Originally, fallibilism (from

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 functione ...

: ''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 n ...

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 Descriptive knowledge, awareness of facts or as Procedural knowledge, practical skills, and may also refer to Knowledge by acquaintance, familiarity with objects or situations. Knowledge of facts, also called pro ...

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 take ...

is certain.Hetherington, Stephen
"Fallibilism"
'' Internet Encyclopedia of Philosophy''. 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 t ...

, as a response to foundationalism. Theorists, following Austrian-British philosopher Karl Popper, 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 of

infinite 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 beca ...

. The term, usually attributed to

Pyrrhonist Pyrrho of Elis (; grc, Πύρρων ὁ Ἠλεῖος, Pyrrhо̄n ho Ēleios; ), born in Elis, Greece, was a Greek philosopher of Classical antiquity, credited as being the first Greek skeptic philosopher and founder of Pyrrhonism. Life ...

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 * Agri ...

, 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 In the field of epistemology, the problem of the criterion is an issue regarding the starting point of knowledge. This is a separate and more fundamental issue than the regress argument found in discussions on justification of knowledge. In W ...

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 A cosmological argument, in natural theology, is an argument which claims that the existence of God can be inferred from facts concerning causation, explanation, change, motion, contingency, dependency, or finitude with respect to the universe ...

, 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 influ ...

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 math ...

, 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 ...

, 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 Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earli ...

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, ...

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

, economic growth,

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 su ...

, 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 materia ...

. 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 In economics, hyperinflation is a very high and typically accelerating inflation. It quickly erodes the real value of the local currency, as the prices of all goods increase. This causes people to minimize their holdings in that currency as t ...

, 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 plural ...

. Other relevant examples of potential infinities include

Galileo's paradox Galileo's paradox is a demonstration of one of the surprising properties of infinite sets. In his final scientific work, ''Two New Sciences'', Galileo Galilei made apparently contradictory statements about the positive integers. First, some numbers ...

and the paradox of

Hilbert's hotel Hilbert's paradox of the Grand Hotel (colloquial: Infinite Hotel Paradox or Hilbert's Hotel) is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely m ...

. 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, 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 of logical positivism. 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 Erkenntnisthe ...

'' (1934), Karl Popper, the founder of critical rationalism, tried to solve the problem of induction 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 s ...

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 involv ...

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 Causality, cause-and-effect by demonstrating what outcome oc ...

ation, but rather deduced from statements and justified by means of deliberation and intersubjective consensus within a particular

scientific community The scientific community is a diverse network of interacting scientists. It includes many " sub-communities" working on particular scientific fields, and within particular institutions; interdisciplinary and cross-institutional activities are als ...

. 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's ...

have been turned into

methodological In its most common sense, methodology is the study of research methods. However, the term can also refer to the methods themselves or to the philosophical discussion of associated background assumptions. A method is a structured procedure for bri ...

refutations.Thornton, Stephen (2021)
"Karl Popper"
'' Stanford Encyclopedia of Philosophy''. The claim that all assertions are provisional and thus open to revision in light of new evidence 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 repeatab ...

. 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 An illusion is a distortion of the senses, which can reveal how the mind normally organizes and interprets sensory stimulation. Although illusions distort the human perception of reality, they are generally shared by most people. Illusions may oc ...

, because in the action of falsifying an argument, scientists will inevitably verify its negation. Thus, verification and falsification are perfectly symmetrical. It seems, in the

philosophy of logic Philosophy of logic is the area of philosophy that studies the scope and nature of logic. It investigates the philosophical problems raised by logic, such as the presuppositions often implicitly at work in theories of logic and in their application ...

, 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 Objective may refer to: * Objective (optics), an element in a camera or microscope * ''The Objective'', a 2008 science fiction horror film * Objective pronoun, a personal pronoun that is used as a grammatical object * Objective Productions, a Brit ...

, 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''.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 Dogma is a belief or set of beliefs that is accepted by the members of a group without being questioned or doubted. It may be in the form of an official system of principles or doctrines of a religion, such as Roman Catholicism, Judaism, Isla ...

, 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 f ...

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 Susan Haack (born 1945) is a distinguished professor in the humanities, Cooper Senior Scholar in Arts and Sciences, professor of philosophy, and professor of law at the University of Miami in Coral Gables, Florida. Haack has written on logic, ...

, 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 Hans Albert (born 8 February 1921) is a German philosopher. Born in Cologne, he lives in Heidelberg. His fields of research are Social Sciences and General Studies of Methods. He is a critical rationalist, paying special attention to rational ...

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), philosopher

implemented mathematical proofs into what he called Popperian "critical fallibilism". Lakatos's mathematical fallibilism is the general view that all mathematical

theorems In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the ...

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, 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 clo ...

