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Certainty (also known as epistemic certainty or objective certainty) is the epistemic property of beliefs which a person has no rational grounds for doubting.[1] 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.[1]

Importantly, epistemic certainty is not the same thing as psychological certainty (also known as subjective certainty or certitude), which describes the highest degree to which a person could be convinced that something is true. While a person may be completely convinced that a particular belief is true, and might even be psychologically incapable of entertaining its falsity, this does not entail that the belief is itself beyond rational doubt or incapable of being false.[2] While the word "certainty" is sometimes used to refer to a person's subjective certainty about the truth of a belief, philosophers are primarily interested in the question of whether any beliefs ever attain objective certainty.

The philosophical question of whether one can ever be truly certain about anything has been widely debated for centuries. Many proponents of philosophical skepticism deny that certainty is possible, or claim that it is only possible in a priori domains such as logic or mathematics. Historically, many philosophers have held that knowledge requires epistemic certainty, and therefore that one must have infallible justification in order to count as knowing the truth of a proposition. However, many philosophers such as René Descartes were troubled by the resulting skeptical implications, since all of our experiences at least seem to be compatible with various skeptical scenarios. It is generally accepted today that most of our beliefs are compatible with their falsity and are therefore fallible, although the status of being certain is still often ascribed to a limited range of beliefs (such as "I exist"). The apparent fallibility of our beliefs has led many contemporary philosophers to deny that knowledge requires certainty.[1]

Ludwig Wittgenstein – 20th century

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If you tried to doubt everything you would not get as far as doubting anything. The game of doubting itself presupposes certainty.

On Certainty is a series of notes made by Ludwig Wittgenstein just prior to his death. The main theme of the work is that context plays a role in epistemology. Wittgenstein asserts an anti-foundationalist message throughout the work: that every claim can be doubted but certainty is possible in a framework. "The function [propositions] serve in language is to serve as a kind of framework within which empirical propositions can make sense".[3]

Degrees of certainty

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See also: Inductive reasoning, Probability interpretations, and Philosophy of statistics

Physicist Lawrence M. Krauss suggests that the need for identifying degrees of certainty is under-appreciated in various domains, including policy-making and the understanding of science. This is because different goals require different degrees of certainty – and politicians are not always aware of (or do not make it clear) how much certainty we are working with.[4]

Rudolf Carnap viewed certainty as a matter of degree ("degrees of certainty") which could be objectively measured, with degree one being certainty. Bayesian analysis derives degrees of certainty which are interpreted as a measure of subjective psychological belief.

Alternatively, one might use the legal degrees of certainty. These standards of evidence ascend as follows: no credible evidence, some credible evidence, a preponderance of evidence, clear and convincing evidence, beyond reasonable doubt, and beyond any shadow of a doubt (i.e. undoubtable – recognized as an impossible standard to meet – which serves only to terminate the list).

If knowledge requires absolute certainty, then knowledge is most likely impossible, as evidenced by the apparent fallibility of our beliefs.

Foundational crisis of mathematics

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Main article: Foundations of mathematics § Foundational crisis

The foundational crisis of mathematics was the early 20th century's term for the search for proper foundations of mathematics.

After several schools of the philosophy of mathematics ran into difficulties one after the other in the 20th century, the assumption that mathematics had any foundation that could be stated within mathematics itself began to be heavily challenged.

One attempt after another to provide unassailable foundations for mathematics was found to suffer from various paradoxes (such as Russell's paradox) and to be inconsistent.

Various schools of thought were opposing each other. The leading school was that of the formalist approach, of which David Hilbert was the foremost proponent, culminating in what is known as Hilbert's program, which sought to ground mathematics on a small basis of a formal system proved sound by metamathematical finitistic means. The main opponent was the intuitionist school, led by L.E.J. Brouwer, which resolutely discarded formalism as a meaningless game with symbols.[5] The fight was acrimonious. In 1920 Hilbert succeeded in having Brouwer, whom he considered a threat to mathematics, removed from the editorial board of Mathematische Annalen, the leading mathematical journal of the time.

Gödel's incompleteness theorems, proved in 1931, showed that essential aspects of Hilbert's program could not be attained. In Gödel's first result he showed how to construct, for any sufficiently powerful and consistent finitely axiomatizable system – such as necessary to axiomatize the elementary theory of arithmetic – a statement that can be shown to be true, but that does not follow from the rules of the system. It thus became clear that the notion of mathematical truth cannot be reduced to a purely formal system as envisaged in Hilbert's program. In a next result Gödel showed that such a system was not powerful enough for proving its own consistency, let alone that a simpler system could do the job. This proves that there is no hope to prove the consistency of any system that contains an axiomatization of elementary arithmetic, and, in particular, to prove the consistency of Zermelo–Fraenkel set theory (ZFC), the system which is generally used for building all mathematics.

