Fallibilism

Last updated
Charles Sanders Peirce around 1900. Peirce is said to have initiated fallibilism. Charles Sanders Peirce.jpg
Charles Sanders Peirce around 1900. Peirce is said to have initiated fallibilism.

Originally, fallibilism (from Medieval Latin: fallibilis, "liable to error") is the philosophical principle that propositions can be accepted even though they cannot be conclusively proven or justified, [1] [2] or that neither knowledge nor belief is certain. [3] The term was coined in the late nineteenth century by the American philosopher Charles Sanders Peirce, 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. [4] Furthermore, fallibilism is said to imply corrigibilism, the principle that propositions are open to revision. [5] Fallibilism is often juxtaposed with infallibilism.

Contents

Infinite regress and infinite progress

According to philosopher Scott F. Aikin, fallibilism cannot properly function in the absence of infinite regress. [6] The term, usually attributed to Pyrrhonist philosopher Agrippa, is argued to be the inevitable outcome of all human inquiry, since every proposition requires justification. [7] Infinite regress, also represented within the regress argument, is closely related to the problem of the criterion and is a constituent of the Münchhausen trilemma. 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 set forth the concept of "infinite progress". With this term, Hobbes had captured the human proclivity to strive for perfection. [8] Philosophers like Gottfried Wilhelm Leibniz, Christian Wolff, and Immanuel Kant, would elaborate further on the concept. Kant even went on to speculate that immortal species should hypothetically be able to develop their capacities to perfection. [9]

Already in 350 B.C.E, Greek philosopher Aristotle 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 sets indefinitely. It eventually led him to refute some of Zeno's paradoxes. [10] 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, but rather necessary antecedents for knowledge acquisition. They allow us to live functional and meaningful lives. [11]

Critical rationalism

The founder of critical rationalism: Karl Popper Karl Popper2.jpg
The founder of critical rationalism: Karl Popper

In the mid-twentieth century, several important philosophers began to critique the foundations of logical positivism. In his work The Logic of Scientific Discovery (1934), Karl Popper, the founder of critical rationalism, argued that scientific knowledge grows from falsifying conjectures rather than any inductive principle and that falsifiability is the criterion of a scientific proposition. 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. [12]

Furthermore, Popper defended his critical rationalism as a normative and methodological theory, that explains how objective, and thus mind-independent, knowledge ought to work. [13] Hungarian philosopher Imre Lakatos 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. [14] When Lakatos urged Popper to admit that the falsification principle cannot be justified without embracing induction, Popper did not succumb. [15] Lakatos' critical attitude towards rationalism has become emblematic for his so called critical fallibilism. [16] [17] While critical fallibilism strictly opposes dogmatism, critical rationalism is said to require a limited amount of dogmatism. [18] [19] 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 can be justified by the truth of their consequences. [16] 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. [15] [17] [18] [20]

Fallibilism has also been employed by philosopher Willard V. O. Quine to attack, among other things, the distinction between analytic and synthetic statements. [21] 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. [22] Critical rationalist Hans Albert argues that it is impossible to prove any truth with certainty, not only in logic, but also in mathematics. [23]

Mathematical fallibilism

Imre Lakatos, in the 1960s, known for his contributions to mathematical fallibilism Professor Imre Lakatos, c1960s.jpg
Imre Lakatos, in the 1960s, known for his contributions to mathematical fallibilism

In Proofs and Refutations: The Logic of Mathematical Discovery (1976), philosopher Imre Lakatos implemented mathematical proofs into what he called Popperian "critical fallibilism". [24] Lakatos's mathematical fallibilism is the general view that all mathematical theorems are falsifiable. [25] Mathematical fallibilism deviates from traditional views held by philosophers like Hegel, Peirce, and Popper. [16] [25] Although Peirce introduced fallibilism, he seems to preclude the possibility of us being mistaken in our mathematical beliefs. [2] 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 may provide us with consistency amidst an inherent incompleteness in mathematics. [26] 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. [27]

In the philosophy of mathematics, a central tenet of fallibilism is undecidability (which bears resemblance to the notion of isostheneia, or "equal veracity"). [25] Two distinct types of the word "undecidable" are currently being applied. The first one relates, most notably, to the continuum hypothesis, which was proposed by mathematician Georg Cantor in 1873. [28] [29] The continuum hypothesis represents a tendency for infinite sets to allow for undecidable solutions — solutions which are true in one constructible universe and false in another. Both solutions are independent from the axioms in Zermelo–Fraenkel set theory combined with the axiom of choice (also called ZFC). This phenomenon has been labeled the independence of the continuum hypothesis. [30] Both the hypothesis and its negation are thought to be consistent with the axioms of ZFC. [31] Many noteworthy discoveries have preceded the establishment of the continuum hypothesis.