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 ...

amidst an inherent incompleteness in mathematics. Mathematical fallibilism differs from quasi-empiricism, to the extent that the latter does not incorporate

inductivism Inductivism is the traditional and still commonplace philosophy of scientific method to develop scientific theories.James Ladyman, ''Understanding Philosophy of Science'' (London & New York: Routledge, 2002), p51��58 Inductivism aims to neutrally ...

, 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 conce ...

. 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 peop ...

, 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 A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A for ...

.Gödel, Kurt (1940).
The Consistency of the Continuum-Hypothesis
'. Princeton University Press. Vol. 3. The continuum 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 of ...

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 ...

(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 **Tarski ...

to prove that the cardinality of two finite sets is equal, by putting them into a

one-to-one correspondence In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...

. 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 post ...

of any countable set 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 In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory. In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: :\forall x \, \exists y \, \forall z \, \in y \iff \forall w \ ...

; a vital part of Zermelo–Fraenkel set theory. Moreover, in 1899,

Cantor's paradox In set theory, Cantor's paradox states that there is no set of all cardinalities. This is derived from the theorem that there is no greatest cardinal number. In informal terms, the paradox is that the collection of all possible "infinite sizes" is ...

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, ...

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 In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains ...

, 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 In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any ...

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 In mathematical logic, the diagonal lemma (also known as diagonalization lemma, self-reference lemma or fixed point theorem) establishes the existence of self-referential sentences in certain formal theories of the natural numbers—specificall ...

, that the continuum hypothesis cannot be refuted, and after 1963, that fellow mathematician

Paul Cohen Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician. He is best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was award ...

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 Sc ...

''. 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 theoretical computer science, a computational problem is a problem that may be solved by an algorithm. For example, the problem of factoring :"Given a positive integer ''n'', find a nontrivial prime factor of ''n''." is a computational probl ...

in which there are countably infinite 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 or ...

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 ...

Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...

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 Undecidable may refer to: * Undecidable problem in computer science and mathematical logic, a decision problem that no algorithm can decide, formalized as an undecidable language or undecidable set * "Undecidable", sometimes also used as a synony ...

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 ...

and the

Entscheidungsproblem In mathematics and computer science, the ' (, ) is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. The problem asks for an algorithm that considers, as input, a statement and answers "Yes" or "No" according to whether the state ...

. Conventionally, an undecidable problem is derived from a

recursive set In computability theory, a set of natural numbers is called computable, recursive, or decidable if there is an algorithm which takes a number as input, terminates after a finite amount of time (possibly depending on the given number) and correctly ...

, formulated in

undecidable language 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 ...

, and measured by the

Turing degree In computer science and mathematical logic the Turing degree (named after Alan Turing) or degree of unsolvability of a set of natural numbers measures the level of algorithmic unsolvability of the set. Overview The concept of Turing degree is fund ...

. 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 anc ...

Philosophical skepticism

Fallibilism improves upon the ideas associated with

philosophical 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 uncertainty ...

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 phil ...

'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 ...

, 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 ''The Open Society and Its Enemies'' is a work on political philosophy by the philosopher Karl Popper, in which the author presents a "defence of the open society against its enemies", and offers a critique of theories of teleological historicism ...

'' (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 Academic skepticism refers to the skeptical period of ancient Platonism dating from around 266 BCE, when Arcesilaus became scholarch of the Platonic Academy, until around 90 BCE, when Antiochus of Ascalon rejected skepticism, although indi ...

(also called ''global skepticism'', ''absolute skepticism'', ''universal skepticism'', ''

radical skepticism Radical skepticism (or radical scepticism in British English) is the philosophical position that knowledge is most likely impossible. Radical skeptics hold that doubt exists as to the veracity of every belief and that certainty is therefore n ...

'', or ''

epistemological nihilism 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 ...

'') 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 Pyrrhonism is a school of philosophical skepticism founded by Pyrrho in the fourth century BCE. It is best known through the surviving works of Sextus Empiricus, writing in the late second century or early third century CE. History Pyrrho of E ...

, 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 Local skepticism is the view that one cannot possess knowledge in some particular domain. It contrasts with global skepticism (also known as absolute skepticism or universal skepticism), the view that one cannot know anything at all. Examples ...

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"
''

''. 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 institution ...

Saint Augustine Augustine of Hippo ( , ; la, Aurelius Augustinus Hipponensis; 13 November 354 – 28 August 430), also known as Saint Augustine, was a theologian and philosopher of Berber origin and the bishop of Hippo Regius in Numidia, Roman North Afr ...

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. Ma ...

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.

References

External links

*
"Fallibilism"
by Stephen Hetherington in the '' Internet Encyclopedia of Philosophy''
"Fallibilism"
by

Nicholas Rescher Nicholas Rescher (; ; born 15 July 1928) is a German-American philosopher, polymath, and author, who has been a professor of philosophy at the University of Pittsburgh since 1961. He is chairman of the Center for Philosophy of Science and was fo ...

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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