However, if ZFC is not consistent, there exists a proof of both a theorem and its negation, and this would imply a proof of all theorems and all their negations. As, despite the large number of mathematical areas that have been deeply studied, no such contradiction has ever been found, this provides an almost certainty of mathematical results. Moreover, if such a contradiction would eventually be found, most mathematicians are convinced that it will be possible to resolve it by a slight modification of the axioms of ZFC.

Moreover, the method of forcing allows proving the consistency of a theory, provided that another theory is consistent. For example, if ZFC is consistent, adding to it the continuum hypothesis or a negation of it defines two theories that are both consistent (in other words, the continuum is independent from the axioms of ZFC). This existence of proofs of relative consistency implies that the consistency of modern mathematics depends weakly on a particular choice on the axioms on which mathematics are built.

In this sense, the crisis has been resolved, as, although consistency of ZFC is not provable, it solves (or avoids) all logical paradoxes at the origin of the crisis, and there are many facts that provide a quasi-certainty of the consistency of modern mathematics.

See also

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Certainty

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Certainty is a fundamental concept in , denoting the absence of regarding the truth of a , and it encompasses both subjective and objective dimensions. Psychologically, certainty involves a state of complete or assurance without reservation, as when an individual is utterly convinced of a fact based on personal conviction. Epistemically, it represents the highest attainable status of justification for a , where the renders the immune to rational or error. This distinction highlights how certainty can be felt internally while also demanding robust external warranting conditions. Historically, the pursuit of certainty has shaped philosophical thought, beginning with ancient definitions of knowledge as demonstrative certainty. Aristotle characterized scientific knowledge () as the syllogistic proof of essential truths, establishing a foundational ideal of unassailable reasoning. In the modern era, elevated certainty to the cornerstone of in his (1641), where he employed methodical doubt to identify indubitable truths, such as the cogito ("I think, therefore I am"), as the bedrock for all knowledge, immune to skeptical challenges. John Locke further developed the notion of moral certainty, describing it as a high degree of probability sufficient for guiding practical actions, even if not absolutely infallible, thereby bridging theoretical rigor with everyday decision-making. These views influenced subsequent thinkers, including , who critiqued absolute certainty in favor of probabilistic reasoning based on . In contemporary , certainty remains a contested ideal, particularly in debates over its necessity for . Fallibilists argue that knowledge does not require certainty, allowing for justified true beliefs that are potentially revisable, while infallibilists maintain that genuine knowledge entails epistemic certainty to exclude error possibilities. This tension appears in discussions of , where radical doubt questions whether certainty is ever achievable beyond trivial cases. Additionally, moral certainty—defined as probability so overwhelming as to preclude —persists in applied contexts like and , where it justifies actions without demanding metaphysical absolutes. Recent scholarship, such as Bob Beddor's , revives certainty's by linking it to evidential support, assertion norms, and contextual variations in epistemic standards, underscoring its ongoing relevance in understanding belief and justification.

Core Concepts

Epistemic Certainty

Epistemic certainty refers to the epistemic property of a that attains the highest possible justification, characterized by the complete absence of rational grounds for regarding a proposition's truth. This concept is fundamentally tied to epistemology's core concerns with truth and justification, where a is epistemically certain only if it is indubitable and necessarily true given the subject's cognitive faculties. In relation to the justified true belief (JTB) account of , epistemic certainty demands more than mere justification, truth, and ; it requires , ensuring the cannot possibly be false under the conditions of justification. Gettier's 1963 paper demonstrated that JTB can fail to constitute in cases involving epistemic , such as when a justified true rests on a false intermediate premise, thereby challenging the sufficiency of JTB and highlighting the need for certainty's stricter standard to secure genuine . Some responses to Gettier problems, such as indefeasibility (justification without defeaters), aim to exclude such ; epistemic certainty, by requiring , would also address them but imposes a stricter standard. A classic example of epistemic certainty is ' formulation in his : "" (I think, therefore I am), which emerges as indubitable even amid radical , as the very act of thinking guarantees the thinker's existence through immediate . This foundational truth exemplifies how epistemic certainty serves as an unshakeable starting point for further claims. Unlike mere probabilistic , which allows for degrees of likelihood, epistemic certainty insists on absolute , precluding any rational error possibility. In contrast, psychological certainty involves a subjective sense of that may not align with objective justification.