In 1877, Cantor introduced the diagonal argument to prove that the cardinality of two finite sets is equal, by putting them into a one-to-one correspondence. [32] Diagonalization reappeared in Cantors theorem, in 1891, to show that the power set of any countable set must have strictly higher cardinality. [33] 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. [33] Two years later, polymath Bertrand Russell would invalidate the existence of the universal set 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. [34] 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, [28] and after 1963, that fellow mathematician Paul Cohen revealed, through the method of forcing, that the continuum hypothesis cannot be proved either. [30] In spite of the undecidability, both Gödel and Cohen suspected dependence of 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. [35] 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. [36]

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 is a type of computational problem in which there are countably infinite sets of questions, each requiring an effective method 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 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. [37] Famous examples of undecidable problems are the halting problem, the Entscheidungsproblem, and the unsolvability of the Diophantine equation. Conventionally, an undecidable problem is derived from a recursive set, formulated in undecidable language, and measured by the Turing degree. [38] [39] Undecidability, with respect to computer science and mathematical logic, is also called unsolvability or non-computability.

Undecidability and uncertainty are not one and the same phenomenon. Mathematical theorems which can be formally proved, will, according to mathematical fallibilists, nevertheless remain inconclusive. [40] Take for example proof of the independence of the continuum hypothesis or, even more fundamentally, proof of the diagonal argument. In the end, both types of undecidability add further nuance to fallibilism, by providing these fundamental thought-experiments. [41]

Philosophical skepticism

Fallibilism should not be confused with local or global skepticism, which is the view that some or all types of knowledge are unattainable.

But the fallibility of our knowledge — or the thesis that all knowledge is guesswork, though some consists of guesses which have been most severely tested — must not be cited in support of scepticism or relativism. From the fact that we can err, and that a criterion of truth which might save us from error does not exist, it does not follow that the choice between theories is arbitrary, or non-rational: that we cannot learn, or get nearer to the truth: that our knowledge cannot grow.

— Karl Popper

Fallibilism claims that legitimate epistemic justifications can lead to false beliefs, whereas academic skepticism claims that no legitimate epistemic justifications exist (acatalepsy). Fallibilism is also different to epoché, a suspension of judgement, often accredited to Pyrrhonian skepticism.

Criticism

Nearly all philosophers today are fallibilists in some sense of the term. [3] 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. [42] [43] [44] Historically, many Western philosophers from Plato to Saint Augustine to René Descartes have argued that some human beliefs are infallibly known. John Calvin espoused a theological fallibilism towards others beliefs. [45] [46] 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. [22]

See also

Related Research Articles

In mathematics, specifically set theory, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. It states that

there is no set whose cardinality is strictly between that of the integers and the real numbers,

<span class="mw-page-title-main">Charles Sanders Peirce</span> American thinker who founded pragmatism (1839–1914)

Charles Sanders Peirce was an American scientist, mathematician, logician, and philosopher who is sometimes known as "the father of pragmatism". According to philosopher Paul Weiss, Peirce was "the most original and versatile of America's philosophers and America's greatest logician". Bertrand Russell wrote "he was one of the most original minds of the later nineteenth century and certainly the greatest American thinker ever".

<span class="mw-page-title-main">Empiricism</span> Idea that knowledge comes only/mainly from sensory experience

In philosophy, empiricism is an epistemological view which holds that true knowledge or justification comes only or primarily from sensory experience and empirical evidence. It is one of several competing views within epistemology, along with rationalism and skepticism. Empiricists argue that empiricism is a more reliable method of finding the truth than purely using logical reasoning, because humans have cognitive biases and limitations which lead to errors of judgement. Empiricism emphasizes the central role of empirical evidence in the formation of ideas, rather than innate ideas or traditions. Empiricists may argue that traditions arise due to relations of previous sensory experiences.

