannot be any computer program
A computer program is a sequence or set of instructions in a programming language for a computer to execute. Computer programs are one component of software, which also includes documentation and other intangible components.
A computer program ...

or Turing machine that will always provide the correct answer. Any program would occasionally give a wrong answer or run forever without giving any answer. Famous examples of undecidable problems are the halting problem
In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. Alan Turing proved in 1936 that a g ...

and the Entscheidungsproblem. Conventionally, an undecidable problem is derived from a recursive set, formulated in undecidable language, and measured by the Turing degree. Practically all undecidable problems are unsolved, but not all unsolved problems are undecidable. Undecidability, with respect to computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...

and mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...

, is also called ''unsolvability'' or ''non-computability''. In the end, both types of undecidability can help to build a case for fallibilism, by providing these fundamental thought experiment
A thought experiment is a hypothetical situation in which a hypothesis, theory, or principle is laid out for the purpose of thinking through its consequences.
History
The ancient Greek ''deiknymi'' (), or thought experiment, "was the most anci ...

s.
Philosophical skepticism

Fallibilism improves upon the ideas associated withphilosophical skepticism
Philosophical skepticism ( UK spelling: scepticism; from Greek σκέψις ''skepsis'', "inquiry") is a family of philosophical views that question the possibility of knowledge. It differs from other forms of skepticism in that it even reject ...

. According to philosophy professor Richard Feldman, nearly all versions of ancient and modern skepticism depend on the mistaken assumption that justification, and thus knowledge, requires conclusive evidence or certainty. An exception can be made for mitigated skepticism. In philosophical parlance, mitigated skepticism is an attitude which supports doubt
Doubt is a mental state in which the mind remains suspended between two or more contradictory propositions, unable to be certain of any of them.
Doubt on an emotional level is indecision between belief and disbelief. It may involve uncertai ...

in knowledge. This attitude is conserved in philosophical endeavors like scientific skepticism
Scientific skepticism or rational skepticism (also spelled scepticism), sometimes referred to as skeptical inquiry, is a position in which one questions the veracity of claims lacking empirical evidence. In practice, the term most commonly refe ...

(or ''rational skepticism'') and David Hume
David Hume (; born David Home; 7 May 1711 NS (26 April 1711 OS) – 25 August 1776) Cranston, Maurice, and Thomas Edmund Jessop. 2020 999br>David Hume" ''Encyclopædia Britannica''. Retrieved 18 May 2020. was a Scottish Enlightenment philo ...

's inductive skepticism (or ''inductive fallibilism''). Scientific skepticism questions the veracity of claims lacking empirical evidence
Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences and ...

, while inductive skepticism avers that inductive inference in forming predictions and generalizations cannot be conclusively justified or proven. Mitigated skepticism is also evident in the philosophical journey of Karl Popper. Furthermore, Popper demonstrates the value of fallibilism in his book '' The Open Society and Its Enemies'' (1945) by echoing the third maxim inscribed in the forecourt of the Temple of Apollo at Delphi: "surety brings ruin".
Fallibilism differs slightly from academic skepticism (also called ''global skepticism'', ''absolute skepticism'', ''universal skepticism'', '' radical skepticism'', or '' epistemological nihilism'') in the sense that fallibilists assume that no beliefs are certain (not even when established a priori), while proponents of academic skepticism advocate that no beliefs exist. In order to defend their position, these skeptics will either engage in epochē, a suspension of judgement, or they will resort to acatalepsy
In philosophy, acatalepsy (from the Greek ἀκαταληψία "inability to comprehend" from alpha privative and καταλαμβάνειν, "to seize") is incomprehensibleness, or the impossibility of comprehending or conceiving a thing. It ...

, a rejection of all knowledge. The concept of epoché is often accredited to Pyrrhonian skepticism, while the concept of acatalepsy can be traced back to multiple branches of skepticism. Acatalepsy is also closely related to the Socratic paradox. Nonetheless, epoché and acatalepsy are respectively self-contradictory and self-refuting, namely because both concepts rely (be it logically or methodologically) on its existence to serve as a justification. Lastly, local skepticism is the view that people cannot obtain knowledge of a particular area or subject (e.g. morality, religion, or metaphysics).
Criticism

Nearly all philosophers today are fallibilists in some sense of the term. Few would claim that knowledge requires absolute certainty, or deny that scientific claims are revisable, though in the 21st century some philosophers have argued for some version of infallibilist knowledge.Benton, Matthew (2021)"Knowledge, hope, and fallibilism"

''

Synthese
''Synthese'' () is a scholarly periodical specializing in papers in epistemology, methodology, and philosophy of science, and related issues. Its subject area is divided into four specialties, with a focus on the first three: (1) "epistemology, me ...

''. Vol. 198. pp. 1673–1689. Historically, many Western philosophers from Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institutio ...

to Saint Augustine to René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...

have argued that some human beliefs are infallibly known. Plausible candidates for infallible beliefs include logical truths ("Either Jones is a Democrat or Jones is not a Democrat"), immediate appearances ("It seems that I see a patch of blue"), and incorrigible beliefs (i.e., beliefs that are true in virtue of being believed, such as Descartes' "I think, therefore I am"). Many others, however, have taken even these types of beliefs to be fallible.
See also

* Defeasible reasoning * Logical holism * Pancritical rationalism * Perspectivism *Probabilism
In theology and philosophy, probabilism (from Latin ''probare'', to test, approve) is an ancient Greek doctrine of Academic skepticism. It holds that in the absence of certainty, plausibility or truth-likeness is the best criterion. The term can a ...

References

Further reading

*''Charles S. Peirce: Selected Writings'', by Philip P. Wiener (Dover, 1980) *''Charles S. Peirce and the Philosophy of Science'', by Edward C. Moore (Alabama, 1993) *''Treatise on Critical Reason'', by Hans Albert (Tübingen, 1968; English translation, Princeton, 1985)External links

*"Fallibilism"

by Stephen Hetherington in the ''

Internet Encyclopedia of Philosophy
The ''Internet Encyclopedia of Philosophy'' (''IEP'') is a scholarly online encyclopedia, dealing with philosophy, philosophical topics, and philosophers. The IEP combines open access publication with peer reviewed publication of original papers ...

''"Fallibilism"

by Nicholas Rescher in the ''

Routledge Encyclopedia of Philosophy
The ''Routledge Encyclopedia of Philosophy'' is an encyclopedia of philosophy edited by Edward Craig that was first published by Routledge in 1998 (). Originally published in both 10 volumes of print and as a CD-ROM, in 2002 it was made availabl ...

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