Psychological Certainty

Psychological certainty refers to the subjective feeling of or in one's beliefs, judgments, or decisions, distinct from objective justification or truth. This is often characterized as an emotional or cognitive assurance that operates independently of external , influencing how individuals perceive and act upon their convictions. In , it is typically defined as the degree of perceived validity or stability in one's mental representations, such as attitudes or memories, without requiring epistemic warrant. In , psychological certainty is commonly assessed through self-reported scales that capture individuals' subjective levels. For instance, the Certainty About Mental States (CAMSQ) measures perceived capacity to understand one's own and others' mental states on a Likert-type scale, where higher scores indicate greater subjective certainty. Other instruments, such as attitude certainty scales, evaluate the perceived clarity and correctness of beliefs via items like "I feel certain about this ," allowing researchers to quantify this internal conviction reliably. These tools highlight certainty as a metacognitive , where people rate their assurance post-decision or judgment. Cognitive biases significantly shape psychological certainty, often leading to inflated perceptions of confidence. The , a pervasive bias, causes individuals to overestimate the accuracy of their knowledge or predictions, manifesting as excessive certainty in judgments despite frequent errors. This effect is robust across domains, with studies showing that people typically exhibit about 30% overprecision in their belief accuracy estimates. Similarly, the enhances perceived certainty by fostering an unwarranted belief in personal influence over random or uncontrollable outcomes, such as in skill-based versus chance events. Ellen Langer's seminal experiments demonstrated this bias, where participants assigned higher value to lottery tickets they selected themselves compared to those assigned randomly, reflecting heightened subjective assurance. Neurologically, certainty judgments involve key brain regions, particularly the , which integrates evidence and modulates signals. The (vmPFC) plays a central role in encoding decision , as evidenced by neuroimaging studies showing its activation correlates with subjective certainty ratings during perceptual tasks. Damage to the vmPFC, as in patients with injuries, impairs the ability to form appropriate certainty assessments, leading to erratic levels in choices. The medial prefrontal cortex further distinguishes from mere decision value, supporting metacognitive evaluations of certainty. Everyday decision-making illustrates these dynamics through illusions of certainty. In , the often engenders strong psychological certainty, prompting persistent betting despite losses; for example, gamblers may feel assured of influencing dice rolls by throwing them themselves, a linked to near-misses that reinforce perceived predictability. In , witnesses frequently express high certainty in their recollections, which juries weigh heavily, yet this subjective conviction can stem from post-event rather than accurate , as shown in studies where confidence-accuracy correlations weaken under suggestive influences. These examples underscore how psychological certainty can drive maladaptive behaviors by prioritizing internal feelings over probabilistic realities.

Historical Perspectives

Ancient and Medieval Views

In , conceived of certainty as arising from knowledge of the eternal Forms, which are immutable, perfect ideals transcending the sensible world of changing particulars. These Forms, such as Beauty Itself or Justice Itself, provide absolute epistemic certainty because they are wholly what they are, free from the compresence of opposites that plagues material objects, and are grasped through intellectual recollection and rather than sensory perception. In contrast, grounded certainty in empirical observation and demonstrative syllogisms, viewing scientific knowledge (epistêmê) as certain when derived from true, primary premises that reveal causes and necessities, thus achieving understanding through logical deduction from self-evident first principles known intuitively (nous). Pyrrhonian skepticism, as articulated by , mounted significant challenges to claims of absolute certainty by emphasizing the equipollence of opposing arguments and appearances, leading to (epochê) on non-evident matters to attain tranquility (ataraxia). Through modes like perceptual relativity—such as honey appearing sweet to humans but bitter to those with —and the regress of justifications, Pyrrhonists argued that dogmatic assertions of certainty are undecidable, undermining the possibility of secure knowledge about the external world. During the medieval period, Augustine advanced introspective certainty as a foundation immune to , famously arguing in works like Contra Academicos that self-knowledge is indubitable: if one knows something, one knows that one knows it, extending to the certainty of one's own existence ("si fallor, sum") and inner states like willing or perceiving. synthesized Aristotelian reason with Christian , positing that certainty in natural truths is attainable through demonstrative reasoning from self-evident principles, while supernatural truths require ; he incorporated a moderated form of , where God's light enables the intellect to abstract universals from particulars, ensuring reliable cognition without direct vision of divine ideas in this life.