<span class="mw-page-title-main">Falsifiability</span> Property of a statement that can be logically contradicted

Falsifiability is a deductive standard of evaluation of scientific theories and hypotheses, introduced by the philosopher of science Karl Popper in his book The Logic of Scientific Discovery (1934). A theory or hypothesis is falsifiable if it can be logically contradicted by an empirical test.

<span class="mw-page-title-main">Karl Popper</span> Austrian–British philosopher of science (1902–1994)

Sir Karl Raimund Popper was an Austrian–British philosopher, academic and social commentator. One of the 20th century's most influential philosophers of science, Popper is known for his rejection of the classical inductivist views on the scientific method in favour of empirical falsification. According to Popper, a theory in the empirical sciences can never be proven, but it can be falsified, meaning that it can be scrutinised with decisive experiments. Popper was opposed to the classical justificationist account of knowledge, which he replaced with critical rationalism, namely "the first non-justificational philosophy of criticism in the history of philosophy".

<span class="mw-page-title-main">Imre Lakatos</span> Hungarian philosopher of mathematics and science

Imre Lakatos 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 pre-axiomatic stages of development, and also for introducing the concept of the "research programme" in his methodology of scientific research programmes.

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's lives.

Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be vague. Foundations of mathematics can be conceived as the study of the basic mathematical concepts and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics. The search for foundations of mathematics is a central question of the philosophy of mathematics; the abstract nature of mathematical objects presents special philosophical challenges.

Critical rationalism is an epistemological philosophy advanced by Karl Popper on the basis that, if a statement cannot be logically deduced, it might nevertheless be possible to logically falsify it. Following Hume, Popper rejected any inductive logic that is ampliative, i.e., any logic that can provide more knowledge than deductive logic. This led Popper to his falsifiability criterion.

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 only if the person holding that belief could not be mistaken in holding that belief. Other common definitions of certainty involve the indubitable nature of such beliefs or define certainty as a property of those beliefs with the greatest possible justification. Certainty is closely related to knowledge, although contemporary philosophers tend to treat knowledge as having lower requirements than certainty.

Verificationism, also known as the verification principle or the verifiability criterion of meaning, is the philosophical doctrine which asserts that a statement is meaningful only if it is either empirically verifiable or a truth of logic.

<span class="mw-page-title-main">Münchhausen trilemma</span> A thought experiment used to demonstrate the impossibility of proving any truth

In epistemology, the Münchhausen trilemma is a thought experiment intended to demonstrate the theoretical impossibility of proving any truth, even in the fields of logic and mathematics, without appealing to accepted assumptions. If it is asked how any given proposition is known to be true, proof in support of that proposition may be provided. Yet that same question can be asked of that supporting proof, and any subsequent supporting proof. The Münchhausen trilemma is that there are only three ways of completing a proof:

Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.

The following outline is provided as an overview of and topical guide to epistemology:

Inductivism is the traditional and still commonplace philosophy of scientific method to develop scientific theories. Inductivism aims to neutrally observe a domain, infer laws from examined cases—hence, inductive reasoning—and thus objectively discover the sole naturally true theory of the observed.

Conventionalism is the philosophical attitude that fundamental principles of a certain kind are grounded on agreements in society, rather than on external reality. Unspoken rules play a key role in the philosophy's structure. Although this attitude is commonly held with respect to the rules of grammar, its application to the propositions of ethics, law, science, biology, mathematics, and logic is more controversial.

Epistemology or theory of knowledge is the branch of philosophy concerned with the nature and scope (limitations) of knowledge. It addresses the questions "What is knowledge?", "How is knowledge acquired?", "What do people know?", "How do we know what we know?", and "Why do we know what we know?". Much of the debate in this field has focused on analyzing the nature of knowledge and how it relates to similar notions such as truth, belief, and justification. It also deals with the means of production of knowledge, as well as skepticism about different knowledge claims.

A timeline of mathematical logic; see also history of logic.

Bold hypothesis or bold conjecture is a concept in the philosophy of science of Karl Popper, first explained in his debut The Logic of Scientific Discovery (1935) and subsequently elaborated in writings such as Conjectures and Refutations: The Growth of Scientific Knowledge (1963). The concept is nowadays widely used in the philosophy of science and in the philosophy of knowledge. It is also used in the social and behavioural sciences.