Enlightenment and Modern Foundations

The Enlightenment marked a pivotal shift in philosophical inquiries into certainty, emphasizing reason, empirical observation, and skepticism as tools to establish reliable knowledge foundations. René Descartes, a foundational rationalist, initiated this era with his Meditations on First Philosophy (1641), where he systematically applied the method of doubt to dismantle all potentially uncertain beliefs, including sensory perceptions and mathematical truths, positing scenarios like dreams or an evil deceiver to test their reliability. This hyperbolic doubt aimed to identify indubitable truths, culminating in the cogito ergo sum—"I think, therefore I am"—as the first certain proposition, immune to deception because the act of doubting affirms the existence of a thinking self. Descartes viewed this as an Archimedean point for rebuilding knowledge, where clear and distinct perceptions, once validated through proofs of God's non-deceptive nature, guarantee epistemic certainty. In contrast, John Locke advanced an empiricist framework in An Essay Concerning Human Understanding (1689), rejecting Descartes' reliance on innate ideas and asserting that the mind begins as a tabula rasa, acquiring all knowledge through sensory experience and internal reflection. Locke argued that certainty arises from the perception of agreement or disagreement among ideas derived from sensation, such as simple ideas of colors or shapes directly imprinted by external objects, though he acknowledged limitations: knowledge of substances remains probable rather than absolutely certain due to the indirect nature of sensory evidence. By denying innate principles—evidenced by the absence of universal assent among children or the illiterate—Locke positioned empirical observation as the sole path to reliable, though not infallible, certainty, influencing the scientific method's emphasis on evidence over speculation. David Hume extended empiricism into profound skepticism in A Treatise of Human Nature (1739–40) and An Enquiry Concerning Human Understanding (1748), challenging the certainty of and causation central to both rationalist and empiricist claims. Hume contended that causal inferences, based on observed constant conjunctions (e.g., one striking another), lack rational justification, as no necessary connection is observable or deducible a priori; instead, they stem from and custom, rendering predictions about unobserved events merely probable, not certain. This undermines absolute certainty in natural laws or future events, as the uniformity of cannot be proven without , exposing the limits of human knowledge to impressions and ideas without deeper metaphysical guarantees. Immanuel Kant sought to reconcile these tensions in Critique of Pure Reason (1781/1787), responding to Hume's skepticism by positing synthetic a priori judgments as the certain structures underlying experience. Unlike analytic judgments (true by definition) or synthetic a posteriori ones (derived from experience), synthetic a priori judgments—such as "every event has a cause"—extend knowledge universally and necessarily, rooted in the mind's innate forms of intuition (space and time) and categories of understanding, which organize sensory data into coherent experience. Kant argued these judgments provide objective certainty for sciences like mathematics and physics, as they are not derived from but imposed upon experience, ensuring necessity in phenomena like causation while limiting certainty to the phenomenal realm, beyond which metaphysics falters. This transcendental idealism thus preserved epistemic foundations against Humean doubt, framing certainty as a product of human cognition's a priori architecture.

Epistemological Dimensions

Certainty and Knowledge

In epistemological theories of knowledge, the debate between infallibilism and centers on whether certainty is a necessary condition for knowing. Infallibilism posits that genuine requires infallible justification, meaning the belief must be such that it could not possibly be false, thereby equating certainty with an absence of any error possibility. This view, associated with philosophers like , demands that the knower's evidence or cognitive state guarantees truth, as seen in the argument where self-evident beliefs achieve such certainty. In contrast, , defended by thinkers such as Keith Lehrer and Richard Feldman, allows to exist without absolute certainty, permitting beliefs that are justified but potentially defeasible by unforeseen evidence. Fallibilists argue that everyday knowledge claims, like knowing one is not a , succeed despite lingering skeptical possibilities, emphasizing practical reliability over indubitable certainty. Reliabilism offers an alternative framework where certainty emerges from the reliability of belief-forming processes rather than subjective indubitability. According to process reliabilism, developed by , a constitutes if it is true and produced by a cognitive process that reliably yields true beliefs across possible circumstances, such as or under normal conditions. This approach measures certainty probabilistically: a process is reliable if it has a high propensity (greater than 50%) for truth, though not necessarily perfect, thereby avoiding the strict demands of infallibilism while addressing Gettier-style counterexamples to traditional justified true accounts. For instance, a visual about an object's color gains certainty through the track record of human sight, even if the subject lacks introspective access to that reliability. The externalism/internalism debate further illuminates certainty's role in justification, particularly regarding the knower's access to factors that confer epistemic warrant. Internalists, like Earl Conee and Richard Feldman, maintain that justification—and thus certainty—must be accessible to the subject's reflective , requiring the knower to have internal reasons or that support the 's truth. Externalists, including Goldman, counter that justification can depend on external relations, such as causal reliability, without the subject's awareness, allowing certainty to stem from objective factors beyond . This tension arises in cases where a is reliably formed but the subject cannot articulate why, challenging internalist demands for conscious certainty while enabling externalist accounts to accommodate intuitive attributions. Contemporary views, particularly epistemic contextualism, treat certainty standards as context-sensitive, varying with conversational or practical stakes rather than fixed across all scenarios. Proponents like Keith DeRose and David Lewis argue that attributions of —and the implied certainty—shift based on the context: low-stakes everyday discussions permit fallibilist with modest evidence, while high-stakes or skeptical inquiries demand near-infallible justification. This approach reconciles with ordinary language by allowing "S knows that p" to be true in casual contexts despite uneliminated error possibilities, yet false in philosophical ones raising radical doubt. Such variability underscores certainty's function not as an absolute epistemic property but as a pragmatic component in claims, influencing responses to challenges like .

Skepticism and Responses

Skepticism poses profound challenges to the possibility of achieving certainty, particularly through global skeptical scenarios that question the reliability of all sensory experience and empirical . One such argument, the dream hypothesis, posits that since dreams can produce vivid illusions indistinguishable from , one cannot be certain that one's current experiences are not merely a dream, thereby undermining claims to certain of the external . Similarly, the brain-in-a-vat scenario suggests that an individual's brain could be disconnected from the body and stimulated by scientists to simulate a false , rendering all perceptual beliefs uncertain since there is no way to verify the authenticity of one's environment. These arguments target traditional criteria for , such as justified true belief, by implying that no evidence can conclusively rule out such deceptions, thus casting doubt on the attainability of epistemic certainty. In response to such radical skepticism, G.E. Moore advanced a common-sense defense, arguing that everyday certainties about the external world provide a more secure foundation than abstract philosophical doubts. In his 1939 paper "Proof of an External World," Moore famously held up his hands and declared, "Here is one hand, and here is another," asserting that this direct perceptual knowledge is indubitable and serves as proof of an external reality, thereby prioritizing intuitive certainties over skeptical hypotheses. Moore maintained that while skeptics may question the premises, the self-evident nature of such observations defeats the need for further justification, preserving certainty in basic empirical claims without engaging in infinite regress. Pragmatist philosophers offered an alternative reply, emphasizing fallible yet practically sufficient certainty over absolute indubitability. Charles Sanders Peirce, in his 1868 essay "Some Consequences of Four Incapacities," introduced fallibilism as a core tenet, contending that human cognition is inherently limited and error-prone, but that inquiry progresses through self-correcting methods that yield reliable beliefs approaching certainty in practice. Peirce argued that true certainty is unattainable due to the provisional nature of all knowledge, yet pragmatic verification—testing beliefs against their real-world consequences—provides a workable form of assurance, countering skepticism by focusing on the utility and convergence of scientific reasoning rather than infallible foundations. Reformed epistemology provides another robust counter to skepticism by redefining the conditions under which beliefs achieve warrant without requiring certainty as a prerequisite. Alvin Plantinga, developing this approach in works such as Faith and Rationality: Reason and Belief in God (1983) and Warranted Christian Belief (2000), posits that certain beliefs, including theistic ones, can be "properly basic"—rationally held without evidential support—if formed by reliable cognitive faculties in appropriate conditions. Plantinga contends that skepticism's demand for certainty ignores the proper basicality of perceptual and memory beliefs, which are warranted directly by noetic structures designed for truth-tracking, thus defending epistemic security against global doubts without conceding to fallible uncertainty.

Certainty in Formal Systems

Foundational Crisis in Mathematics

The foundational crisis in mathematics emerged in the early as efforts to establish rigorous foundations for the discipline revealed deep inconsistencies, particularly in . discovered what became known as in the spring of 1901 while working on his Principles of Mathematics, which he announced in a letter to on June 16, 1902. This paradox arises from the naive comprehension principle in , positing a set RR of all sets that are not members of themselves; if RR is a member of itself, it is not, and vice versa, leading to a contradiction. By undermining the unrestricted formation of sets, especially infinite ones central to Cantor's work, shattered the certainty of , exposing vulnerabilities in the logical and prompting a reevaluation of absolute security in mathematical reasoning. In response to such paradoxes and the ensuing foundational instability, launched his formalist program in the early 1920s, aiming to secure through a complete axiomatization and proof of absolute consistency using finitary methods. Motivated by the crisis, including doubts about infinite totalities raised by intuitionists like , Hilbert sought to formalize all of in a finite system where every proof could be verified contentually, thereby restoring certainty without reliance on problematic infinities. Key formulations appeared in Hilbert's 1921 lectures and his 1925 address "On the Infinite," emphasizing that consistency proofs would provide an indubitable basis, protecting classical from internal contradictions. Kurt Gödel's incompleteness theorems, published in 1931, delivered a profound blow to these ambitions, demonstrating inherent limits to s. The first theorem states that any consistent capable of expressing basic arithmetic, such as Peano arithmetic, is incomplete: there exist true statements within the system that cannot be proved or disproved. The second theorem asserts that if the system is consistent, it cannot prove its own consistency, directly thwarting Hilbert's goal of absolute self-verification. These results shifted mathematical foundations from aspirations of absolute certainty to a recognition of relative certainty, where consistency must be assumed or established externally, influencing subsequent developments in and axiomatic .