References

  1. ↑ Peirce, Charles S. (1896–1899) "The Scientific Attitude and Fallibilism". In Buchler, Justus (1940). Philosophical Writings of Peirce. Routledge. p. 59.
  2. 1 2 Haack, Susan (1979). "Fallibilism and Necessity". Synthese , Vol. 41, No. 1, pp. 37–63.
  3. 1 2 Hetherington, Stephen. "Fallibilism". Internet Encyclopedia of Philosophy .
  4. ↑ Anastas, Jeane W. (1999). Research Design for Social Work and the Human Services. Columbia University Press. p. 19.
  5. ↑ Levi, Isaac (1984). Messianic vs Myopic Realism. The University of Chicago Press. Vol. 2. pp. 617–636.
  6. ↑ Aikin, Scott F. (2014). "Prospects for Moral Epistemic Infinitism". Metaphilosophy . Vol. 45. No. 2. pp. 172–181.
  7. ↑ Annas, Julia & Barnes, Jonathan (2000). Sextus Empiricus: Outlines of Scepticism . Cambridge University Press.
  8. ↑ Hobbes, Thomas (1974). De homine . Cambridge University Press. Vol. 20.
  9. ↑ Rorty, Amélie; Schmidt, James (2009). Kant's Idea for a Universal History with a Cosmopolitan Aim . Cambridge University Press.
  10. ↑ Aristotle (350 B.C.E.). Physics . Massachusetts Institute of Technology.
  11. ↑ Cooke, Elizabeth F. (2006). Peirce's Pragmatic Theory of Inquiry: Fallibilism and Indeterminacy. Continuum.
  12. ↑ Kuhn, Thomas S. (1962). The Structure of Scientific Revolutions. University of Chicago Press.
  13. ↑ Thornton, Stephen (2022). Karl Popper. Stanford Encyclopedia of Philosophy.
  14. ↑ Forrai, Gábor (2002). Lakatos, Reason and History. p. 6–7. In Kampis, George; Kvasz, Ladislav & Stöltzner, Michael (2002). Appraising Lakatos: Mathematics, Methodology, and the Man. Vienna Circle Institute Library. Vol. 1.
  15. 1 2 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.
  16. 1 2 3 Musgrave, Alan; Pigden, Charles (2021). "Imre Lakatos". Stanford Encyclopedia of Philosophy .
  17. 1 2 Kiss, Ogla (2006). Heuristic, Methodology or Logic of Discovery? Lakatos on Patterns of Thinking. MIT Press Direct. p. 314.
  18. 1 2 Lakatos, Imre (1978). Mathematics, Science and Epistemology. Cambridge University Press. Vol. 2. p. 9–23.
  19. ↑ Popper, Karl (1995). The Myth of the Framework: In Defence of Science and Rationality. Routledge. p. 16.
  20. ↑ Kockelmans, Joseph J. Reflections on Lakatos’ Methodology of Scientific Research Programs. Boston Studies in the Philosophy of Science. Vol. 59. pp. 187–203. In Radnitzky, Gerard & Andersson, Gunnar. The Structure and Development of Science. D. Reidel Publishing Company.
  21. ↑ Quine, Willard. V. O. (1951). "Two Dogmas of Empiricism". The Philosophical Review Vol. 60, No. 1, pp. 20–43.
  22. 1 2 Haack, Susan (1978). Philosophy of Logics . Cambridge University Press. pp. 234; Chapter 12.
  23. ↑ Niemann, Hans-Joachim (2000). Hans Albert - critical rationalist . Opensociety.
  24. ↑ Lakatos, Imre (1962). "Infinite Regress and Foundations of Mathematics (in Symposium: The Foundations of Mathematics)". Proceedings of the Aristotelian Society, Supplementary Volumes. 36: 145–184 (165). JSTOR   4106691. Popperian critical fallibilism takes the infinite regress in proofs and definitions seriously, does not have illusions about 'stopping' them, accepts the sceptic criticism of any infallible truth-injection. However, Lakatos' interpretation of Popper was not equivalent to Popper's philosophy: Ravn, Ole; Skovsmose, Ole (2019). "Mathematics as Dialogue". Connecting Humans to Equations: A Reinterpretation of the Philosophy of Mathematics. History of Mathematics Education. Cham: Springer-Verlag. pp. 107–119 (110). doi:10.1007/978-3-030-01337-0_8. ISBN   9783030013363. S2CID   127561458. Lakatos also refers to the scepticist programme as a 'Popperian critical fallibilism.' However, we find that this labelling could be a bit misleading as the programme includes a good deal of Lakatos' own philosophy.
  25. 1 2 3 Kadvany, John (2001). Imre Lakatos and the Guises of Reason. Duke University Press. pp. 45, 109, 155, 323.
  26. ↑ Falguera, José L.; Rivas, Uxía; Sagüillo, José M. (1999). Analytic Philosophy at the Turn of the Millennium: Proceedings of the International Congress: Santiago de Compostela, 1–4 December, 1999 . Servicio de Publicacións da Universidade de Santiago de Compostela. pp. 259–261.
  27. ↑ Dove, Ian J. (2004). Certainty and error in mathematics: Deductivism and the claims of mathematical fallibilism . Dissertation, Rice University. pp. 120-122.
  28. 1 2 Gödel, Kurt (1940). The Consistency of the Continuum-Hypothesis . Princeton University Press. Vol. 3.
  29. ↑ Enderton, Herbert. "Continuum hypothesis". Encyclopædia Britannica .
  30. 1 2 Cohen, Paul (1963). "The Independence of the Continuum Hypothesis". Proceedings of the National Academy of Sciences of the United States of America . Vol. 50, No. 6. pp. 1143–1148.
  31. ↑ Goodman, Nicolas D. (1979) "Mathematics as an Objective Science". The American Mathematical Monthly . Vol. 86, No. 7. pp. 540-551.
  32. ↑ Cantor, Georg (1877). "Ein Beitrag zur Mannigfaltigkeitslehre". Journal für die reine und angewandte Mathematik . Vol. 84. pp. 242-258.
  33. 1 2 Hosch, William L. "Cantor's theorem". Encyclopædia Britannica .
  34. ↑ Forster, Thomas. E. (1992). Set Theory with a Universal Set: Exploring an Untyped Universe . Clarendon Press.
  35. ↑ Goodman, Nicolas D. (1979) "Mathematics as an Objective Science". The American Mathematical Monthly . Vol. 86, No. 7. pp. 540-551.
  36. ↑ Marciszewskip, Witold (2015). "On accelerations in science driven by daring ideas: Good messages from fallibilistic rationalism". Studies in Logic, Grammar and Rhetoric 40, no. 1. p. 39.
  37. ↑ Rogers, Hartley (1971). "Theory of Recursive Functions and Effective Computability". Journal of Symbolic Logic . Vol. 36. pp. 141–146.
  38. ↑ Post, Emil L. (1944). Recursively enumerable sets of positive integers and their decision problems . Bulletin of the American Mathematical Society.
  39. ↑ Kleene, Stephen C.; Post, Emil L. (1954). "The upper semi-lattice of degrees of recursive unsolvability". Annals of Mathematics . Vol. 59. No. 3. pp. 379–407.
  40. ↑ Dove, Ian J. (2004). Certainty and error in mathematics: Deductivism and the claims of mathematical fallibilism . Dissertation, Rice University. pp. 120-122.
  41. ↑ Lakatos, Imre; Worrall, John & Zahar, Elie (1976). "Proofs and Refutations". Cambridge University Press. p. 9.
  42. ↑ Moon, Andrew (2012). "Warrant does entail truth". Synthese . Vol. 184. No. 3. pp. 287–297.
  43. ↑ Dutant, Julien (2016). "How to be an infallibilist". Philosophical Issues . Vol 26. pp. 148–171.
  44. ↑ Benton, Matthew (2021). "Knowledge, hope, and fallibilism". Synthese . Vol. 198. pp. 1673–1689.
  45. ↑ Friedman, J. (2022). Intolerance: Premodern Roots and Modern Manifestations. Taylor & Francis. p. 12. ISBN   978-1-000-80294-8 . Retrieved 2023-06-01.
  46. ↑ Helm, P. (2010). Calvin at the Centre. OUP Oxford. p. 92. ISBN   978-0-19-953218-6 . Retrieved 2023-06-01.

Further reading

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.