Axiomatic and Logical Frameworks

In the aftermath of foundational challenges in mathematics, axiomatic systems emerged as a means to establish structured certainty by specifying primitive notions and inference rules that avoid contradictions. Zermelo-Fraenkel set theory (ZFC), the predominant framework for modern mathematics, comprises a collection of axioms designed to formalize set operations while preventing paradoxes such as unrestricted comprehension. Introduced initially by in 1908 with axioms of extensionality, separation, , union, infinity, and choice (with and pairing derivable), the system was refined by and in 1922 through the addition of the replacement axiom schema, which allows for the substitution of sets within formulas to generate new sets. The axiom of foundation, to eliminate infinite descending membership chains, was added later by . These axioms ensure consistency by limiting set formation to bounded schemas, thereby providing a rigorous basis for deriving theorems with certainty within the system's boundaries, as evidenced by the absence of known contradictions in ZFC despite extensive consistency proofs relative to weaker theories. Formal logic complements axiomatic by offering deductive mechanisms for achieving certain from . In propositional logic, truth tables systematically enumerate all possible assignments to atomic propositions and compound formulas built using connectives like conjunction (∧), disjunction (∨), and (¬), revealing tautologies or contradictions with absolute certainty; for instance, the formula p∨¬pp \lor \neg p evaluates to true across all rows, confirming its status as a . Deduction rules, such as (from pp and p→qp \to q, infer qq) and universal instantiation in predicate logic, enable step-by-step proofs where each preserves truth, ensuring that theorems derived from axioms are necessarily true in any interpretation satisfying the . Predicate logic extends this to quantified statements, with rules like existential generalization allowing certain conclusions about objects in domains, thus underpinning the certainty of mathematical reasoning in systems. Model theory provides a semantic foundation for certainty in axiomatic frameworks by defining interpretations—structures consisting of a domain and assignments to non-logical symbols—that satisfy a set of s if every holds true in that model. Developed through Alfred Tarski's foundational work on and truth in the 1930s, this approach guarantees that a consistent theory admits at least one model, affirming the non-contradictory nature of its s and enabling the classification of theories as categorical (unique up to isomorphism) or complete. For example, the s of Peano arithmetic have non-standard models, illustrating how model-theoretic certainty verifies the coherence of formal systems without requiring exhaustive proof enumeration. However, limits to certainty persist in sufficiently expressive systems. Alan Turing's 1936 demonstration of the undecidability of the shows that no exists to determine, for every and input, whether the computation terminates, implying inherent in predicting behavior within Turing-complete formal systems despite their axiomatic underpinnings. This result underscores that while axiomatic and logical frameworks secure certainty for decidable fragments, broader mathematical inquiries confront undecidable propositions, necessitating alternative strategies for partial assurance.

Gradations of Certainty

Degrees and Probabilistic Measures

In Bayesian epistemology, certainty is conceptualized as the of a approaching 1 given , providing a quantifiable measure of belief strength that updates iteratively with new data. This framework formalizes through , which calculates the probability of a HH given EE: P(H∣E)=P(E∣H)P(H)P(E)P(H|E) = \frac{P(E|H) P(H)}{P(E)} Here, P(H)P(H) is the of the , P(E∣H)P(E|H) is the likelihood of the under the , and P(E)P(E) is the marginal probability of the ; as P(H∣E)P(H|E) nears 1, the gains near-certain . This approach treats degrees of certainty as continuous probabilities between 0 and 1, avoiding binary absolutes and enabling rational in uncertain environments. Rudolf Carnap advanced this quantification through his theory of logical probability, defining degrees of confirmation as objective measures of how evidence supports hypotheses within formal languages. In his system, confirmation c(h,e)c(h,e) represents the logical probability that hypothesis hh is true given evidence ee, satisfying axioms akin to those of classical probability, such as c(h,e)=1c(h,e) = 1 if hh logically follows from ee, and approaching 0 for incompatible cases. Carnap's cc-functions, introduced in works like Logical Foundations of Probability, aim to provide a neutral, language-based metric for inductive support, influencing later probabilistic logics by emphasizing symmetry and simplicity in assigning confirmation values. In legal contexts, probabilistic measures operationalize high thresholds of certainty without demanding absolutes, as seen in the "beyond " standard, which approximates a exceeding 0.95 for guilt based on evidential likelihoods. This threshold balances error minimization—false convictions versus acquittals—using Bayesian-like updating of priors with trial evidence, though courts avoid explicit numerical assignments to maintain qualitative judgment. integrates such certainty levels via expected utility, where rational choices maximize EU(a)=∑ipiu(xi)EU(a) = \sum_i p_i u(x_i), with pip_i as outcome probabilities reflecting evidential and u(xi)u(x_i) as utilities; higher certainty (elevated pip_i near 1 for preferred outcomes) amplifies the appeal of actions with low-risk payoffs. This formulation, rooted in von Neumann-Morgenstern axioms, quantifies trade-offs between probabilistic certainty and potential gains in uncertain scenarios.

Subjective and Objective Variants

Objective certainty in pertains to beliefs that achieve justification through intersubjective standards, such as evidential consensus among experts, rendering them independent of individual biases or intuitions. This form of certainty is often exemplified in shared scientific facts, where propositions like the heliocentric model of the solar system are deemed certain due to rigorous, communal verification processes that transcend personal perspectives. In contrast, it aligns with objective epistemic norms that tie justification to truth-conduciveness, ensuring that beliefs meet criteria applicable across rational agents rather than varying by subjective . Subjective certainty, however, arises from an individual's internal of , rooted in personal experiences, intuitions, or cultural backgrounds, without necessitating external validation. This variant can differ markedly across individuals or groups; for instance, one's cultural upbringing might foster unshakeable certainty in traditional ethical norms that another culture views as contingent. Philosophers have described it as the highest degree of personal confidence in a proposition's truth, often serving as a basis for formation in everyday contexts like testimonial acceptance, where a speaker's displayed assurance transmits certainty to the hearer absent independent . A key distinction appears in examples from ethics and history. Moral certainty, historically developed in scholastic thought and refined by Descartes, denotes a practical assurance sufficient for ethical action—such as deeming an act morally obligatory based on probable evidence short of absolute proof—allowing decisions amid minor doubts without full metaphysical indubitability. In contrast, factual certainty in history relies on objective corroboration, like accepting the occurrence of the Battle of Waterloo through converging archival testimonies and artifacts, achieving intersubjective agreement that elevates it beyond personal intuition. These examples highlight how subjective certainty fuels individual moral intuitions, while objective certainty underpins communal historical narratives. Challenges to objective certainty emerge from , which posits that epistemic justifications are framework-dependent, potentially eroding claims to universal consensus by rendering scientific or historical facts relative to cultural or historical epistemic systems. This view, explored in relation to Wittgenstein's propositions, suggests that what one holds as objectively certain may be appraised differently—or not at all—in another, complicating intersubjective standards without resorting to probabilistic quantification.

Applications in Science

Certainty in the Scientific Method

In the , certainty is pursued through empirical , experimentation, and iterative refinement rather than absolute proof, emphasizing provisional conclusions that can be revised with new . formulate hypotheses based on existing and test them against observable data, aiming to build confidence in explanatory models while acknowledging the tentative nature of findings. This process relies on statistical tools to quantify the reliability of results, ensuring that claims are grounded in reproducible rather than or . Hypothesis testing plays a central role in assessing provisional certainty, using to evaluate the probability that observed data occurred by chance under a . A below a conventional threshold, such as 0.05, indicates that the results are statistically significant, suggesting the can be rejected in favor of the alternative, though this does not prove the true. Complementing , provide a range of plausible values for an estimated parameter, typically at 95% , offering a measure of precision and around the point estimate. For instance, a 95% that does not include the null value supports rejecting the , allowing scientists to express certainty in the direction and magnitude of effects while highlighting the limits of the sample data. Karl Popper's principle of further qualifies certainty by arguing that scientific theories gain credibility not through confirmation but by surviving rigorous attempts at refutation. A theory is scientific if it makes testable predictions that could potentially be disproven; repeated failure to falsify it increases in its validity, though it remains conjectural and open to future challenges. This demarcation criterion distinguishes empirical from , emphasizing critical testing over inductive accumulation of supportive evidence. Communal mechanisms like and replication enhance collective certainty by subjecting findings to external scrutiny and independent verification. , conducted by experts before publication, evaluates methodological soundness, potential biases, and logical coherence, helping to filter unreliable claims. Replication involves repeating experiments under similar conditions to confirm results, building broader when consistent outcomes emerge across studies and labs. These practices mitigate individual errors and foster a self-correcting . A landmark historical example is the 1919 solar eclipse expedition led by , which tested Albert Einstein's general by measuring the deflection of starlight passing near the Sun's gravitational field. Observations from and Sobral confirmed the predicted 1.75 arcsecond deflection—twice that expected under Newtonian gravity—providing strong empirical support for relativity and elevating its certainty within physics. This falsifiable prediction, successfully withstanding the test, marked a pivotal moment in scientific validation through observation.

Uncertainty Principles and Limits

In quantum mechanics, the Heisenberg uncertainty principle establishes a fundamental limit on the precision with which certain pairs of physical properties can be simultaneously measured. Formulated by in 1927, the principle states that the product of the uncertainties in position (Δx\Delta x) and momentum (Δp\Delta p) of a particle satisfies the inequality ΔxΔp≥ℏ2\Delta x \Delta p \geq \frac{\hbar}{2}, where ℏ\hbar is the reduced Planck's constant. This relation arises from the wave-particle duality inherent in quantum theory, implying that increasing the accuracy of one measurement necessarily broadens the uncertainty in the conjugate variable. Consequently, absolute certainty in both position and momentum cannot be achieved simultaneously, capping the achievable knowledge in quantum systems. Chaos theory reveals another inherent limit to predictive certainty in classical deterministic systems, particularly those governed by nonlinear dynamics. Pioneered by Edward Lorenz in his 1963 paper, is characterized by extreme sensitivity to initial conditions, where minuscule differences in starting states can lead to vastly divergent outcomes over time—a phenomenon often described as the "butterfly effect." In such systems, like weather patterns modeled by the Lorenz equations, long-term forecasts become practically impossible despite the underlying , as uncertainties amplify exponentially. This sensitivity underscores that uncertainties in initial conditions, however small, amplify exponentially over time, limiting long-term predictive certainty in practice despite the underlying of chaotic systems. In , entropy quantifies the intrinsic uncertainty arising from the vast number of microscopic configurations consistent with a macroscopic state. introduced this concept in the late 19th century, defining entropy SS as S=kln⁡WS = k \ln W, where kk is Boltzmann's constant and WW represents the number of microstates corresponding to a given macrostate. This formulation interprets entropy not as disorder but as a measure of the probabilistic ignorance about individual particle positions and velocities in a system at equilibrium. As a result, while macroscopic properties like can be predicted with high , the detailed microscopic behavior remains fundamentally uncertain due to the of possible arrangements. These principles collectively demonstrate that scientific inquiry, even within rigorous frameworks, delivers high degrees of certainty but falls short of absolutes due to irreducible uncertainties embedded in nature's laws. In , chaos, and statistical ensembles, the limits highlight the probabilistic fabric of reality, guiding experimental design and theoretical interpretations toward probabilistic rather than deterministic conclusions.

20th-Century Philosophical Views

Wittgenstein's Framework

In On Certainty, a collection of Wittgenstein's posthumously published notes from 1949 to 1951 edited by and G. H. von Wright, certainty is portrayed not as an epistemic achievement grounded in or deduction but as a practical embedded in human activity. Wittgenstein critiques the quest for indubitable foundations, arguing instead that certainty operates as the unarticulated background against which claims make sense. At the core of this framework are "hinge propositions," such as "My body will obey my orders" or "The earth has existed for more than five minutes," which function as rigid pivots immune to or justification. These propositions are not hypotheses tested empirically or logically but foundational assumptions that "stand fast" for us, enabling the possibility of error, inquiry, and learning in specific contexts. Wittgenstein emphasizes that attempting to prove or disprove them would dissolve the framework of meaning itself, as they are removed from the riverbed of rational argumentation and instead shape it (OC §§94–99, 341–343). Certainty, in Wittgenstein's view, arises within "language games"—the rule-governed activities of speaking and acting that constitute our shared forms of life—rather than from abstract philosophical scaffolding. Hinge propositions derive their unassailability not from innate or universal truths but from their role in these communal practices, where doubting them would render everyday assertions incoherent (OC §§6, 358). For instance, the certainty that "I am sitting in a chair" is not a theoretical but a practical orientation embedded in the of describing one's surroundings, presupposed by any meaningful challenge to it. This approach yields a robust anti-skeptical stance, as Wittgenstein contends that skepticism presupposes the very certainties it seeks to undermine, making global doubt logically and practically impossible. , like that of Descartes, fails because it ignores the contextual limits of doubt: one cannot coherently question everything without relying on unquestioned hinges to formulate the doubt (OC §§114–115, 456). Wittgenstein captures this insight succinctly: "If you tried to doubt everything you would not get as far as doubting anything. The game of doubting itself presupposes certainty" (OC §115). Thus, certainty is not a static property of isolated propositions but a dynamic feature of our lived, social world, where what is certain shifts with the contours of our forms of life.

Other Key Thinkers

, a leading figure in , argued that certainty in knowledge derives from statements that are either analytically true through logical necessity or empirically verifiable via observation. He initially adhered to a strict verification principle, where meaningful assertions must be reducible to sensory experiences, dismissing metaphysical claims as unverifiable and thus cognitively insignificant. Later, Carnap refined this view by introducing degrees of in his inductive logic, positing that gain probabilistic support rather than absolute certainty from evidence, with confirmation measured as the logical probability of a hypothesis given empirical data. Willard Van Orman Quine challenged traditional notions of certainty by rejecting the analytic-synthetic distinction in his 1951 essay "Two Dogmas of Empiricism," asserting that no sharp boundary exists between statements true by meaning alone and those true by empirical fact. Instead, Quine proposed a holistic "web of belief," where certainty emerges from the interconnected structure of scientific theories, all of which are revisable in light of experience, with adjustments occurring at the periphery to preserve central tenets. In his naturalized epistemology, outlined in subsequent works, Quine further contended that epistemological inquiry should be integrated into empirical science, replacing a priori pursuits of certainty with psychological and sociological explanations of belief formation. Thomas Kuhn's analysis of scientific progress in "" (1962) portrayed certainty as relative to dominant s during periods of normal science, where researchers solve puzzles within an accepted framework, achieving reliable knowledge under those assumptions. Anomalies that resist resolution lead to crises, culminating in paradigm shifts through scientific revolutions, which replace old certainties with new ones incompatible with the prior framework, thus rendering absolute or cumulative certainty illusory across revolutionary divides. Richard Rorty's , as developed in "" (1989), undermined claims to absolute certainty by emphasizing the contingency of , selfhood, and liberal communities, arguing that vocabularies and beliefs are historical products without foundational grounding in . He advocated an ironic stance, where individuals recognize the contingency of their convictions without seeking final justifications, rejecting philosophical mirrors of that promise objective certainty in favor of edifying conversations that foster through shared narratives rather than truth.

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